diff --git a/Algebra-and-Number-Theory/Awards-Algebra-and-Number-Theory-2024.csv b/Algebra-and-Number-Theory/Awards-Algebra-and-Number-Theory-2024.csv index 0541fc4..cfd9da0 100644 --- a/Algebra-and-Number-Theory/Awards-Algebra-and-Number-Theory-2024.csv +++ b/Algebra-and-Number-Theory/Awards-Algebra-and-Number-Theory-2024.csv @@ -1,42 +1,45 @@ "AwardNumber","Title","NSFOrganization","Program(s)","StartDate","LastAmendmentDate","PrincipalInvestigator","State","Organization","AwardInstrument","ProgramManager","EndDate","AwardedAmountToDate","Co-PIName(s)","PIEmailAddress","OrganizationStreet","OrganizationCity","OrganizationState","OrganizationZip","OrganizationPhone","NSFDirectorate","ProgramElementCode(s)","ProgramReferenceCode(s)","ARRAAmount","Abstract" "2411537","Conference: Comparative Prime Number Theory Symposium","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","04/05/2024","Wanlin Li","MO","Washington University","Standard Grant","Adriana Salerno","04/30/2025","$10,000.00","","wanlin@wustl.edu","ONE BROOKINGS DR","SAINT LOUIS","MO","63110","3147474134","MPS","126400","7556","$0.00","The workshop Comparative Prime Number Theory Symposium, which is the first scientific event to focus predominantly on this subject, will take place on the UBC--Vancouver campus from June 17--21, 2024. One of the first and central topics in the research of number theory is to study the distribution of prime numbers. In 1853, Chebyshev observed that there seems to be more primes taking the form of a multiple of four plus three than a multiple of four plus one. This phenomenon is now referred to as Chebyshev's bias and its study led to a new branch of number theory, comparative prime number theory. As a subfield of analytic number theory, research in this area focuses on examining how prime counting functions and other arithmetic functions compare to one another. This field has witnessed significant growth and activity in the last three decades, especially after the publication of the influential article on Chebyshev's bias by Rubinstein and Sarnak in 1994. The primary goal of this award will be to provide participant support and fund US-based early career researchers to attend this unique event, giving them the opportunity to discuss new ideas, advance research projects, and interact with established researchers.

The symposium will bring together many leading and early-career researchers with expertise and interest in comparative prime number theory to present and discuss various aspects of current research in the field, with special emphasis on results pertaining to the distribution of counting functions in number theory and zeros of L-functions, consequences of quantitative Linear Independence, oscillations of the Mertens sum, and the frequency of sign changes. Through this symposium, we will advertise the recently disseminated survey ""An Annotated Bibliography for Comparative Prime Number Theory"" by Martin et al which aims to record every publication within the topic of comparative prime number theory, together with a summary of results, and presenting a unified system of notation and terminology for referring to the quantities and hypotheses that are the main objects of study. Another important outcome of the symposium will be compiling and publicizing a problem list, with the hope of stimulating future research and providing young researchers with potential projects. Information about the conference can be found at the website: https://sites.google.com/view/crgl-functions/comparative-prime-number-theory-symposium

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2401321","Euler Systems, Iwasawa Theory, and the Arithmetic of Elliptic Curves","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/05/2024","Francesc Castella","CA","University of California-Santa Barbara","Continuing Grant","Adriana Salerno","06/30/2027","$74,832.00","","castella@ucsb.edu","3227 CHEADLE HALL","SANTA BARBARA","CA","931060001","8058934188","MPS","126400","","$0.00","Elliptic curves are a class of polynomial equations (of degree three in two variables) that have been studied for centuries, yet for which many basic questions remain open. For instance, at present there is no proven algorithm to decide whether or not a given elliptic curve has finite or infinitely many rational solutions. Over the past century, mathematicians conjectured that an answer to these questions could be extracted from certain functions of a complex variable, namely the L-function of the elliptic curve. Euler systems and Iwasawa theory are two of the most powerful tools available to date for the study of these and related conjectured links between arithmetic and analysis. This award will advance our understanding of the arithmetic of elliptic curves by developing new results and techniques in Euler systems and Iwasawa theory. The award will also support several mentoring, training, dissemination, and outreach activities.

More specifically, the research to be pursued by the PI and his collaborators will largely focus on problems whose solutions will significantly advance our understanding of issues at the core of the Birch and Swinnerton-Dyer conjecture and related questions in situations of analytic rank 1, and shed new light on the much more mysterious cases of analytic rank 2 and higher. In rank 1, they will prove the first p-converse to the celebrated theorem of Gross-Zagier and Kolyvagin in the case of elliptic curves defined over totally real fields. In rank 2, they will continue their investigations of the generalized Kato classes introduced a few years ago by Darmon-Rotger, establishing new nonvanishing results in the supersingular case. They will also study a systematic p-adic construction of Selmer bases for elliptic curves over Q of rank 2 in connection with the sign conjecture of Mazur-Rubin. For elliptic curves of arbitrary rank, they will establish various non-triviality results of associated Euler systems and Kolyvagin systems, as first conjectured by Kolyvagin and Mazur-Tate.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2402553","Torsors under Reductive Groups and Dualities for Hitchin Systems","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Roman Fedorov","PA","University of Pittsburgh","Standard Grant","Tim Hodges","06/30/2027","$250,000.00","","fedorov@pitt.edu","4200 FIFTH AVENUE","PITTSBURGH","PA","152600001","4126247400","MPS","126400","","$0.00","The study of torsors (also known as principal bundles) began in the early 20th century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that torsors became a basis for the so-called Standard Model - a physical theory describing all fundamental forces except for gravitation. The standard model predicted the existence of various particles, the last of which, called the Higgs boson, was found in a Large Hadron Collider experiment in 2012. In 1950's Fields medalist Jean-Pierre Serre recognized the importance of torsors in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a torsor and formulated a certain deep conjecture. The first part of this project is aimed at proving this conjecture, which is among the oldest unsolved foundational questions in mathematics. The second part of the project is related to the so-called Higgs bundles, which can be thought of as mathematical incarnations of the Higgs bosons. More precisely, the PI proposes to prove a certain duality for the spaces parameterizing Higgs bundles. This duality is a vast generalization of the fact that the Maxwell equations describing electromagnetic fields are symmetric with respect to interchanging electrical and magnetic fields. The duality is a part of the famous Langlands program unifying number theory, algebraic geometry, harmonic analysis, and mathematical physics. This award will support continuing research in these areas. Advising students and giving talks at conferences will also be part of the proposed activity.

In more detail, a conjecture of Grothendieck and Serre predicts that a torsor under a reductive group scheme over a regular scheme is trivial locally in the Zariski topology if it is rationally trivial. This conjecture was settled by Ivan Panin and the PI in the equal characteristic case. The conjecture is still far from resolution in the mixed characteristic case, though there are important results in this direction. The PI proposes to resolve the conjecture in the unramified case; that is, for regular local rings whose fibers over the ring of integers are regular. A more ambitious goal is to prove the purity conjecture for torsors, which is, in a sense, the next step after the Grothendieck?Serre conjecture. The second project is devoted to Langlands duality for Hitchin systems, predicting that moduli stacks of Higgs bundles for Langlands dual groups are derived equivalent. This conjecture may be viewed as the classical limit of the geometric Langlands duality. By analogy with the usual global categorical Langlands duality, the PI formulates a local version of the conjecture and the basic compatibility between the local and the global conjecture. The PI will attempt to give a proof of the local conjecture based on the geometric Satake equivalence for Hodge modules constructed by the PI.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401548","Topics in automorphic Forms and Algebraic Cycles","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Wei Zhang","MA","Massachusetts Institute of Technology","Continuing Grant","Andrew Pollington","06/30/2029","$110,398.00","","wz2113@mit.edu","77 MASSACHUSETTS AVE","CAMBRIDGE","MA","021394301","6172531000","MPS","126400","","$0.00","This awards concern research in Number Theory. Solving polynomial equations in rational numbers dates back to Diophantus in the 3rd century and has been a central subject in mathematics for generations. The modern study of Diophantine equations has incorporated the revolutionary idea of Riemann from his use of a class of special functions called ""zeta functions? or ""L-functions"". Such special functions are built up on counting the numbers of solutions of polynomial equations in the much simpler setting of modular arithmetic. In the 1960s, Birch and Swinnerton-Dyer came up with a remarkable conjecture revealing a relation between the zeros of L-functions and the solutions to a special class of polynomial equations in the rationals. Later Beilinson and Bloch conjectured that, for general polynomial equations in the rationals, there should always be a relation between the zeros of L-functions and algebraic cycles which are ?parameter solutions to polynomial equations?.

The project will study the zeros of L-functions through automorphic forms and special cycles on modular varieties. The theory of automorphic form provides a fruitful way to access the zeros of L-functions. The modular varieties are either Shimura varieties over number fields or moduli spaces of Shtukas over function fields. They play a central role in modern number theory and arithmetic geometry, and they often come with a great supply of algebraic cycles. The project aims to prove results relating zeros of L-functions and algebraic cycles on modular varieties, including new cases of the arithmetic Gan?Gross?Prasad conjecture for Shimura varieties associated to unitary groups, certain Higher Gross?Zagier formula over function fields, and the function field analog of Kudla?s program with an emphasis on the modularity of generating series of special cycles and the arithmetic Siegel?Weil formula. The project will also develop new relative trace formula, a powerful equation connecting spectral information and geometric structure, to study general automorphic period integral including the unitary Friedberg?Jacquet period. The broader impacts of this project include mentoring of graduate students and seminar organization.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348833","Studies in Categorical Algebra","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","04/03/2024","Chelsea Walton","TX","William Marsh Rice University","Continuing Grant","Tim Hodges","04/30/2027","$119,965.00","","notlaw@rice.edu","6100 MAIN ST","Houston","TX","770051827","7133484820","MPS","126400","","$0.00","Algebraic structures have been employed for nearly two centuries to understand the behavior, particularly the symmetry, of various entities in nature. Now with the current technology of category theory (i.e., the study of objects and how they are transported), classical algebraic structures can be upgraded to provide information on natural phenomena that was not previously understood. This yields significant consequences in quantum physics. The work sponsored by this grant lies in the framework of monoidal categories, which are categories that come equipped with a way of combining objects and combining maps between objects. Several projects are earmarked for partial work by undergraduate and graduate students. Moreover, the PI will make significant progress on completing a three-volume, user-friendly textbook series on quantum algebra. The PI is also an active mentor for numerous members of underrepresented groups, particularly for those in groups to which the PI belongs (women, African-Americans, and first generation college students).

The first research theme of the projects sponsored by this grant is on algebras in monoidal categories. The PI will extend classical properties of algebras over a field to the monoidal context, and will also study properties that only have significant meaning in the categorical setting. In addition, the PI will examine other algebraic structures (e.g., Frobenius algebras) in monoidal categories, especially those tied to Topological Quantum Field Theories (TQFTs). Another theme of the PI's sponsored research work is on representations of certain monoidal categories that play a crucial role in 2-dimensional Conformal Field Theory (2d-CFTs), and that correspond to 3d-TQFTs. Of particular interest are representations of modular tensor categories, and the PI's work here will build on recent joint work with R. Laugwitz and M. Yakimov that constructs canonical representations of braided monoidal categories.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2420166","Conference: The Mordell conjecture 100 years later","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Bjorn Poonen","MA","Massachusetts Institute of Technology","Standard Grant","Andrew Pollington","06/30/2025","$29,970.00","","poonen@math.mit.edu","77 MASSACHUSETTS AVE","CAMBRIDGE","MA","021394301","6172531000","MPS","126400","7556","$0.00","The award will support a conference, ``The Mordell conjecture 100 years later'', at the Massachusetts Institute of Technology during the week July 8-12, 2024. The conference website, showing the list of invited speakers, is https://mordell.org/ . The Mordell conjecture, proved in 1983, is one of the landmarks of modern number theory. A conference on this topic is needed now, because in recent years, there have been advances on different aspects of the conjecture, while other key questions remain unsolved. This would be the first conference to bring together all the researchers coming from these different perspectives. The conference will feature 16 hour-long lectures, with speakers ranging from the original experts to younger mathematicians at the forefront of current research. Some lectures will feature surveys of the field, which have educational value especially for the next generation of researchers. The conference will also feature a problem session and many 5-minute lightning talk slots, which will give junior participants an opportunity to showcase their own research on a wide variety of relevant topics. The award will support the travel and lodging of a variety of mathematicians including those from underrepresented groups in mathematics and attendees from colleges and universities where other sources of funding are unavailable. Materials from the lectures, problem session, and lightning talks will be made publicly available on the website, to reach an audience broader than just conference attendees.

The Mordell conjecture motivated much of the development of arithmetic geometry in the 20th century, both before and after its resolution by Faltings. The conference will feature lectures covering a broad range of topics connected with the Mordell conjecture, its generalizations, and other work it has inspired. In particular, it will build on recent advances in the following directions: 1) nonabelian analogues of Chabauty's p-adic method; 2) the recent proof via p-adic Hodge theory; 3) uniform bounds on the number of rational points; 4) generalizations to higher-dimensional varieties, studied by various methods: analytic, cohomological, and computational.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2401554","Functoriality for Relative Trace Formulas","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Ioannis Sakellaridis","MD","Johns Hopkins University","Continuing Grant","Andrew Pollington","06/30/2027","$106,185.00","","sakellar@jhu.edu","3400 N CHARLES ST","BALTIMORE","MD","212182608","4439971898","MPS","126400","","$0.00","The Langlands functoriality conjecture, that ""different arithmetic drums share some common eigenfrequencies,"" has immense applications in number theory, among others to the century-old conjectures due to Ramanujan and others about the size of coefficients of special functions called automorphic forms. The PI and his collaborators have broadened this conjecture to the so-called ""relative"" setting, which includes methods of studying special values of L-functions (also called zeta functions), such as in the prominent, and more recent, conjectures of Gan, Gross, and Prasad. The main tool for proving important instances of functoriality so far has been the trace formula, but in its current form it has nearly reached its limits. This project will examine ways to prove these conjectures by use of the idea of quantization, whose origins lie in mathematical physics. This idea will be used to construct novel ways of comparing (relative) trace formulas, drastically expanding their potential reach and applicability. The broader impacts of the project include conference organization and mentoring of graduate students.

The PI has already shown, in prior work, that in some low-rank cases one can establish relative functoriality via some novel ""transfer operators"" between relative trace formulas. Such non-standard comparisons of trace formulas were envisioned in Langlands's ""Beyond Endoscopy"" proposal; the ""relative"" setting allows for more flexibility, and more potential applications, for the exploration of such comparisons. Prior work was focused mostly on the case when the L-groups associated to the relative trace formulas are of rank one. The main goal of this project will be to examine ways to generalize the construction of transfer operators to higher rank. The main idea is to view a trace formula as the quantization of its cotangent stack, which in turn is largely controlled by the L-group. Using natural correspondences between such cotangent stacks, the project aims to construct transfer operators between their quantizations. On a separate track, the project will continue work on the duality of Hamiltonian spaces conjectured in the PI's recent work with Ben-Zvi and Venkatesh, with the aim of extending this duality beyond the hyperspherical setting, and exploring applications for the representation theory of p-adic groups.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2422557","Conference: Resolution of Singularities, Valuation Theory and Related Topics","DMS","ALGEBRA,NUMBER THEORY,AND COM","08/01/2024","04/03/2024","Steven Cutkosky","MO","University of Missouri-Columbia","Standard Grant","Tim Hodges","01/31/2025","$17,520.00","","cutkoskys@missouri.edu","121 UNIVERSITY HALL","COLUMBIA","MO","652113020","5738827560","MPS","126400","7556","$0.00","This award supports US-based participants in a conference on ``Resolution of Singularities, Valuation Theory and Related Topics'' which will be held from August 5 - 9, 2024 in Morelia, Mexico. The conference will be held at Centro de Ciencias Matematicas, UNAM, Morelia. NSF will provide significant travel and lodging support for 12 U.S. participants to the conference. The funding will be for students, postdoctoral scholars and other U.S. participants who do not have other federal support. A particular emphasis will be on supporting a diversity of participants, especially from under-represented groups.

The focus of the conference is on applications of valuation theory to resolution of singularities in positive characteristic and to other areas of algebraic geometry, commutative algebra and singularity theory. Recently, there have been significant advances in this area, and this conference will cover this progress in talks by the authors of this work. The proposed gathering will provide an opportunity for researchers from diverse fields to interact and establish research connections with each other; in particular, the participants will benefit from this interaction and from seeing recent developments in the field and its relationships with other areas. The conference webpage is
https://sites.google.com/view/spivakovsky60thbirthday/home/authuser=0

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2347850","Structure theory for measure-preserving systems, additive combinatorics, and correlations of multiplicative functions","DMS","ALGEBRA,NUMBER THEORY,AND COM, ANALYSIS PROGRAM","07/01/2024","04/02/2024","Terence Tao","CA","University of California-Los Angeles","Continuing Grant","Wing Suet Li","06/30/2027","$244,222.00","","tao@math.ucla.edu","10889 WILSHIRE BLVD STE 700","LOS ANGELES","CA","900244200","3107940102","MPS","126400, 128100","","$0.00","Consider a stream of digital data - a sequence of zeroes and ones. This sequence could be highly structured - for instance, it could alternate periodically between 0 and 1. Or it could be completely random, with the value of each member of the sequence having no relation whatsoever to the next. It could also be ""pseudorandom"" - described by a deterministic algorithm, but yet statistically indistinguishable from a genuinely random sequence. Or it could be some complex mixture of structure and (pseudo)randomness. Can one define precisely what structure and randomness mean and describe arbitrary data as combinations of these two different components? Such questions are of importance in cryptography, computer science, combinatorics, dynamics, and number theory, as they allow one to mathematically determine whether certain patterns in arbitrary streams of data are guaranteed to occur or not. For instance, in 2004, Ben Green and the PI were able to settle a long-standing conjecture in number theory that the prime numbers contained arbitrarily long arithmetic progressions, with the key idea being to break up the prime numbers into structured and random components and study the contribution of each component. In computer science, this theory has led, for instance, to efficient ways to generate pseudorandom bits for several types of applications. In the subsequent twenty years, much progress has been made in quantifying more precisely what structure and randomness mean, particularly in the area of mathematics now known as higher-order Fourier analysis. More understanding has been gained on the precise way in which number-theoretic structures, such as the primes, exhibit (pseudo-)random behavior at both large and small scales. There has been steady progress in this direction in recent years, in which the scale on which one is able to definitively demonstrate various types of pseudorandomness has narrowed over time, and further work will be carried out in this project, in particular, it is tantalizingly near to resolve (a version) of a well-known conjecture in number theory - the Chowla conjecture - which could be in turn a stepping stone to even more famous conjectures such as the twin prime conjecture. This project provides research training opportunities for graduate students.


In this project, the PI (in conjunction with collaborators) plans to work on two related projects. Firstly, the PI will continue recent work on developing general inverse theorems for the Gowers uniformity norms in additive combinatorics on one hand and the Host--Kra uniformity seminorms in ergodic theory on the other. Secondly, the PI will continue building upon recent breakthroughs in the understanding of multiplicative functions, to make further progress towards the (logarithmically averaged) Chowla and Elliott conjectures for such functions, and to apply these results to related problems in analytic number theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2341365","Conference: Southern Regional Number Theory Conference","DMS","ALGEBRA,NUMBER THEORY,AND COM","02/01/2024","01/19/2024","Gene Kopp","LA","Louisiana State University","Standard Grant","James Matthew Douglass","01/31/2026","$35,000.00","Fang-Ting Tu","gkopp@lsu.edu","202 HIMES HALL","BATON ROUGE","LA","708030001","2255782760","MPS","126400","9150","$0.00","Southern Regional Number Theory Conferences (SRNTCs) are planned to be held in the Gulf Coast region March 9?11, 2024, and in Spring 2025, at Louisiana State University in Baton Rouge. The 2024 conference will be the 10th anniversary of the conference series. The SRNTC series serves as an annual number theory event for the Gulf Coast region. It brings together researchers from the region and beyond to disseminate and discuss fundamental research in various branches of number theory, in turn fostering communication and collaboration between researchers. Local students and early-career researchers attending the conferences are exposed to a wide array of problems and techniques, including specialized topics that may have no local experts at their home institutions. Students and early-career researchers are given opportunities to present their research through contributed talks and to expand their professional network.

SRNTC 2024 will feature about ten invited talks by established experts from four countries, speaking on topics in algebraic number theory, analytic number theory, and automorphic forms. It will also feature about twenty-five contributed talks, mostly by regional graduate students and early-career researchers. Information about SRNTC 2024 and SRNTC 2025, including a registration form and the schedule for each conference, is available at the conference website (https://www.math.lsu.edu/srntc).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2334874","Conference: Pittsburgh Links among Analysis and Number Theory (PLANT)","DMS","ALGEBRA,NUMBER THEORY,AND COM, ANALYSIS PROGRAM","02/01/2024","01/19/2024","Carl Wang Erickson","PA","University of Pittsburgh","Standard Grant","James Matthew Douglass","01/31/2025","$20,000.00","Theresa Anderson, Armin Schikorra","carl.wang-erickson@pitt.edu","4200 FIFTH AVENUE","PITTSBURGH","PA","152600001","4126247400","MPS","126400, 128100","7556","$0.00","This award will support the four-day conference ""Pittsburgh Links among Analysis and Number Theory (PLANT)"" that will take place March 4-7, 2024 in Pittsburgh, PA. The purpose of the conference is to bring together representatives of two disciplines with a shared interface: number theory and analysis. There is a large potential for deeper collaboration between these fields resulting in new and transformative mathematical perspectives, and this conference aims at fostering such an interchange. In particular, the conference is designed to attract PhD students and post-doctoral scholars into working on innovations at this interface.

To encourage the development of new ideas, the conference speakers, collectively, represent many subfields that have developed their own distinctive blend of analysis and number theory, such as analytic number theory, arithmetic statistics, analytic theory of modular and automorphic forms, additive combinatorics, discrete harmonic analysis, and decoupling. While there have been a wide variety of conferences featuring these subfields in relative isolation, the PIs are excited at PLANT's potential for sparking links among all of these subfields and giving early-career participants the opportunity to be part of this exchange. The conference website is https://sites.google.com/view/plant24/home.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2408333","Conference: GAeL XXXI (Geometrie Algebrique en Liberte)","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","03/15/2024","Jose Rodriguez","WI","University of Wisconsin-Madison","Standard Grant","Tim Hodges","03/31/2025","$16,255.00","","jrodriguez43@wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","126400","7556","$0.00","This award funds participation of junior US mathematicians in the 31st edition of Gael (Géométrie Algébrique en Liberté) from June 17-21, 2024 at Turin, Italy, held jointly by Politecnico di Torino and Universitŕ di Torino. Géométrie Algébrique en Liberté is a series of annual meetings organized for and by junior researchers in algebraic geometry with a long tradition, drawing in 70-90 participants each year. There are both casual and structured career opportunities for junior mathematicians to interact with speakers and other attendees.

GAeL XXXI will bring together leading experts on a range of topics within Algebraic Geometry, providing an excellent opportunity for junior mathematicians to learn about major new developments. There will be three senior speakers giving mini-courses covering cutting edge results from a wide variety of topics so that GAeL appeals to all PhD students and junior postdocs in algebraic geometry. The rest of the talks are chosen from among the junior participants, often providing the first opportunity for many of these individuals to speak in front of an international audience. More information about GAeL XXXI may be found on the event website: https://sites.google.com/view/gaelxxxi

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2401049","Conference: Representation Theory and Related Geometry","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/15/2024","04/08/2024","Laura Rider","GA","University of Georgia Research Foundation Inc","Standard Grant","James Matthew Douglass","12/31/2024","$46,000.00","Mee Seong Im","laurajoymath@gmail.com","310 E CAMPUS RD RM 409","ATHENS","GA","306021589","7065425939","MPS","126400","7556","$0.00","This is a grant to support participation in the conference ""Representation theory and related geometry: progress and prospects"" that will take place May 27-31, 2024 at the University of Georgia in Athens, GA. This conference will bring together a diverse set of participants to discuss two key areas of mathematics and their interplay. Talks will include historical perspectives on the area as well as the latest mathematical breakthroughs. A goal of the conference is to facilitate meetings between graduate students, junior mathematicians, and seasoned experts to share knowledge and inspire new avenues of research. In addition to the formally invited talks, the conference will include opportunities for contributed talks and discussion.

The interplay of representation theory and geometry is fundamental to many of the recent breakthroughs in representation theory. Topics will include the representation theory of Lie (super)algebras, and finite, algebraic, and quantum groups; cohomological methods in representation theory; modular representation theory; geometric representation theory; categorification; tensor triangular geometry and related topics in noncommutative algebraic geometry; among others. More specific topics of interest may include support varieties, cohomology and extensions, endotrivial modules, Schur algebras, tensor triangular geometry, and categorification. The conference website can be found at https://sites.google.com/view/representation-theory-geometry/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2401321","Euler Systems, Iwasawa Theory, and the Arithmetic of Elliptic Curves","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/05/2024","Francesc Castella","CA","University of California-Santa Barbara","Continuing Grant","Adriana Salerno","06/30/2027","$74,832.00","","castella@ucsb.edu","3227 CHEADLE HALL","SANTA BARBARA","CA","931060001","8058934188","MPS","126400","","$0.00","Elliptic curves are a class of polynomial equations (of degree three in two variables) that have been studied for centuries, yet for which many basic questions remain open. For instance, at present there is no proven algorithm to decide whether or not a given elliptic curve has finite or infinitely many rational solutions. Over the past century, mathematicians conjectured that an answer to these questions could be extracted from certain functions of a complex variable, namely the L-function of the elliptic curve. Euler systems and Iwasawa theory are two of the most powerful tools available to date for the study of these and related conjectured links between arithmetic and analysis. This award will advance our understanding of the arithmetic of elliptic curves by developing new results and techniques in Euler systems and Iwasawa theory. The award will also support several mentoring, training, dissemination, and outreach activities.

More specifically, the research to be pursued by the PI and his collaborators will largely focus on problems whose solutions will significantly advance our understanding of issues at the core of the Birch and Swinnerton-Dyer conjecture and related questions in situations of analytic rank 1, and shed new light on the much more mysterious cases of analytic rank 2 and higher. In rank 1, they will prove the first p-converse to the celebrated theorem of Gross-Zagier and Kolyvagin in the case of elliptic curves defined over totally real fields. In rank 2, they will continue their investigations of the generalized Kato classes introduced a few years ago by Darmon-Rotger, establishing new nonvanishing results in the supersingular case. They will also study a systematic p-adic construction of Selmer bases for elliptic curves over Q of rank 2 in connection with the sign conjecture of Mazur-Rubin. For elliptic curves of arbitrary rank, they will establish various non-triviality results of associated Euler systems and Kolyvagin systems, as first conjectured by Kolyvagin and Mazur-Tate.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2420166","Conference: The Mordell conjecture 100 years later","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Bjorn Poonen","MA","Massachusetts Institute of Technology","Standard Grant","Andrew Pollington","06/30/2025","$29,970.00","","poonen@math.mit.edu","77 MASSACHUSETTS AVE","CAMBRIDGE","MA","021394301","6172531000","MPS","126400","7556","$0.00","The award will support a conference, ``The Mordell conjecture 100 years later'', at the Massachusetts Institute of Technology during the week July 8-12, 2024. The conference website, showing the list of invited speakers, is https://mordell.org/ . The Mordell conjecture, proved in 1983, is one of the landmarks of modern number theory. A conference on this topic is needed now, because in recent years, there have been advances on different aspects of the conjecture, while other key questions remain unsolved. This would be the first conference to bring together all the researchers coming from these different perspectives. The conference will feature 16 hour-long lectures, with speakers ranging from the original experts to younger mathematicians at the forefront of current research. Some lectures will feature surveys of the field, which have educational value especially for the next generation of researchers. The conference will also feature a problem session and many 5-minute lightning talk slots, which will give junior participants an opportunity to showcase their own research on a wide variety of relevant topics. The award will support the travel and lodging of a variety of mathematicians including those from underrepresented groups in mathematics and attendees from colleges and universities where other sources of funding are unavailable. Materials from the lectures, problem session, and lightning talks will be made publicly available on the website, to reach an audience broader than just conference attendees.

The Mordell conjecture motivated much of the development of arithmetic geometry in the 20th century, both before and after its resolution by Faltings. The conference will feature lectures covering a broad range of topics connected with the Mordell conjecture, its generalizations, and other work it has inspired. In particular, it will build on recent advances in the following directions: 1) nonabelian analogues of Chabauty's p-adic method; 2) the recent proof via p-adic Hodge theory; 3) uniform bounds on the number of rational points; 4) generalizations to higher-dimensional varieties, studied by various methods: analytic, cohomological, and computational.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2401554","Functoriality for Relative Trace Formulas","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Ioannis Sakellaridis","MD","Johns Hopkins University","Continuing Grant","Andrew Pollington","06/30/2027","$106,185.00","","sakellar@jhu.edu","3400 N CHARLES ST","BALTIMORE","MD","212182608","4439971898","MPS","126400","","$0.00","The Langlands functoriality conjecture, that ""different arithmetic drums share some common eigenfrequencies,"" has immense applications in number theory, among others to the century-old conjectures due to Ramanujan and others about the size of coefficients of special functions called automorphic forms. The PI and his collaborators have broadened this conjecture to the so-called ""relative"" setting, which includes methods of studying special values of L-functions (also called zeta functions), such as in the prominent, and more recent, conjectures of Gan, Gross, and Prasad. The main tool for proving important instances of functoriality so far has been the trace formula, but in its current form it has nearly reached its limits. This project will examine ways to prove these conjectures by use of the idea of quantization, whose origins lie in mathematical physics. This idea will be used to construct novel ways of comparing (relative) trace formulas, drastically expanding their potential reach and applicability. The broader impacts of the project include conference organization and mentoring of graduate students.

The PI has already shown, in prior work, that in some low-rank cases one can establish relative functoriality via some novel ""transfer operators"" between relative trace formulas. Such non-standard comparisons of trace formulas were envisioned in Langlands's ""Beyond Endoscopy"" proposal; the ""relative"" setting allows for more flexibility, and more potential applications, for the exploration of such comparisons. Prior work was focused mostly on the case when the L-groups associated to the relative trace formulas are of rank one. The main goal of this project will be to examine ways to generalize the construction of transfer operators to higher rank. The main idea is to view a trace formula as the quantization of its cotangent stack, which in turn is largely controlled by the L-group. Using natural correspondences between such cotangent stacks, the project aims to construct transfer operators between their quantizations. On a separate track, the project will continue work on the duality of Hamiltonian spaces conjectured in the PI's recent work with Ben-Zvi and Venkatesh, with the aim of extending this duality beyond the hyperspherical setting, and exploring applications for the representation theory of p-adic groups.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2418414","Conference: Additive Combinatorics 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Sarah Peluse","MI","Regents of the University of Michigan - Ann Arbor","Standard Grant","Andrew Pollington","06/30/2025","$12,000.00","","speluse@umich.edu","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","126400","7556","$0.00","This award supports participation of US-based graduate students and postdocs in Additive Combinatorics 2024, the first major general conference in additive combinatorics in 15 years, which will be held 7/22/2024-7/26/2024 at the ICMS in Edinburgh, Scotland. This conference will feature five expository plenary talks by senior experts in the field, around a dozen research talks, and an open problem session.

Additive combinatorics continues to be a highly active and dynamic field, and has many deep connections and applications to other branches of pure mathematics. Since the last major general conference in additive combinatorics, the subject has developed enormously. For example, in recent years we have seen the classification of approximate subgroups of nonabelian groups, proofs of an effective inverse theorems for the Gowers uniformity norms in both the setting of cyclic groups and of vector spaces over finite fields, a dramatic improvement in the best upper bounds in the cap set problem, a proof of near-optimal bounds in Roth's theorem, further progress on the sum-product conjecture, proofs of effective bounds in the polynomial Szemerédi theorem along with applications of these results to pointwise ergodic theory, applications of higher-order Fourier analysis to correlations of multiplicative functions and Sarnak's conjecture on the asymptotic orthogonality of the Möbius function to zero entropy flows, a series of works relating the analytic and partition rank of tensors, and the resolution of the polynomial Freiman--Ruzsa conjecture in vector spaces over finite fields. The expository plenary lectures will survey the current state of various subfields of additive combinatorics, and the research lectures will cover the most exciting recent breakthroughs. The open problem session will suggest future directions for the field, and should be particularly useful for the more junior participants.

Conference website: https://www.icms.org.uk/AdditiveCombinatorics

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401184","Representation Theory and Geometry in Monoidal Categories","DMS","ALGEBRA,NUMBER THEORY,AND COM","09/01/2024","04/04/2024","Daniel Nakano","GA","University of Georgia Research Foundation Inc","Continuing Grant","Tim Hodges","08/31/2027","$88,519.00","","nakano@math.uga.edu","310 E CAMPUS RD RM 409","ATHENS","GA","306021589","7065425939","MPS","126400","","$0.00","The Principal Investigator (PI) will investigate the representation theory of various algebraic objects. A representation of an abstract algebraic object is a realization of the object via matrices of numbers. Often times, it is advantageous to view the entire collection of representations of an algebraic object as a structure known as a tensor category. Tensor categories consist of objects with additive and multiplicative operations like the integers or square matrices. Using the multiplicative operation, one can introduce the spectrum of the tensor category which is a geometric object (like a cone, sphere or torus). The PI will utilize the important connections between the algebraic and geometric properties of tensor categories to make advances in representation theory. The PI will continue to involve undergraduate and graduate students in these projects. He will continue to be an active member of the mathematical community by serving on national committees for the American Mathematical Society (AMS), and as an editor of a major mathematical journal.

The PI will develop new methods to study monoidal triangular geometry. Several central problems will utilize the construction of homological primes in the general monoidal setting and the introduction of a representation theory for MTCs. This representation theory promises to yield new information about the Balmer spectrum of the MTC. In particular, the general MTC theory will be applied to study representations of Lie superalgebras. The PI will also explore new ideas to study representations of classical simple Lie superalgebras. This involves systematically studying various versions of Category O and the rational representations for the associated quasi-reductive supergroups. One of the main ideas entails the use of the detecting and BBW parabolic subgroups/subalgebras. Furthermore, the PI will study the orbit structure of the nilpotent cone and will construct resolutions of singularities for the orbit closures. The PI will study important questions involving representations of reductive algebraic groups. Key questions will focus on the understanding the structures of induced representations, and whether these modules admit p-filtrations. These questions are interrelated with the 30-year-old problem of realizing projective modules for the Frobenius kernels via tilting modules for the reductive algebraic group, and the structure of extensions between simple modules for the first Frobenius kernel.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401117","Arithmetic Structure in Dense Sets","DMS","ALGEBRA,NUMBER THEORY,AND COM","06/01/2024","04/04/2024","Sarah Peluse","MI","Regents of the University of Michigan - Ann Arbor","Continuing Grant","Andrew Pollington","05/31/2027","$117,039.00","","speluse@umich.edu","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","126400","9251","$0.00","This project focuses primarily on three different problems in number theory, combinatorics, and ergodic theory. This includes work in additive combinatorics concerning generalizations of Szemerédi's theorem on arithmetic progressions (sequences of numbers that are all equally spaced, like 4, 6, 8, and 10), which, informally, says that any sufficiently large collection of whole numbers contains a long arithmetic progression. It is a central problem in additive combinatorics to determine how large ""sufficiently large"" is. The investigator will study versions of this question involving more complicated patterns than arithmetic progressions, and then use the results and techniques developed to make progress on a related problem in ergodic theory. The investigator will also study the size and structure of integer distance sets, which are sets of points whose pairwise distances are all whole numbers. This award will support undergraduate summer research on representation theory and additive combinatorics, and also support the training of graduate students.

More specifically, the investigator will build on her previous work on quantitative bounds for subsets of the integers lacking polynomial progressions of distinct degrees and for subsets of vector spaces over finite fields lacking a certain four-point configuration to tackle more general polynomial, multidimensional, and multidimensional polynomial configurations. The results for multidimensional polynomial configurations of distinct degree will then be used to make progress on the Furstenberg--Bergelson--Leibman conjecture in ergodic theory, which concerns the pointwise almost everywhere convergence of certain nonconventional ergodic averages. She will also investigate the size and structure of integer distance sets, in both the Euclidean plane and in higher dimensions, by encoding them as subsets of rational points on certain families of varieties and then studying these varieties. With her undergraduate students, the investigator will study the distribution of entries in the character tables of symmetric groups and some algorithmic problems in higher-order Fourier analysis.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401106","Analytic Number Theory at the Interface","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","04/04/2024","Maksym Radziwill","IL","Northwestern University","Continuing Grant","Andrew Pollington","04/30/2029","$75,520.00","","maksym.radziwill@northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","126400","","$0.00","The main objective of this award is to develop connections between analytic number theory and other areas of mathematics, specifically dynamics and probability theory. A basic problem in dynamics is to understand how quickly a deterministic process, for example particles in a gas sampled at regularly spaced intervals, randomizes. The regular spacing amounts to sampling the system at the integers. Answering the subtler questions in this area often requires a non-trivial understanding of the properties of the integers. Conversely, a basic problem in number theory is to show that fundamental properties of the integers (e.g their prime factorization) randomize, and here techniques from dynamics can be useful. As for probability, questions from mathematical physics led to the development of techniques for the study of interacting systems, for example the macroscopic properties of a gas of electrons constrained to a surface. It has been recently understood that these techniques are applicable in a number theoretic context, for example in the study of the Riemann zeta-function, a basic function governing the finer properties of the integers. A second objective of this proposal is to develop such techniques further in a number theoretic context. The PI will continue training graduate students and mentor postdocs on topics related to this research.

In dynamics the focus of this project is to understand the convergence of dynamical systems over sparse sets (primes, squares) at every point in the space, with a particular focus on horocycle flow. A second goal is to reexamine the work of Elkies-McMullen on the gap distribution of square-roots of integers modulo one from the perspective of the circle method. In probability theory, the focus is on developing further connections with branching random walks and statistical mechanics, specifically by focusing on the Fyodorov-Hiary-Keating conjecture and its other avatars. A final goal of the project is to develop our understanding of automorphic forms of fractional weight, with applications to concrete number theoretic problems (e.g the equidistribution of Kummer sums).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2416129","Local Geometric Langlands Correspondence and Representation Theory","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","01/12/2024","Sam Raskin","CT","Yale University","Standard Grant","James Matthew Douglass","06/30/2024","$47,858.00","","sam.raskin@yale.edu","150 MUNSON ST","NEW HAVEN","CT","065113572","2037854689","MPS","126400","","$0.00","Representation theory studies the realization of groups as linear symmetries. There are two typical stages: 1) finding the general structure of representations of a given group (e.g., classifying irreducible representations), and 2) applying this to representations of particular interest (e.g., functions on a homogeneous space). This project aims to study higher representation theory, which studies the realization of groups as categorical symmetries. The emphasis of the proposal focuses on loop groups, where the theory remarkably mirrors classical harmonic analysis for p-adic groups. In particular, one finds Langlands-style decompositions here. This project focuses on understanding some key categories of interest in this framework. The investigator will study 3d mirror symmetry conjectures, representations of affine Lie algebras, and moduli spaces of bundles arising in the global geometric Langlands program. This project provides training opportunities for graduate students.

In more detail, 3d mirror symmetry, representations of (reductive) affine Lie algebras, and the geometric Langlands program are the three primary ways actions of loop groups of reductive groups on categories arise. A large class of 3d mirror symmetry conjectures concerns the categorical Plancherel formula for loop group actions on categories of sheaves on loop spaces of particular varieties with group actions. The PI will establish first cases of 3d mirror symmetry and apply the results to give coherent descriptions of some categories of primary interest in geometric representation theory. Representations of Lie algebras concern the action of a group on its category of Lie algebra representations. The PI will extend previous work on critical level localization theory and develop a substitute for Soergel modules that will apply to poorly understood categories in the local geometric Langlands program. The applications to global geometric Langlands concern actions of loop groups of reductive groups on moduli spaces of a global nature, namely bundles with a level structure. The PI will extend the Satake theorem and apply the result to study Eisenstein series in the global geometric Langlands program.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2422557","Conference: Resolution of Singularities, Valuation Theory and Related Topics","DMS","ALGEBRA,NUMBER THEORY,AND COM","08/01/2024","04/03/2024","Steven Cutkosky","MO","University of Missouri-Columbia","Standard Grant","Tim Hodges","01/31/2025","$17,520.00","","cutkoskys@missouri.edu","121 UNIVERSITY HALL","COLUMBIA","MO","652113020","5738827560","MPS","126400","7556","$0.00","This award supports US-based participants in a conference on ``Resolution of Singularities, Valuation Theory and Related Topics'' which will be held from August 5 - 9, 2024 in Morelia, Mexico. The conference will be held at Centro de Ciencias Matematicas, UNAM, Morelia. NSF will provide significant travel and lodging support for 12 U.S. participants to the conference. The funding will be for students, postdoctoral scholars and other U.S. participants who do not have other federal support. A particular emphasis will be on supporting a diversity of participants, especially from under-represented groups.

The focus of the conference is on applications of valuation theory to resolution of singularities in positive characteristic and to other areas of algebraic geometry, commutative algebra and singularity theory. Recently, there have been significant advances in this area, and this conference will cover this progress in talks by the authors of this work. The proposed gathering will provide an opportunity for researchers from diverse fields to interact and establish research connections with each other; in particular, the participants will benefit from this interaction and from seeing recent developments in the field and its relationships with other areas. The conference webpage is
https://sites.google.com/view/spivakovsky60thbirthday/home/authuser=0

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2400550","Splicing Summation Formulae and Triple Product L-Functions","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Jayce Getz","NC","Duke University","Standard Grant","Andrew Pollington","06/30/2027","$220,000.00","","jgetz@math.duke.edu","2200 W MAIN ST","DURHAM","NC","277054640","9196843030","MPS","126400","","$0.00","This award concerns the Langlands program which has been described as a grand unification theory within mathematics. In some sense the atoms of the theory are automorphic representations. The Langlands functoriality conjecture predicts that a collection of natural correspondences preserve these atoms. To even formulate this conjecture precisely, mathematical subjects as diverse as number theory, representation theory, harmonic analysis, algebraic geometry, and mathematical physics are required. In turn, work on the conjecture has enriched these subjects, and in some cases completely reshaped them.

One particularly important example of a correspondence that should preserve automorphic representations is the automorphic tensor product. It has been known for some time that in order to establish this particular case of Langlands functoriality it suffices to prove that certain functions known as L-functions are analytically well-behaved. More recently, Braverman and Kazhdan, Ngo, Lafforgue and Sakellaridis have explained that the expected properties of these L-functions would follow if one could obtain certain generalized Poisson summation formulae. The PI has isolated a particular family of known Poisson summation formulae and proposes to splice them together to obtain the Poisson summation formulae relevant for establishing the automorphic tensor product.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2402637","Conference: Connecticut Summer School in Number Theory 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/15/2024","04/03/2024","Alvaro Lozano-Robledo","CT","University of Connecticut","Standard Grant","Adriana Salerno","03/31/2025","$29,967.00","Keith Conrad, Jennifer Balakrishnan, Christelle Vincent","alvaro.lozano-robledo@uconn.edu","438 WHITNEY RD EXTENSION UNIT 11","STORRS","CT","062699018","8604863622","MPS","126400","7556","$0.00","The Connecticut Summer School in Number Theory (CTNT 2024) is a conference for advanced undergraduate and beginning graduate students, to be followed by a research conference, taking place at at the University of Connecticut, Storrs campus, from June 10 through June 16, 2024. Even though the northeast of the United States is a hotspot for number theory research, there is no instructional school in number theory that occurs in this region. Undergraduate and beginning graduate students who are interested in number theory may only have had an elementary number theory course during college. The CTNT summer school will achieve several outcomes: expose undergraduate and beginning graduate students to accessible topics that are fundamental to contemporary number theory; provide an environment where students interested in number theory can meet each other and network with students, postdocs, and faculty from institutions where number theory is a strong research area; train a diverse group of students on topics of current importance in number theory; allow advanced undergraduates and beginning graduate students to attend a research conference in number theory; videotape the lectures and post them online at a dedicated website to reach as wide of an audience as possible later: https://ctnt-summer.math.uconn.edu/

CTNT 2024 will consist of a 4.5-day summer school followed by a 2-day conference. The summer school will have six mini-courses on topics important to contemporary number theory that are not available in a typical college curriculum, such as elliptic curves, reciprocity, adeles and ideles, and class field theory. The courses will be complemented with course projects, daily invited talks, evening problem sessions, and discussion panels about aspects of graduate school (both for those already in graduate school and those thinking of applying). The conference will consist of several sessions with research talks in number theory, arithmetic geometry, and related topics, and it will be an opportunity for young researchers to present their work.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401337","Algebraic Cycles and L-functions","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/03/2024","Chao Li","NY","Columbia University","Standard Grant","Adriana Salerno","06/30/2027","$230,000.00","","chaoli@math.columbia.edu","615 W 131ST ST","NEW YORK","NY","100277922","2128546851","MPS","126400","","$0.00","The research in this project concerns one of the basic questions in mathematics: solving algebraic equations. The information of the solutions are encoded in various mathematical objects: algebraic cycles, automorphic forms and L-functions. The research will deepen the understanding of these mathematical objects and the connection between them, especially in high dimensions, which requires solving many new problems, developing new tools and interactions in diverse areas, and appealing to new perspectives which may shed new light on old problems. It will also advance the techniques for understanding the arithmetic of elliptic curves, particularly the Birch and Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute. The PI will continue to mentor graduate students, organize conferences and workshops, and write expository articles.

The PI will work on several projects relating arithmetic geometry with automorphic L-function, centered around the common theme of the generalization and applications of the Gross--Zagier formula. The PI will investigate the Kudla--Rapoport conjecture for parahoric levels. The PI will extend the arithmetic inner product formula to orthogonal groups, and study the Bloch--Kato conjecture of symmetric power motives of elliptic curves and endoscopic cases of the arithmetic Gan--Gross--Prasad conjectures. The PI will also investigate a new arithmetic relative trace formula approach towards a Gross--Zagier type formula for orthogonal Shimura varieties.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2416129","Local Geometric Langlands Correspondence and Representation Theory","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","01/12/2024","Sam Raskin","CT","Yale University","Standard Grant","James Matthew Douglass","06/30/2024","$47,858.00","","sam.raskin@yale.edu","150 MUNSON ST","NEW HAVEN","CT","065113572","2037854689","MPS","126400","","$0.00","Representation theory studies the realization of groups as linear symmetries. There are two typical stages: 1) finding the general structure of representations of a given group (e.g., classifying irreducible representations), and 2) applying this to representations of particular interest (e.g., functions on a homogeneous space). This project aims to study higher representation theory, which studies the realization of groups as categorical symmetries. The emphasis of the proposal focuses on loop groups, where the theory remarkably mirrors classical harmonic analysis for p-adic groups. In particular, one finds Langlands-style decompositions here. This project focuses on understanding some key categories of interest in this framework. The investigator will study 3d mirror symmetry conjectures, representations of affine Lie algebras, and moduli spaces of bundles arising in the global geometric Langlands program. This project provides training opportunities for graduate students.

In more detail, 3d mirror symmetry, representations of (reductive) affine Lie algebras, and the geometric Langlands program are the three primary ways actions of loop groups of reductive groups on categories arise. A large class of 3d mirror symmetry conjectures concerns the categorical Plancherel formula for loop group actions on categories of sheaves on loop spaces of particular varieties with group actions. The PI will establish first cases of 3d mirror symmetry and apply the results to give coherent descriptions of some categories of primary interest in geometric representation theory. Representations of Lie algebras concern the action of a group on its category of Lie algebra representations. The PI will extend previous work on critical level localization theory and develop a substitute for Soergel modules that will apply to poorly understood categories in the local geometric Langlands program. The applications to global geometric Langlands concern actions of loop groups of reductive groups on moduli spaces of a global nature, namely bundles with a level structure. The PI will extend the Satake theorem and apply the result to study Eisenstein series in the global geometric Langlands program.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2334874","Conference: Pittsburgh Links among Analysis and Number Theory (PLANT)","DMS","ALGEBRA,NUMBER THEORY,AND COM, ANALYSIS PROGRAM","02/01/2024","01/19/2024","Carl Wang Erickson","PA","University of Pittsburgh","Standard Grant","James Matthew Douglass","01/31/2025","$20,000.00","Theresa Anderson, Armin Schikorra","carl.wang-erickson@pitt.edu","4200 FIFTH AVENUE","PITTSBURGH","PA","152600001","4126247400","MPS","126400, 128100","7556","$0.00","This award will support the four-day conference ""Pittsburgh Links among Analysis and Number Theory (PLANT)"" that will take place March 4-7, 2024 in Pittsburgh, PA. The purpose of the conference is to bring together representatives of two disciplines with a shared interface: number theory and analysis. There is a large potential for deeper collaboration between these fields resulting in new and transformative mathematical perspectives, and this conference aims at fostering such an interchange. In particular, the conference is designed to attract PhD students and post-doctoral scholars into working on innovations at this interface.

To encourage the development of new ideas, the conference speakers, collectively, represent many subfields that have developed their own distinctive blend of analysis and number theory, such as analytic number theory, arithmetic statistics, analytic theory of modular and automorphic forms, additive combinatorics, discrete harmonic analysis, and decoupling. While there have been a wide variety of conferences featuring these subfields in relative isolation, the PIs are excited at PLANT's potential for sparking links among all of these subfields and giving early-career participants the opportunity to be part of this exchange. The conference website is https://sites.google.com/view/plant24/home.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401041","Conference: Singularities in Ann Arbor","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","03/28/2024","Mircea Mustata","MI","Regents of the University of Michigan - Ann Arbor","Standard Grant","Adriana Salerno","04/30/2025","$33,758.00","Qianyu Chen","mmustata@umich.edu","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","126400","7556","$0.00","The conference ""Singularities in Ann Arbor"", scheduled for May 13-17, 2024, at the University of Michigan, Ann Arbor, will explore recent progress in the study of singularities in algebraic geometry. Algebraic geometry, in simple terms, concerns itself with studying geometric objects defined by polynomial equations. This conference will focus on several recent advances concerning singularities: these are points where the geometric objects behave in unexpected ways (such as the bumps or dents on a normally flat surface). Understanding these singularities not only satisfies intellectual curiosity but also plays a crucial role in classifying and comprehending global complex geometric structures. More details about the conference, as well as the list of confirmed lecturers, are available on the conference website, at https://sites.google.com/view/singularitiesinaa.

The conference will feature four lecture series presented by leading experts and rising stars in the field, covering recent advancement related to singularities. These lectures will introduce fresh perspectives and tools, including Hodge Theory, D-modules, and symplectic topology, to address challenging questions in algebraic geometry. The conference aims to make these complex ideas accessible to a younger audience, fostering engagement and understanding among participants. Additionally, the conference will provide a platform for young researchers to showcase their work through a poster session, encouraging collaboration and discussion among participants. This award will provide travel and lodging support for about 35 early-career conference participants.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2412921","Conference: CAAGTUS (Commutative Algebra and Algebraic Geometry in TUcSon)","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","02/14/2024","Debaditya Raychaudhury","AZ","University of Arizona","Standard Grant","Tim Hodges","04/30/2025","$15,000.00","Arvind Suresh, Zhengning Hu","draychaudhury@math.arizona.edu","845 N PARK AVE RM 538","TUCSON","AZ","85721","5206266000","MPS","126400","7556","$0.00","This award will support participation in a weekend conference to be held at the University of Arizona, Tucson on May 4 - 5. The aim of the conference is to establish a solid basis for contacts and collaborations among researchers in Commutative Algebra and Algebraic Geometry located in Arizona and its neighboring states. Its main purposes are to stimulate new directions of research, to provide opportunities to junior researchers to share their work, and to provide a venue for networking and collaboration in the southwest. Its other aim is to expand the network of algebraic and arithmetic geometers by providing an algebro-geometric complement of the Arizona Winter School.

The conference plans to host four leading researchers from Arizona and its neighboring states working in Commutative Algebra and Algebraic Geometry, who will give colloquium-style one-hour lectures on their respective areas of expertise. These hour-long lectures are expected to provide surveys of the current state of the research in these areas, and to provide suggestions for new avenues of research. There will be five or six 30-minute talks given by young researchers, as well as six to eight contributed short 20-minute talks and a poster session. Priority for these contributed talks and posters will be given to recent PhD recipients and members of groups underrepresented in mathematics. Further information is available at the conference website: https://sites.google.com/math.arizona.edu/caagtus/home

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2344680","Conference: Tensor Invariants in Geometry and Complexity Theory","DMS","ALGEBRA,NUMBER THEORY,AND COM, GEOMETRIC ANALYSIS, Algorithmic Foundations","03/15/2024","02/20/2024","Luke Oeding","AL","Auburn University","Standard Grant","James Matthew Douglass","02/28/2025","$40,000.00","","oeding@auburn.edu","321-A INGRAM HALL","AUBURN","AL","368490001","3348444438","MPS","126400, 126500, 779600","7556, 9150","$0.00","The conference Tensor Invariants in Geometry and Complexity Theory will take place May 13-17, 2024 at Auburn University. This conference aims to bring together early-career researchers and experts to study tensor invariants, their appearance in pure algebraic and differential geometry, and their application in Algebraic Complexity Theory and Quantum Information. The workshop will feature talks from both seasoned experts and promising young researchers. The event is designed to facilitate new research connections and to initiate new collaborations. The conference will expose the participants to state-of-the-art research results that touch a variety of scientific disciplines. The activities will support further development of both pure mathematics and the ""down-stream"" applications in each area of scientific focus (Algebraic and Differential Geometry, Algebraic Complexity, Quantum Information).

The conference is centered on invariants in geometry, divided into three themes: Algebraic and Differential Geometry, Tensors and Complexity, and Quantum Computing and Quantum Information. Geometry has long been a cornerstone of mathematics, and invariants are the linchpins. Regarding Algebraic and Differential Geometry, the organizers are inviting expert speakers on topics such as the connections between projective and differential geometry. Considerations in these areas, such as questions about dimensions and defining equations of secant varieties, have led to powerful tools both within geometry and applications in areas such as computational complexity and quantum information. Likewise, the organizers are inviting application-area experts in Algebraic Complexity and Quantum Information. This natural juxtaposition of pure and applied mathematics will lead to new and interesting connections and help initiate new research collaborations. In addition to daily talks by seasoned experts, the conference will include young researchers in a Poster Session and provide networking opportunities, including working group activities, to help early career researchers meet others in the field, which will provide opportunities for new (and ongoing) research collaborations. It is anticipated that these collaborations will continue long after the meeting is over. The conference webpage is: https://webhome.auburn.edu/~lao0004/jmlConference.html.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2400550","Splicing Summation Formulae and Triple Product L-Functions","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","04/04/2024","Jayce Getz","NC","Duke University","Standard Grant","Andrew Pollington","06/30/2027","$220,000.00","","jgetz@math.duke.edu","2200 W MAIN ST","DURHAM","NC","277054640","9196843030","MPS","126400","","$0.00","This award concerns the Langlands program which has been described as a grand unification theory within mathematics. In some sense the atoms of the theory are automorphic representations. The Langlands functoriality conjecture predicts that a collection of natural correspondences preserve these atoms. To even formulate this conjecture precisely, mathematical subjects as diverse as number theory, representation theory, harmonic analysis, algebraic geometry, and mathematical physics are required. In turn, work on the conjecture has enriched these subjects, and in some cases completely reshaped them.

One particularly important example of a correspondence that should preserve automorphic representations is the automorphic tensor product. It has been known for some time that in order to establish this particular case of Langlands functoriality it suffices to prove that certain functions known as L-functions are analytically well-behaved. More recently, Braverman and Kazhdan, Ngo, Lafforgue and Sakellaridis have explained that the expected properties of these L-functions would follow if one could obtain certain generalized Poisson summation formulae. The PI has isolated a particular family of known Poisson summation formulae and proposes to splice them together to obtain the Poisson summation formulae relevant for establishing the automorphic tensor product.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401238","Free Resolutions","DMS","ALGEBRA,NUMBER THEORY,AND COM","06/01/2024","04/03/2024","Irena Peeva","NY","Cornell University","Continuing Grant","Tim Hodges","05/31/2028","$83,779.00","","irena@math.cornell.edu","341 PINE TREE RD","ITHACA","NY","148502820","6072555014","MPS","126400","","$0.00","This project concerns research in Commutative Algebra. A core goal in the subject deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. Closely related to this is the concept of a free resolution. Constructing such a resolution amounts to repeatedly solving systems of polynomial equations. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra. The idea of constructing free resolutions was introduced by Hilbert in a famous paper in 1890. The study of these objects flourished in the second half of the twentieth century and has seen spectacular progress recently. The field is very broad, with strong connections and applications to other mathematical areas. The broader impacts of the project include the writing of an expository paper, organization of professional development workshops for undergraduate students, and organization of mathematical conferences.

The main research goal in this project is to make significant progress in understanding the structure of minimal free resolutions and their numerical invariants. In particular, the PI will: work jointly with M. Mastroeni and J. McCullough on Koszul Algebras; continue work on minimal free resolutions of binomial edge ideals; study the asymptotic structure of minimal free resolutions over exterior algebras.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2422291","CAREER: New methods in curve counting","DMS","ALGEBRA,NUMBER THEORY,AND COM","03/01/2024","02/23/2024","Felix Janda","IL","University of Illinois at Urbana-Champaign","Continuing Grant","James Matthew Douglass","06/30/2028","$144,329.00","","fjanda@illinois.edu","506 S WRIGHT ST","URBANA","IL","618013620","2173332187","MPS","126400","1045","$0.00","The past thirty years have seen a deep and surprising interplay between several branches in pure mathematics, and string theory in physics. In particular, physical predictions have led to the development of mathematical invariants which count algebraic curves in spaces, and conversely, the mathematical study of these invariants has led to advances in string theory. This project further develops two curve counting techniques, the ""logarithmic gauged linear sigma model"" (log GLSM) and ""quasimaps"", and their combination, with the goal of making progress on challenging conjectures from physics, which have appeared out of reach of mathematicians until recently. This project will offer ample training opportunities for graduate students and postdocs. In addition, the PI will organize a yearly intensive weekend learning workshop on a topic of interest, as well as organize events aiming to counter stereotypes in STEM.

More specifically, the project will result in a proof of the localization formula for log GLSM, which is of utmost importance for the application of this technique. In addition, effective invariants, which are a major ingredient of the localization formula, will be studied. In a different direction, the PI will explore applications of log GLSM to the tautological ring, to establish structural predictions observed in physics, such as the ""conifold gap condition"", for the quintic threefold and other one-parameter Calabi-Yau threefolds, and to establish the Landau-Ginzburg/Calabi-Yau correspondence for quintic threefolds in all genera. With regard to quasi-maps, the second main technique employed in this project, the PI will use quasi-maps for explicit computations of Gromov-Witten invariants of non-convex complete intersections. Quasi-maps appear necessary for approaching some of the more mysterious predictions from physics, and hence log GLSM will be extended to allow for quasi-maps.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349623","Invariant Rings, Frobenius, and Differential Operators","DMS","ALGEBRA,NUMBER THEORY,AND COM","06/01/2024","04/03/2024","Anurag Singh","UT","University of Utah","Continuing Grant","Tim Hodges","05/31/2027","$82,544.00","","singh@math.utah.edu","201 PRESIDENTS CIR","SALT LAKE CITY","UT","841129049","8015816903","MPS","126400","","$0.00","This project will investigate several questions in commutative algebra, a field that studies solution sets of polynomial equations. The research will yield concrete information about the properties of solution sets of such equations. Polynomial equations arise in a wide number of applications; one fruitful approach to their study is via studying polynomial functions on their solution sets, that form what is known as a commutative ring. This offers an enormous amount of flexibility in studying solutions sets in various settings, and indeed commutative algebra continues to develop a fascinating interaction with several fields, becoming an increasingly valuable tool in science and engineering. A key component of this project is the training of graduate students in topics connected with the research program.


The focus of the research is on questions related to local cohomology, differential operators, and the property of having finite Frobenius representation type. Local cohomology often provides the best answers to fundamental questions such as the least number of polynomial equations needed to define a solution set; this will be investigated for solution sets related to certain rings of invariants. The differential operators that one encounters in calculus make sense in good generality on solution sets of polynomial equations and are proving to be an increasingly fruitful object of study. Similarly, finite Frobenius representation type, first introduced for the study of differential operators, is proving to be a very powerful property with several applications.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2339274","CAREER: New directions in the study of zeros and moments of L-functions","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","02/12/2024","Alexandra Florea","CA","University of California-Irvine","Continuing Grant","Tim Hodges","06/30/2029","$87,350.00","","alexandra.m.florea@gmail.com","160 ALDRICH HALL","IRVINE","CA","926970001","9498247295","MPS","126400","1045","$0.00","This project focuses on questions in analytic number theory, and concerns properties of the Riemann zeta-function and of more general L-functions. L-functions are functions on the complex plane that often encode interesting information about arithmetic objects, such as prime numbers, class numbers, or ranks of elliptic curves. For example, the Riemann zeta-function (which is one example of an L-function) is closely connected to the question of counting the number of primes less than a large number. Understanding the analytic properties of L-functions, such as the location of their zeros or their rate of growth, often provides insight into arithmetic questions of interest. The main goal of the project is to advance the knowledge of the properties of some families of L-functions and to obtain arithmetic applications. The educational component of the project involves groups of students at different stages, ranging from high school students to beginning researchers. Among the educational activities, the PI will organize a summer school in analytic number theory focusing on young mathematicians, and will run a yearly summer camp at UCI for talented high school students.

At a more technical level, the project will investigate zeros of L-functions by studying their ratios and moments. While positive moments of L?functions are relatively well-understood, much less is known about negative moments and ratios, which have applications to many difficult questions in the field. The planned research will use insights from random matrix theory, geometry, sieve theory and analysis. The main goals fall under two themes. The first theme is developing a general framework to study negative moments of L-functions, formulating full conjectures and proving partial results about negative moments. The second theme involves proving new non-vanishing results about L-functions at special points. Values of L-functions at special points often carry important arithmetic information; the PI plans to show that wide classes of L-functions do not vanish at the central point (i.e., the center of the critical strip, where all the non-trivial zeros are conjectured to be), as well as to study correlations between the values of different L-functions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401152","Conference: Modular forms, L-functions, and Eigenvarieties","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","03/26/2024","John Bergdall","AR","University of Arkansas","Standard Grant","Adriana Salerno","11/30/2024","$15,000.00","","bergdall@uark.edu","1125 W MAPLE ST STE 316","FAYETTEVILLE","AR","727013124","4795753845","MPS","126400","7556, 9150","$0.00","This award supports US-based scientists to attend the conference ""Modular Forms, L-functions, and Eigenvarieties"". The event will take place in Paris, France from June 18, 2024, until June 21, 2024. Whole numbers are the atoms of our mathematical universe. Number theorists study why patterns arise among whole numbers. In the 1970's, Robert Langlands proposed connections between number theory and mathematical symmetry. His ideas revolutionized the field. Some of the most fruitful approaches to his ideas have come via calculus on geometric spaces. The conference funded here will expose cutting edge research on such approaches. The ideas disseminated at the conference will have a broad impact on the field. The presentations of leading figures will propel junior researchers toward new theories. The US-based participants will make a written summary of the conference. The summaries will encourage the next generation to adopt the newest perspectives. Writing them will also engender a spirit of collaboration within the research community. The summaries along with details of the events will be available on the website https://www.eventcreate.com/e/bellaiche/.

The detailed aim of the conference is exposing research on modular forms and L-functions in the context of eigenvarieties. An eigenvariety is a p-adic space that encodes congruence phenomena in number theory. Families of eigenforms, L-functions, and other arithmetic objects find their homes on eigenvarieties. The conference's primary goal is exposing the latest research on such families. The presentations will place new research and its applications all together in one place, under the umbrella of the p-adic Langlands program.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2333970","Conference: Collaborative Workshop in Algebraic Geometry","DMS","ALGEBRA,NUMBER THEORY,AND COM","06/01/2024","03/21/2024","Sarah Frei","NH","Dartmouth College","Standard Grant","Andrew Pollington","05/31/2025","$24,400.00","Ursula Whitcher, Rohini Ramadas, Julie Rana","sarah.frei@dartmouth.edu","7 LEBANON ST","HANOVER","NH","037552170","6036463007","MPS","126400","7556, 9150","$0.00","This award supports participants to attend a collaborative algebraic geometry research workshop at the Institute for Advanced Study (IAS) during the week of June 24-28, 2024. The goals of the workshop are to facilitate significant research in algebraic geometry and to strengthen the community of individuals in the field from underrepresented backgrounds. We will place a particular focus on forming connections across different career stages. Participants will join project groups composed of a leader and co-leader together with two to three junior participants and will spend the workshop engaged in focused and substantive research.

The projects to be initiated during this workshop represent a wide range of subfields of algebraic geometry (e.g. intersection theory, toric geometry and arithmetic geometry), as well as connections to other fields of math (e.g. representation theory). Specifically, topics include: abelian covers of varieties, del Pezzo surfaces over finite fields, positivity of toric vector bundles, Chow rings of Hurwitz spaces with marked ramification, Ceresa cycles of low genus curves, and the geometry of Springer fibers and Hessenberg varieties. More information is available at https://sites.google.com/view/wiag2024/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2408333","Conference: GAeL XXXI (Geometrie Algebrique en Liberte)","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","03/15/2024","Jose Rodriguez","WI","University of Wisconsin-Madison","Standard Grant","Tim Hodges","03/31/2025","$16,255.00","","jrodriguez43@wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","126400","7556","$0.00","This award funds participation of junior US mathematicians in the 31st edition of Gael (Géométrie Algébrique en Liberté) from June 17-21, 2024 at Turin, Italy, held jointly by Politecnico di Torino and Universitŕ di Torino. Géométrie Algébrique en Liberté is a series of annual meetings organized for and by junior researchers in algebraic geometry with a long tradition, drawing in 70-90 participants each year. There are both casual and structured career opportunities for junior mathematicians to interact with speakers and other attendees.

GAeL XXXI will bring together leading experts on a range of topics within Algebraic Geometry, providing an excellent opportunity for junior mathematicians to learn about major new developments. There will be three senior speakers giving mini-courses covering cutting edge results from a wide variety of topics so that GAeL appeals to all PhD students and junior postdocs in algebraic geometry. The rest of the talks are chosen from among the junior participants, often providing the first opportunity for many of these individuals to speak in front of an international audience. More information about GAeL XXXI may be found on the event website: https://sites.google.com/view/gaelxxxi

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349810","Conference: Richmond Geometry Meeting: Geometric Topology and Moduli","DMS","ALGEBRA,NUMBER THEORY,AND COM, TOPOLOGY","05/01/2024","12/07/2023","Nicola Tarasca","VA","Virginia Commonwealth University","Standard Grant","Qun Li","10/31/2025","$26,430.00","Allison Moore, Marco Aldi","tarascan@vcu.edu","910 WEST FRANKLIN ST","RICHMOND","VA","232849005","8048286772","MPS","126400, 126700","7556","$0.00","This award supports the Richmond Geometry Meeting: Geometric Topology and Moduli scheduled for August 9-11, 2024, hosted at Virginia Commonwealth University in Richmond, VA. The conference is designed to unite experts in low-dimensional topology and algebraic geometry, spanning diverse career stages and affiliations. Beyond lectures delivered by internationally recognized experts, vertically integrated participation will be fostered by a poster session showcasing the contributions of early-career researchers and a Career and Mentorship Panel.

The conference will investigate the intersection of low-dimensional topology, algebraic geometry, and mathematical physics. The roots of this interdisciplinary exploration trace back to Witten's groundbreaking work in the late 1980s and the emergence of the Jones polynomial in Chern-Simons theory. Since then, a landscape of profound connections between knot theory, moduli spaces, and string theory has emerged, due to the collective efforts of generations of mathematicians and physicists. Noteworthy developments include the deep ties between Heegaard Floer homology and the Fukaya category of surfaces, the intricate interplay revealed by Khovanov homology, and the correspondence of Gromov-Witten and Donaldson-Thomas theories. The study of moduli spaces of curves, as exemplified in Heegaard Floer homology, has played a pivotal role in several developments. The preceding three editions of the Richmond Geometry Meeting, encompassing both virtual and in-person gatherings, have showcased a wave of collaborative advancements in knot theory, algebraic geometry, and string theory. Topics such as braid varieties, Khovanov homotopy, link lattice homology, and the GW/DT correspondence in families have been explored, unveiling a nexus of interdependent breakthroughs. This award supports the fourth edition of the Richmond Geometry Meeting, providing a vital platform for the dissemination of the latest findings in this dynamic realm of research. For more information, please visit the Richmond Geometry Meeting website: https://math.vcu.edu/rgm

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349888","Conference: International Conference on L-functions and Automorphic Forms","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","03/15/2024","Larry Rolen","TN","Vanderbilt University","Standard Grant","Adriana Salerno","03/31/2025","$25,000.00","Jesse Thorner, Andreas Mono","larry.rolen@vanderbilt.edu","110 21ST AVE S","NASHVILLE","TN","372032416","6153222631","MPS","126400","7556","$0.00","This award provides support for the conference entitled ""International Conference on L-functions and Automorphic Forms'', which will take place at Vanderbilt University in Nashville, Tennessee on May 13--16 2024. This is part of an annual series hosted by Vanderbilt, known as the Shanks conference series. The main theme will be on new developments and recent interactions between the areas indicated in the title. The interplay between automorphic forms and L-functions has a long and very fruitful history in number theory, and bridging both fields is still a very active area of research. This conference is oriented at establishing and furthering dialogue on new developments at the boundary of these areas. This will foster collaboration between researchers working in these fields.

One beautiful feature of modern number theory is that many problems of broad interest, in areas of study as diverse as arithmetic geometry to mathematical physics, can be solved in an essentially optimal way if the natural extension of the Riemann hypothesis holds for L-functions associated to automorphic representations. Although many generalizations and applications around L-functions have have already been worked out, there are still various fundamental open problems among them to tackle, including bounds for and the value distribution of L-functions. The former is related to the pursuit of so-called sub-convexity bounds for L-functions. The latter is related to the Birch and Swinnterton-Dyer conjecture (another ?Millenium problem? posed by the Clay Mathematics institute). These pursuits are closely connected with the Langlands program, a ?grand unifying theory? relating automorphic forms. Further details can be found on the conference website https://my.vanderbilt.edu/shanksseries/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2402436","Conference: Visions in Arithmetic and Beyond","DMS","ALGEBRA,NUMBER THEORY,AND COM","06/01/2024","03/26/2024","Akshay Venkatesh","NJ","Institute For Advanced Study","Standard Grant","Andrew Pollington","05/31/2025","$44,975.00","Alexander Gamburd","akshay@math.ias.edu","1 EINSTEIN DR","PRINCETON","NJ","085404952","6097348000","MPS","126400","7556","$0.00","This award provides funding to help defray the expenses of participants in the conference ""Visions in Arithmetic and Beyond"" (conference website https://www.ias.edu/math/events/visions-in-arithmetic-and-beyond ) to be held at the Institute for Advanced Study and Princeton University from June 3 to June 7, 2024. Those speaking at the meeting include the leading researchers across arithmetic, analysis and geometry.

The conference will provide high-level talks by mathematicians who are both outstanding researchers and excellent speakers. These will synthesize and expose a broad range of recent advances in number theory as well as related developments in analysis and dynamics. In addition to the talks by leading researchers there is also time allotted for a session on the best practices for mentoring graduate students and postdocs.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400006","Conference: Underrepresented Students in Algebra and Topology Research Symposium (USTARS)","DMS","INFRASTRUCTURE PROGRAM, ALGEBRA,NUMBER THEORY,AND COM, TOPOLOGY","03/15/2024","03/12/2024","Ryan Moruzzi","CA","California State University, East Bay Foundation, Inc.","Standard Grant","Adriana Salerno","02/28/2025","$36,000.00","Christopher ONeill, Robyn Brooks","ryan.moruzzi@csueastbay.edu","25800 CARLOS BEE BLVD","HAYWARD","CA","945423000","5108854212","MPS","126000, 126400, 126700","7556","$0.00","This award will support the Underrepresented Students in Topology and Algebra Research Symposium (USTARS). A goal of this conference is to highlight research being conducted by underrepresented students in the areas of algebra and topology. At this unique meeting, attendees are exposed to a greater variety of current research, ideas, and results in their areas of study and beyond. Participants are also given the opportunity to meet and network with underrepresented professors and students who may later become collaborators and colleagues. This is particularly important for students with great academic potential who do not attend top-tier research institutions; students that are often overlooked, despite a strong faculty and graduate student population. Furthermore, USTARS promotes diversity in the mathematical sciences by encouraging women and minorities to attend and give talks. Participants of USTARS continue to influence the next generation of students in positive ways by serving as much needed mentors and encouraging students in the mathematical sciences to advance themselves and participate in research and conference events. USTARS exposes all participants to the research and activities of underrepresented mathematicians, encouraging a more collaborative mathematics community.

The Underrepresented Students in Topology and Algebra Research Symposium (USTARS) is a project proposed by a group of underrepresented young mathematicians. The conference organizing committee is diverse in gender, ethnicity, and educational background, and is well-positioned to actively encourage participation by women and minorities. The symposium includes networking sessions along with research presentations. Speakers will give 30-minute parallel research talks. Graduate students will give at least 75% of these presentations. Two distinguished graduate students and one invited faculty member are chosen to give 1-hour presentations and a poster session featuring invited undergraduates is also planned. Additionally, a discussion panel and creative math session will provide networking, guidance, and mentorship opportunities from past USTARS participants that have transitioned to full-time faculty positions. The conference website is https://www.ustars.org/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2342225","RTG: Numbers, Geometry, and Symmetry at Berkeley","DMS","ALGEBRA,NUMBER THEORY,AND COM, WORKFORCE IN THE MATHEMAT SCI","08/01/2024","03/18/2024","Tony Feng","CA","University of California-Berkeley","Continuing Grant","Andrew Pollington","07/31/2029","$1,196,058.00","Martin Olsson, David Nadler, Sug Woo Shin, Yunqing Tang","fengt@berkeley.edu","1608 4TH ST STE 201","BERKELEY","CA","947101749","5106433891","MPS","126400, 733500","","$0.00","The project will involve a variety of activities organized around the research groups in number theory, geometry, and representation theory at UC Berkeley. These subjects study the structure and symmetries of mathematical equations, and have applications to (for example) cryptography, codes, signal processing, and physics. There will be an emphasis on training graduate students to contribute to society as scientists, educators, and mentors.

More precisely, RTG will be used to organize annual graduate research workshops, with external experts, as well as weekly research seminars to keep up-to-date on cutting-edge developments. Summer programs on Research Experiences of Undergraduates will provide valuable research exposure to undergraduates, and also mentorship training to graduate students. Postdocs will be hired to help lead these activities. Finally, the project will fund outreach to local schools. At the core of all these activities is the goal of training students and postdocs as strong workforce in their dual roles as mentors and mentees, recruiting students into the ?eld with a good representation of underrepresented groups, and providing a setting for collaboration between all levels and across di?erent areas.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2409946","Conference: URiCA 2024 and 2025","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","03/15/2024","Mark Walker","NE","University of Nebraska-Lincoln","Standard Grant","Tim Hodges","03/31/2026","$30,000.00","","mark.walker@unl.edu","2200 VINE ST","LINCOLN","NE","685032427","4024723171","MPS","126400","7556, 9150","$0.00","This award supports participation of graduate students and postdoctoral researchers in the conference ""Upcoming Researchers in Commutative Algebra"" (URiCA) to be held in Lincoln, Nebraska, May 11-12, 2024, and in May 2025. URiCA (formerly known as KUMUNU Jr), is an annual conference hosted by the Department of Mathematics at the University of Nebraska-Lincoln designed to promote visibility and interactions between graduate students and postdocs working in commutative algebra and related fields.

The conference will feature ten talks from graduate students and postdocs over the span of two days. These talks will be on original research done in commutative algebra and related fields of mathematics. While URiCA was initially focused on connecting mathematical communities in the Great Plains region, previous success has allowed us to extend our outreach and include more early career researchers from outside this region. The goal of the conference is to help foster a diverse and welcoming community of early researchers leading to increased collaborations and exposure to different areas of research being done in commutative algebra. This conference will continue to provide an opportunity for early researchers and underrepresented groups in math to share their work and gain visibility among their peers.

Conference Website: https://urica-unl.github.io/

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2337830","CAREER: Quantifying congruences between modular forms","DMS","ALGEBRA,NUMBER THEORY,AND COM","08/15/2024","01/18/2024","Preston Wake","MI","Michigan State University","Continuing Grant","Tim Hodges","07/31/2029","$85,593.00","","wakepres@msu.edu","426 AUDITORIUM RD RM 2","EAST LANSING","MI","488242600","5173555040","MPS","126400","1045","$0.00","Number theory is the study of the most basic mathematical objects, whole numbers. Because whole numbers are so fundamental, number theory has connections with all major areas of mathematics. For instance, consider the problem of finding the whole-number solutions to a given equation. One can consider the shape given by the graph of that equation, or the set of symmetries that the equation has, or the function whose coefficients come from counting the number of solutions over a variety of number systems. The geometric properties of this shape, the algebraic properties of these symmetries, and the analytic properties of this function are all intimately related to the behavior of the equation?s whole-number solutions. Number theorists use techniques from each of these mathematical areas, but also, in the process, uncover surprising connections between the areas whereby discoveries in one area can lead to growth in another. One part of number theory where the connections between geometry, algebra, and analysis are particularly strong is in the field of modular forms. The proposed research focuses on an important and well-known type of relation between different modular forms called congruence and aims to compute the number of forms that are congruent to a given modular form and uncover the number-theoretic significance of this computation. Many of the conjectures that drive this project were found experimentally, through computer calculations. The main educational objective is to contribute to the training of the next generation of theoretical mathematicians in computational and experimental methods. To achieve this, the Principal Investigator (PI) will design software modules for a variety of undergraduate algebra and number theory courses that provide hands-on experience with computation. In addition, the PI will supervise undergraduates in computational research experiments designed to numerically verify conjectures made in the project and to explore new directions.

Congruences between modular forms provide a link between two very different types of objects in number theory: algebraic objects, like Galois representations, and analytic objects, like L-functions. This link has been used as a tool for proving some of the most celebrated results in modern number theory, such as the Main Conjecture of Iwasawa theory. The proposed research pushes the study of congruences in a new, quantitative direction by counting the number of congruences, not just determining when a congruence exists. The central hypothesis is that this quantitative structure of congruences contains finer information about the algebraic and analytic quantities involved than the Main Conjecture and its generalizations (such as the Bloch?Kato conjecture) can provide.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2347095","Collaborative Research: Conference: Texas-Oklahoma Representations and Automorphic forms (TORA)","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","12/19/2023","Kimball Martin","OK","University of Oklahoma Norman Campus","Standard Grant","Andrew Pollington","12/31/2026","$20,000.00","Ameya Pitale","kmartin@math.ou.edu","660 PARRINGTON OVAL RM 301","NORMAN","OK","730193003","4053254757","MPS","126400","7556, 9150","$0.00","This award supports the TORA mathematics conference series. This series consists of annual meetings hosted by the University of North Texas, Oklahoma State University, and the University of Oklahoma on a rotating basis. This award provides support for three weekend conferences, one at the University of North Texas in Spring 2024 (TORA XIII), one at Oklahoma State University in Spring 2025 (TORA XIV), and another at the University of Oklahoma in Spring 2026 (TORA XV). Each conference will feature three prominent guest speakers from outside the Texas-Oklahoma region, in addition to other participants including students, post-doctoral researchers, and junior faculty. Regional graduate students and researchers will also give talks describing their work. These conferences will facilitate collaborations and interactions among the students and researchers in the region who work in the areas of Automorphic Forms, Representation Theory, and Number Theory.

Over the last century, the theories of automorphic forms and representations have grown enormously. Important applications impact various fields of research, ranging from number theory, coding theory, algebraic geometry, and topology to Kac-Moody algebras and quantum field theory. The interplay of automorphic forms and representation theory has been especially fruitful, and many surprising and deep results have emerged. The TORA conference series will emphasize the interplay between automorphic forms and representations, both in the classical and adelic languages, and related topics like analytic number theory and harmonic analysis.



The conference Texas-Oklahoma Representations and Automorphic forms XIII will take place on April 12-14, 2024, at the University of North Texas. Additional information can be found on the conference website: https://www.math.unt.edu/~richter/TORA/TORA13.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2347097","Collaborative Research: Conference: Texas-Oklahoma Representations and Automorphic forms (TORA)","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","12/19/2023","Melissa Emory","OK","Oklahoma State University","Standard Grant","Andrew Pollington","12/31/2026","$20,000.00","Maria Fox, Mahdi Asgari","melissa.emory@okstate.edu","401 WHITEHURST HALL","STILLWATER","OK","740781031","4057449995","MPS","126400","7556, 9150","$0.00","This award supports the TORA mathematics conference series. This series consists of annual meetings hosted by the University of North Texas, Oklahoma State University, and the University of Oklahoma on a rotating basis. This award provides support for three weekend conferences, one at the University of North Texas in Spring 2024 (TORA XIII), one at Oklahoma State University in Spring 2025 (TORA XIV), and another at the University of Oklahoma in Spring 2026 (TORA XV). Each conference will feature three prominent guest speakers from outside the Texas-Oklahoma region, in addition to other participants including students, post-doctoral researchers, and junior faculty. Regional graduate students and researchers will also give talks describing their work. These conferences will facilitate collaborations and interactions among the students and researchers in the region who work in the areas of Automorphic Forms, Representation Theory, and Number Theory.

Over the last century, the theories of automorphic forms and representations have grown enormously. Important applications impact various fields of research, ranging from number theory, coding theory, algebraic geometry, and topology to Kac-Moody algebras and quantum field theory. The interplay of automorphic forms and representation theory has been especially fruitful, and many surprising and deep results have emerged. The TORA conference series will emphasize the interplay between automorphic forms and representations, both in the classical and adelic languages, and related topics like analytic number theory and harmonic analysis.



The conference Texas-Oklahoma Representations and Automorphic forms XIII will take place on April 12-14, 2024, at the University of North Texas. Additional information can be found on the conference website: https://www.math.unt.edu/~richter/TORA/TORA13.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401444","Conference: Workshop on Automorphic Forms and Related Topics","DMS","ALGEBRA,NUMBER THEORY,AND COM","03/01/2024","02/27/2024","Melissa Emory","OK","Oklahoma State University","Standard Grant","Andrew Pollington","02/28/2025","$24,800.00","Kimberly Logan, Liyang Yang, Jonathan Cohen","melissa.emory@okstate.edu","401 WHITEHURST HALL","STILLWATER","OK","740781031","4057449995","MPS","126400","7556, 9150","$0.00","The 36th Annual Workshop on Automorphic Forms and Related Topics (AFW) will take place May 20-24, 2024, at Oklahoma State University in Stillwater, OK. The AFW is an internationally recognized, well-respected conference on topics related to automorphic forms, which have played a key role in many recent breakthroughs in mathematics. The AFW will bring together a geographically diverse group of participants at a wide range of career stages, from graduate students to senior professors. Typically, about half of the attendees at the AFW are at early stages of their careers, and about one quarter to one third of participants are women. The AFW will continue to provide a supportive and encouraging environment for giving talks, exchanging ideas, and beginning new collaborations. This is the first time the AFW will meet in Oklahoma where many experts on automorphic forms and closely related topics are nearby. Thus, in addition to attracting speakers who participate annually, the workshop is likely to draw a mix of new attendees who will contribute new perspectives and energy and benefit from the workshop. The workshop is known for its inclusive, encouraging atmosphere, particularly to early career researchers and to those from underrepresented groups in the number theory community. The workshop has traditionally been a fruitful place for these researchers to connect with potential collaborators and mentors at other institutions, working on related topics. To help achieve this goal, the 2024 AFW will feature five expository talks on various fundamental topics in the theory of automorphic forms, aimed at the graduate student level. There will also be two panel discussions focused on mathematical career questions.

Automorphic forms play a central role in number theory, being integral to the proofs of many groundbreaking theorems, including Fermat's Last Theorem (by Andrew Wiles), the Sato-Tate Conjecture (by Thomas Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor), Serre's Conjecture (by Chandrashekhar Khare, Mark Kisin, and Jean-Pierre Wintenberger), the Sato-Tate Conjecture (by Thomas Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor), Serre's Uniformity Conjecture (by Yuri Bilu and Pierre Parent), and the Fundamental Lemma (for which Ngo Bau Chau was awarded the Fields Medal). Automorphic forms are the subject of many important ongoing conjectures, among them the Langlands program, connections to random matrix theory, and the generalized Riemann hypothesis. They also appear in many areas of mathematics outside number theory, most notably in mathematical physics. The topics covered in this year's workshop are likely to include elliptic, Siegel, Hilbert, and Bianchi modular forms, elliptic curves and abelian varieties, special values of L-functions, p-adic aspects of L-functions and automorphic forms, connections with representation theory, mock modular forms, quadratic forms, connections with mathematical physics, monstrous moonshine, and additional related areas of research.


Additional information can be found on the conference website: http://automorphicformsworkshop.org/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401305","Conference: ANTS XVI: Algorithmic Number Theory Symposium 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM, Secure &Trustworthy Cyberspace","07/01/2024","02/27/2024","Andrew Sutherland","MA","Massachusetts Institute of Technology","Standard Grant","Andrew Pollington","06/30/2025","$36,000.00","","drew@math.mit.edu","77 MASSACHUSETTS AVE","CAMBRIDGE","MA","021394301","6172531000","MPS","126400, 806000","7556","$0.00","This award provides funds for early-career researchers (graduate students, postdocs, and tenure-track faculty not having other NSF support) to attend the sixteenth edition of the Algorithmic Number Theory Symposium (ANTS-XVI) held July 15-19, 2024 at the Massachusetts Institute of Technology (MIT). The ANTS meetings, held biannually since 1994, are the premier international forum for new research in computational number theory. As an established conference series, ANTS attracts invited and contributed lectures of the highest quality, and serves as a forum for dissemination of new ideas and techniques throughout the research community in the area of computational number theory and number-theoretic aspects of cryptography. In addition to numerous applications to theoretical mathematics, these fields have immense importance through real world connections to computer security.

The ANTS meetings are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, arithmetic algebraic geometry, modular forms, finite fields, and applications of number theory to cryptography. Participants include academic researchers in both mathematics and computer science, as well as mathematicians in industry who work on cryptography and other areas of application; similarly, the topics presented include both pure and applied topics. The review process for contributed lectures and the subsequent production of a proceedings volume provides documentation of the presented results at a quality level comparable to an international research journal in mathematics. This award funds lodging and US-based travel for researchers who might not otherwise be able to participate in this premier event. Funding priority will be given to those contributing papers or posters; the organizers also seek to actively promote participation by women and underrepresented minorities.

More information about the conference can be found at https://antsmath.org/ANTSXVI/

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2412921","Conference: CAAGTUS (Commutative Algebra and Algebraic Geometry in TUcSon)","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","02/14/2024","Debaditya Raychaudhury","AZ","University of Arizona","Standard Grant","Tim Hodges","04/30/2025","$15,000.00","Arvind Suresh, Zhengning Hu","draychaudhury@math.arizona.edu","845 N PARK AVE RM 538","TUCSON","AZ","85721","5206266000","MPS","126400","7556","$0.00","This award will support participation in a weekend conference to be held at the University of Arizona, Tucson on May 4 - 5. The aim of the conference is to establish a solid basis for contacts and collaborations among researchers in Commutative Algebra and Algebraic Geometry located in Arizona and its neighboring states. Its main purposes are to stimulate new directions of research, to provide opportunities to junior researchers to share their work, and to provide a venue for networking and collaboration in the southwest. Its other aim is to expand the network of algebraic and arithmetic geometers by providing an algebro-geometric complement of the Arizona Winter School.

The conference plans to host four leading researchers from Arizona and its neighboring states working in Commutative Algebra and Algebraic Geometry, who will give colloquium-style one-hour lectures on their respective areas of expertise. These hour-long lectures are expected to provide surveys of the current state of the research in these areas, and to provide suggestions for new avenues of research. There will be five or six 30-minute talks given by young researchers, as well as six to eight contributed short 20-minute talks and a poster session. Priority for these contributed talks and posters will be given to recent PhD recipients and members of groups underrepresented in mathematics. Further information is available at the conference website: https://sites.google.com/math.arizona.edu/caagtus/home

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2339274","CAREER: New directions in the study of zeros and moments of L-functions","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","02/12/2024","Alexandra Florea","CA","University of California-Irvine","Continuing Grant","Tim Hodges","06/30/2029","$87,350.00","","alexandra.m.florea@gmail.com","160 ALDRICH HALL","IRVINE","CA","926970001","9498247295","MPS","126400","1045","$0.00","This project focuses on questions in analytic number theory, and concerns properties of the Riemann zeta-function and of more general L-functions. L-functions are functions on the complex plane that often encode interesting information about arithmetic objects, such as prime numbers, class numbers, or ranks of elliptic curves. For example, the Riemann zeta-function (which is one example of an L-function) is closely connected to the question of counting the number of primes less than a large number. Understanding the analytic properties of L-functions, such as the location of their zeros or their rate of growth, often provides insight into arithmetic questions of interest. The main goal of the project is to advance the knowledge of the properties of some families of L-functions and to obtain arithmetic applications. The educational component of the project involves groups of students at different stages, ranging from high school students to beginning researchers. Among the educational activities, the PI will organize a summer school in analytic number theory focusing on young mathematicians, and will run a yearly summer camp at UCI for talented high school students.

At a more technical level, the project will investigate zeros of L-functions by studying their ratios and moments. While positive moments of L?functions are relatively well-understood, much less is known about negative moments and ratios, which have applications to many difficult questions in the field. The planned research will use insights from random matrix theory, geometry, sieve theory and analysis. The main goals fall under two themes. The first theme is developing a general framework to study negative moments of L-functions, formulating full conjectures and proving partial results about negative moments. The second theme involves proving new non-vanishing results about L-functions at special points. Values of L-functions at special points often carry important arithmetic information; the PI plans to show that wide classes of L-functions do not vanish at the central point (i.e., the center of the critical strip, where all the non-trivial zeros are conjectured to be), as well as to study correlations between the values of different L-functions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2337178","Conference: Fairfax Algebra Days 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM","02/01/2024","12/27/2023","Rebecca Rebhuhn-Glanz","VA","George Mason University","Standard Grant","Tim Hodges","01/31/2025","$25,000.00","Neil Epstein, Ela Celikbas, Hugh Geller","rrebhuhn@gmu.edu","4400 UNIVERSITY DR","FAIRFAX","VA","220304422","7039932295","MPS","126400","7556","$0.00","This award supports participation in the ?Fairfax Algebra Days 2024? (FAD 2024), a commutative algebra conference which is to take place at George Mason University on the weekend of March 9-10, 2024. FAD is a continuation of Morgantown Algebra Days (MAD), which started running in 2019 to promote interaction among faculty, postdoctoral researchers, and graduate students in commutative algebra in the Eastern US region. The first two meetings united faculty and students from a variety of institutions for inspiring talks and poster presentations. As in 2019 and 2023, the primary objective of the conference is to give participants at all career stages a chance to share their research with others, learn about recent developments in the field, develop new collaborations, and strengthen their ties to the mathematical community.

FAD 2024 will feature seven research talks from a diverse group of speakers and a poster session allowing for junior mathematicians to present their research. All talks will be concerned with commutative algebra and will involve subjects such as minimal free resolutions, homological invariants of rings, and complete intersections. In addition, the talks will cover connections between commutative algebra and other areas of mathematics such as representation theory, combinatorics, topology and graph theory

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401662","Conference: Southern Regional Algebra Conference 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM","03/15/2024","01/18/2024","Jean Nganou","TX","University of Houston - Downtown","Standard Grant","Tim Hodges","02/28/2025","$14,990.00","","nganouj@uhd.edu","1 MAIN ST","HOUSTON","TX","770021014","7132218005","MPS","126400","7556","$0.00","This award supports participation in the Southern Regional Algebra Conference (SRAC). The SRAC is a yearly weekend conference that has been in existence since 1988. Its first edition was held at the University of Southern Mississippi in the Spring of 1988. This spring the SRAC will be held at the University of Houston-Downtown, March 22-24, 2024. The SRAC brings together mathematicians that carry out research in the area of algebra and closely related areas for a full weekend of lectures, short presentations and discussions. The conference attracts researchers from many undergraduate institutions in the Gulf Coast Region that usually do not have sufficient funding to support their research activities, especially long-distance meetings. It is also an important platform for graduate students and early career mathematicians to present their research in algebra and be exposed to a community of algebraists outside their respective home institutions.

The main themes of the conference are Lie/Leibniz Algebras and their representation theory; and the theory of nearrings and other generalizations of rings. On Friday March 22, there will be a single session on topics in algebra that lie either at the intersection of two themes of the conference or outside of their union. On Saturday March 23, the conference will begin with an hour-long plenary session on Leibniz algebras and the rest of the day will be split into two parallel sessions of 25-min talks, with each session focusing on one of the main themes. On Sunday March 24, the conference will start with an hour-long plenary session on the near-rings theory, and the rest of the morning will be split into two parallel sessions of 25-min talks, with each session focusing on one of the main themes. There will be plenty of opportunity for informal follow-up discussions. Further information is available at the conference website:
https://www.uhd.edu/academics/sciences/mathematics-statistics/southern-regional-algebra-conference.aspx

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349244","Conference: Texas Algebraic Geometry Symposium (TAGS) 2024-2026","DMS","ALGEBRA,NUMBER THEORY,AND COM","04/01/2024","01/19/2024","Frank Sottile","TX","Texas A&M University","Continuing Grant","James Matthew Douglass","03/31/2027","$15,000.00","","sottile@tamu.edu","400 HARVEY MITCHELL PKY S STE 30","COLLEGE STATION","TX","778454375","9798626777","MPS","126400","","$0.00","The Texas Algebraic Geometry Symposium (TAGS) will be held at Texas A&M University April 5,6, and 7, 2024, and in Spring 2025 and Spring 2026. TAGS is an annual regional conference which is jointly organized by faculty at Rice University, Texas A&M University, and the University of Texas at Austin. The conference series began in 2005, and serves to enhance the educational and research environment in Texas and the surrounding states, providing an important opportunity for interaction and sharing of ideas for students and researchers in this region.

TAGS serves to ensure that members of the algebraic geometry community in the Texas region stay in regular contact and brings distinguished mathematicians and rising stars to an area with no other comparable regular gatherings in algebraic geometry. The 2024 TAGS will have nine lectures delivered by a diverse group of speakers, and will include accessible lectures for graduate students and a juried poster session for students and junior researchers. It will be held in conjunction with the annual Maxson lectures at Texas A&M the week before and delivered by Prof. David Eisenbud. The TAGS website is https://franksottile.github.io/conferences/TAGS24/index.html.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." @@ -44,11 +47,9 @@ "2400553","Conference: Arithmetic quantum field theory","DMS","ALGEBRA,NUMBER THEORY,AND COM","03/01/2024","02/12/2024","Daniel Freed","MA","Harvard University","Standard Grant","Andrew Pollington","02/28/2025","$45,000.00","David Ben-Zvi","dafr@math.harvard.edu","1033 MASSACHUSETTS AVE STE 3","CAMBRIDGE","MA","021385366","6174955501","MPS","126400","7556","$0.00","The conference Arithmetic Quantum Field Theory will be held at the Harvard Center of Mathematical Sciences and Applications (CMSA) on March 25-29 2024. This will be an in-person gathering of approximately 70 researchers - graduate students, postdocs, and faculty in mathematics and physics, available in hybrid mode to an unlimited number of outside participants. A central focus of the conference - and the dedicated aim of its first day - is to encourage a high level of participation by women in math and physics. The first day is designed to encourage junior researchers to come and network, give talks in a friendly environment, and participate without concern over the precise fit of their research to the narrow theme of the workshop.

The conference Arithmetic Quantum Field Theory, and the two-month program of the same title it concludes, are aimed at catalyzing interactions between mathematicians and physicists by disseminating exciting new connections emerging between quantum field theory and algebraic number theory, and in particular between the fundamental invariants of each: partition functions and L-functions. On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds. On the other hand, the arithmetic topology dictionary provides a sturdy bridge between the topology of manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory. The program will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, harmonic analysis and L-functions, and conversely to explore potential geometric and physical consequences of arithmetic ideas.
The conference website is https://cmsa.fas.harvard.edu/event/aqftconf/ where recordings of the talks and notes from lectures will be made widely available.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2338485","CAREER: Moduli Spaces, Fundamental Groups, and Asphericality","DMS","ALGEBRA,NUMBER THEORY,AND COM, TOPOLOGY","07/01/2024","02/02/2024","Nicholas Salter","IN","University of Notre Dame","Continuing Grant","Swatee Naik","06/30/2029","$67,067.00","","nsalter@nd.edu","836 GRACE HALL","NOTRE DAME","IN","465566031","5746317432","MPS","126400, 126700","1045","$0.00","This NSF CAREER award provides support for a research program at the interface of algebraic geometry and topology, as well as outreach efforts aimed at improving the quality of mathematics education in the United States. Algebraic geometry can be described as the study of systems of polynomial equations and their solutions, whereas topology is the mathematical discipline that studies notions such as ?shape? and ?space"" and develops mathematical techniques to distinguish and classify such objects. A notion of central importance in these areas is that of a ?moduli space? - this is a mathematical ?world map? that gives a complete inventory and classification of all instances of a particular mathematical object. The main research objective of the project is to better understand the structure of these spaces and to explore new phenomena, by importing techniques from neighboring areas of mathematics. While the primary aim is to advance knowledge in pure mathematics, developments from these areas have also had a long track record of successful applications in physics, data science, computer vision, and robotics. The educational component includes an outreach initiative consisting of a ?Math Circles Institute? (MCI). The purpose of the MCI is to train K-12 teachers from around the country in running the mathematical enrichment activities known as Math Circles. This annual 1-week program will pair teachers with experienced instructors to collaboratively develop new materials and methods to be brought back to their home communities. In addition, a research conference will be organized with the aim of attracting an international community of researchers and students and disseminating developments related to the research objectives of the proposal.

The overall goal of the research component is to develop new methods via topology and geometric group theory to study various moduli spaces, specifically, (1) strata of Abelian differentials and (2) families of polynomials. A major objective is to establish ?asphericality"" (vanishing of higher homotopy) of these spaces. A second objective is to develop the geometric theory of their fundamental groups. Asphericality occurs with surprising frequency in spaces coming from algebraic geometry, and often has profound consequences. Decades on, asphericality conjectures of Arnol?d, Thom, and Kontsevich?Zorich remain largely unsolved, and it has come to be regarded as a significantly challenging topic. This project?s goal is to identify promising-looking inroads. The PI has developed a method called ""Abel-Jacobi flow"" that he proposes to use to establish asphericality of some special strata of Abelian differentials. A successful resolution of this program would constitute a major advance on the Kontsevich?Zorich conjecture; other potential applications are also described. The second main focus is on families of polynomials. This includes linear systems on algebraic surfaces; a program to better understand the fundamental groups is outlined. Two families of univariate polynomials are also discussed, with an eye towards asphericality conjectures: (1) the equicritical stratification and (2) spaces of fewnomials. These are simple enough to be understood concretely, while being complex enough to require new techniques. In addition to topology, the work proposed here promises to inject new examples into geometric group theory. Many of the central objects of interest in the field (braid groups, mapping class groups, Artin groups) are intimately related to algebraic geometry. The fundamental groups of the spaces the PI studies here should be just as rich, and a major goal of the project is to bring this to fruition.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2338942","CAREER: Algebraicity and Integral Models of Shimura Varieties","DMS","ALGEBRA,NUMBER THEORY,AND COM","07/01/2024","01/12/2024","Ananth Shankar","IL","Northwestern University","Continuing Grant","Tim Hodges","06/30/2029","$73,080.00","","ananth@northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","126400","1045, 9251","$0.00","This project concerns the study of Shimura varieties. These are geometric objects that are defined as solutions to polynomial equations with coefficients that are rational numbers. Shimura varieties have played a crucial role in settling several long standing conjectures, including the Mordel Conjecture. The PI and his collaborators propose to work on the question of finding polynomial equations with integer coefficients which define Shimura varieties. This question is fundamental to the study of Number Theory and Arithmetic Geometry and has broad applications to several important and well-known conjectures. The educational component of the project includes a workshop targeted at early-stage graduate students looking to work in Arithmetic Geometry, aimed at helping these students acquire background to start working on research problems in this field. The project also provides opportunities for undergraduate students to work on research problems, as well as thesis-problems for graduate students.

The PI will work on the fundamental problems of studying integral models and the p-adic geometry of Shimura varieties. Specifically, the PI and his collaborators will work on studying integral models of exceptional Shimura varieties, and studying questions pertaining to p-adic transcendence on Shimura varieties (including a p-adic analogue of Borel's algebraicity theorem, and questions pertaining to p-adic bi-algebraicity on Shimura varieties).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2337830","CAREER: Quantifying congruences between modular forms","DMS","ALGEBRA,NUMBER THEORY,AND COM","08/15/2024","01/18/2024","Preston Wake","MI","Michigan State University","Continuing Grant","Tim Hodges","07/31/2029","$85,593.00","","wakepres@msu.edu","426 AUDITORIUM RD RM 2","EAST LANSING","MI","488242600","5173555040","MPS","126400","1045","$0.00","Number theory is the study of the most basic mathematical objects, whole numbers. Because whole numbers are so fundamental, number theory has connections with all major areas of mathematics. For instance, consider the problem of finding the whole-number solutions to a given equation. One can consider the shape given by the graph of that equation, or the set of symmetries that the equation has, or the function whose coefficients come from counting the number of solutions over a variety of number systems. The geometric properties of this shape, the algebraic properties of these symmetries, and the analytic properties of this function are all intimately related to the behavior of the equation?s whole-number solutions. Number theorists use techniques from each of these mathematical areas, but also, in the process, uncover surprising connections between the areas whereby discoveries in one area can lead to growth in another. One part of number theory where the connections between geometry, algebra, and analysis are particularly strong is in the field of modular forms. The proposed research focuses on an important and well-known type of relation between different modular forms called congruence and aims to compute the number of forms that are congruent to a given modular form and uncover the number-theoretic significance of this computation. Many of the conjectures that drive this project were found experimentally, through computer calculations. The main educational objective is to contribute to the training of the next generation of theoretical mathematicians in computational and experimental methods. To achieve this, the Principal Investigator (PI) will design software modules for a variety of undergraduate algebra and number theory courses that provide hands-on experience with computation. In addition, the PI will supervise undergraduates in computational research experiments designed to numerically verify conjectures made in the project and to explore new directions.

Congruences between modular forms provide a link between two very different types of objects in number theory: algebraic objects, like Galois representations, and analytic objects, like L-functions. This link has been used as a tool for proving some of the most celebrated results in modern number theory, such as the Main Conjecture of Iwasawa theory. The proposed research pushes the study of congruences in a new, quantitative direction by counting the number of congruences, not just determining when a congruence exists. The central hypothesis is that this quantitative structure of congruences contains finer information about the algebraic and analytic quantities involved than the Main Conjecture and its generalizations (such as the Bloch?Kato conjecture) can provide.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2337451","CAREER: Higgs bundles and Anosov representations","DMS","ALGEBRA,NUMBER THEORY,AND COM, GEOMETRIC ANALYSIS","07/01/2024","02/02/2024","Brian Collier","CA","University of California-Riverside","Continuing Grant","Swatee Naik","06/30/2029","$79,647.00","","brian.collier@ucr.edu","200 UNIVERSTY OFC BUILDING","RIVERSIDE","CA","925210001","9518275535","MPS","126400, 126500","1045","$0.00","This project focuses on the mathematical study of curved surfaces by connecting algebraic objects to them and thereby generalizing the scope of their application. One of the main notions used is that of a surface group representation, a concept which connects surfaces to generalizations of classical geometries such as Euclidean and hyperbolic geometry. The study of surfaces has surprising applications throughout many fields of mathematics and physics. Consequently, the project lies at the intersection of multiple disciplines. In addition to cutting edge mathematical research, the project will promote the progress of science and mathematics through different workshops aimed at graduate students as well as community outreach events. The educational component will also focus on creating an engaging and inclusive place for mathematical interactions for students and early career researchers.

In the past decades, both the theories of Higgs bundles and Anosov dynamics have led to significant advancements in our understanding of the geometry of surface groups. Recent breakthroughs linking these approaches are indirect and mostly involve higher rank generalizations of hyperbolic geometry known as higher rank Teichmuller spaces. The broad aim of this project is to go beyond higher rank Teichmuller spaces by using Higgs bundles to identify subvarieties of surface group representations which generalize the Fuchsian locus in quasi-Fuchsian space. The cornerstone for the approach is the role of Slodowy slices for Higgs bundles. Specifically, the PI aims to establish Anosov properties of surface group representations associated to Slodowy slices in the Higgs bundle moduli space. This approach will significantly extend applications of Higgs bundles to both Anosov representations and (G,X) geometries. It will complete the component count for moduli of surface group representations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2338933","CAREER: Topology, Spectral Geometry, and Arithmetic of Locally Symmetric Spaces","DMS","ALGEBRA,NUMBER THEORY,AND COM","05/01/2024","01/12/2024","Michael Lipnowski","OH","Ohio State University","Continuing Grant","Tim Hodges","04/30/2029","$106,804.00","","michaellipnowski@gmail.com","1960 KENNY RD","COLUMBUS","OH","432101016","6146888735","MPS","126400","1045","$0.00","This project investigates problems in algebraic number theory. Algebraic numbers are the roots of polynomials with integer coefficients. Symmetries inherent to these numbers are fundamental in number theory, the branch of mathematics devoted to the study of integers. Deep conjectures of Langlands predict that the symmetries of algebraic numbers are apparent in very specific geometric spaces, known as locally symmetric spaces, which are associated to large groups of integer matrices. Near every point, locally symmetric spaces are exceptionally symmetric, and their shapes nearby any two points are indistinguishable. However, the large-scale geometry of these spaces is disordered and chaotic owing to one jarring geometric feature: straight lines emanating from the same point in different directions tend to diverge from each other at an exponential rate. Additionally, the latter property makes these number theoretic worlds difficult to chart. This project aims to systematically organize locally symmetric spaces arising from arithmetic in order to distill inherent structure thereon. The blend of probabilistic, geometric, and algorithmic methods underlying this project lends itself well to an outreach program for middle school and high school students which the PI has piloted, designed to foster outside-the-box mathematical thinking.

From several perspectives, this project will probe the topology, geometry, and arithmetic of positive fundamental rank locally symmetric spaces of number theoretic origin. First, it will chart these spaces using an expanding ball algorithm to construct point grids, akin to mapping the world by progressively building a network of cell towers and regularly transmitting signal to detect other towers nearby. Second, it will study the bass notes of hyperbolic manifolds via relationships between spectrum and cycle complexity. Third, it will attempt to overcome the absence of complex analytic structure on the overlying archimedean symmetric space by systematic use of associated p-adic symmetric spaces. Construction of attendant rigid meromorphic cocycles for associated p-arithmetic groups give possible inroads to Hilbert's twelfth problem, regarding explicit class field theory, in new contexts beyond CM number fields and the classical theory of complex multiplication.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2338345","CAREER: Interpolation, stability, and rationality","DMS","ALGEBRA,NUMBER THEORY,AND COM","09/01/2024","01/11/2024","Isabel Vogt","RI","Brown University","Continuing Grant","Tim Hodges","08/31/2029","$113,376.00","","isabel_vogt@brown.edu","1 PROSPECT ST","PROVIDENCE","RI","029129100","4018632777","MPS","126400","1045, 9150, 9251","$0.00","This project develops techniques to understand the geometry of the solutions to systems of polynomial equations. Of particular interest is when the set of solutions are one-dimensional, in which case it is called an algebraic curve. Since polynomials are ubiquitous in science and engineering, such solution sets arise in many different contexts. One important example is the interpolation problem: given a collection of general points, when is there a fixed type of algebraic curve passing through these points? This problem has applications to cryptography and information theory. The research projects will shed light on the possible realizations of an algebraic curve by polynomial equations, as well as new and important cases of the interpolation problem. This will be complemented by educational and outreach activities, including a sequence of workshops designed for early-to-mid career women and nonbinary graduate students in algebraic geometry and mentoring undergraduate research.

More specifically, the research projects are in the following three directions. The first focuses on the natural stratification of the space of vector bundles on a curve equipped with a fixed dominant map to another curve by the stability of the pushforward. The second focuses on the interpolation problem in other settings, including when the ambient variety is a homogeneous space. The last focuses on the rationality problem for varieties over nonclosed ground fields, and in particular upon generalizations of the intermediate Jacobian torsor obstruction.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2347097","Collaborative Research: Conference: Texas-Oklahoma Representations and Automorphic forms (TORA)","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","12/19/2023","Melissa Emory","OK","Oklahoma State University","Standard Grant","Andrew Pollington","12/31/2026","$20,000.00","Maria Fox, Mahdi Asgari","melissa.emory@okstate.edu","401 WHITEHURST HALL","STILLWATER","OK","740781031","4057449995","MPS","126400","7556, 9150","$0.00","This award supports the TORA mathematics conference series. This series consists of annual meetings hosted by the University of North Texas, Oklahoma State University, and the University of Oklahoma on a rotating basis. This award provides support for three weekend conferences, one at the University of North Texas in Spring 2024 (TORA XIII), one at Oklahoma State University in Spring 2025 (TORA XIV), and another at the University of Oklahoma in Spring 2026 (TORA XV). Each conference will feature three prominent guest speakers from outside the Texas-Oklahoma region, in addition to other participants including students, post-doctoral researchers, and junior faculty. Regional graduate students and researchers will also give talks describing their work. These conferences will facilitate collaborations and interactions among the students and researchers in the region who work in the areas of Automorphic Forms, Representation Theory, and Number Theory.

Over the last century, the theories of automorphic forms and representations have grown enormously. Important applications impact various fields of research, ranging from number theory, coding theory, algebraic geometry, and topology to Kac-Moody algebras and quantum field theory. The interplay of automorphic forms and representation theory has been especially fruitful, and many surprising and deep results have emerged. The TORA conference series will emphasize the interplay between automorphic forms and representations, both in the classical and adelic languages, and related topics like analytic number theory and harmonic analysis.



The conference Texas-Oklahoma Representations and Automorphic forms XIII will take place on April 12-14, 2024, at the University of North Texas. Additional information can be found on the conference website: https://www.math.unt.edu/~richter/TORA/TORA13.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2346615","Conference: Zassenhaus Groups and Friends Conference 2024","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","11/08/2023","Yong Yang","TX","Texas State University - San Marcos","Standard Grant","Tim Hodges","12/31/2024","$18,000.00","Thomas Keller","yy10@txstate.edu","601 UNIVERSITY DR","SAN MARCOS","TX","786664684","5122452314","MPS","126400","7556","$0.00","This award supports participation in the 2024 Zassenhaus Groups and Friends Conference which will be held at Texas State University in San Marcos, TX. It will take place on the campus of the university from noon of Friday, May 31, 2024, to the early afternoon on Sunday, June 2, 2024. It is expected that about 40 researchers will attend the conference, many of whom will give a talk.

The Zassenhaus Groups and Friends Conference, formerly known as Zassenhaus Group Theory Conference, is a series of yearly conferences that has served the mathematical community since its inception in the 1960s. The speakers are expected to come from all over the country and will cover a broad spectrum of topics related to the study of groups, such as representations of solvable groups, representations of simple groups, character theory, classes of groups, groups and combinatorics, recognizing simple groups from group invariants, p-groups, and fusion systems.

The conference will provide group theory researchers in the US a forum to disseminate their own research as well as to learn about new and significant results in the area. The conference will provide a particularly inviting environment to young mathematicians and will inspire future cooperation and collaborations among the participants. It is expected that it will have great impacts on the group theory research community. The organizers will make great effort to attract a demographically diverse group of participants including women and racial and ethnic minorities. More information can be found at the conference website, https://zassenhausgroupsandfriends.wp.txstate.edu/

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2401104","Combinatorics of Complex Curves and Surfaces","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/15/2024","01/24/2024","Philip Engel","IL","University of Illinois at Chicago","Standard Grant","James Matthew Douglass","07/31/2025","$67,057.00","","pengel@uic.edu","809 S MARSHFIELD AVE M/C 551","CHICAGO","IL","606124305","3129962862","MPS","126400","","$0.00","Two-dimensional tilings lie at a fulcrum connecting many areas of mathematics and physics. Easy to visualize and appealing in their simplicity, tilings have fascinated mathematicians at all levels, artists, architects, and the general public. This goals of this project are (1) to study tilings in the context of recent mathematical developments about algebraic curves and surfaces, exploring their connections to algebra, geometry, and representation theory, (2) to disseminate mathematical ideas to a wide audience and increase aesthetic and intellectual appreciation of mathematics in the general public, and (3) to develop an active and diverse community of young researchers, postdocs, and PhD students focusing on this circle of ideas.

One primary area of research will be modular toroidal compactifications of spaces of K3 surfaces. This project, joint with V. Alexeev, seeks to build extensions of the universal family of polarized K3 surfaces to the boundary of a toroidal compactification, extending previous work on degree 2 and elliptic K3 surfaces. The approach employs tilings of integral-affine structures on the sphere. The second primary research topic is moduli spaces of higher differentials. This project aims to study strata of higher differentials, their volumes, and the connection with enumeration of tilings. Joint work with P. Smillie explores decompositions of flat surfaces into Penrose-like tiles. The approach is novel, requiring a generalization of Hurwitz theory to one complex-dimensional leaf spaces.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2347096","Collaborative Research: Conference: Texas-Oklahoma Representations and Automorphic forms (TORA)","DMS","ALGEBRA,NUMBER THEORY,AND COM","01/01/2024","12/19/2023","Lea Beneish","TX","University of North Texas","Standard Grant","Andrew Pollington","12/31/2026","$20,000.00","Anne Shepler, Olav Richter","lea.beneish@unt.edu","1112 DALLAS DR STE 4000","DENTON","TX","762051132","9405653940","MPS","126400","7556","$0.00","This award supports the TORA mathematics conference series. This series consists of annual meetings hosted by the University of North Texas, Oklahoma State University, and the University of Oklahoma on a rotating basis. This award provides support for three weekend conferences, one at the University of North Texas in Spring 2024 (TORA XIII), one at Oklahoma State University in Spring 2025 (TORA XIV), and another at the University of Oklahoma in Spring 2026 (TORA XV). Each conference will feature three prominent guest speakers from outside the Texas-Oklahoma region, in addition to other participants including students, post-doctoral researchers, and junior faculty. Regional graduate students and researchers will also give talks describing their work. These conferences will facilitate collaborations and interactions among the students and researchers in the region who work in the areas of Automorphic Forms, Representation Theory, and Number Theory.

Over the last century, the theories of automorphic forms and representations have grown enormously. Important applications impact various fields of research, ranging from number theory, coding theory, algebraic geometry, and topology to Kac-Moody algebras and quantum field theory. The interplay of automorphic forms and representation theory has been especially fruitful, and many surprising and deep results have emerged. The TORA conference series will emphasize the interplay between automorphic forms and representations, both in the classical and adelic languages, and related topics like analytic number theory and harmonic analysis.



The conference Texas-Oklahoma Representations and Automorphic forms XIII will take place on April 12-14, 2024, at the University of North Texas. Additional information can be found on the conference website: https://www.math.unt.edu/~richter/TORA/TORA13.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." diff --git a/Analysis/Awards-Analysis-2024.csv b/Analysis/Awards-Analysis-2024.csv index 9cdd6d6..90bcef3 100644 --- a/Analysis/Awards-Analysis-2024.csv +++ b/Analysis/Awards-Analysis-2024.csv @@ -1,38 +1,40 @@ "AwardNumber","Title","NSFOrganization","Program(s)","StartDate","LastAmendmentDate","PrincipalInvestigator","State","Organization","AwardInstrument","ProgramManager","EndDate","AwardedAmountToDate","Co-PIName(s)","PIEmailAddress","OrganizationStreet","OrganizationCity","OrganizationState","OrganizationZip","OrganizationPhone","NSFDirectorate","ProgramElementCode(s)","ProgramReferenceCode(s)","ARRAAmount","Abstract" +"2350079","RUI: Nonuniformly Hyperbolic and Extended Dynamical Systems","DMS","ANALYSIS PROGRAM","09/01/2024","04/08/2024","Mark Demers","CT","Fairfield University","Standard Grant","Jan Cameron","08/31/2027","$242,456.00","","mdemers@fairfield.edu","1073 N BENSON RD","FAIRFIELD","CT","068245171","2032544000","MPS","128100","9229","$0.00","The PI will investigate the properties of chaotic dynamical systems that are out of equilibrium due to the influence of either external forces or interconnected components. Research in dynamical systems is often focused on closed systems in which the dynamics are self-contained. In many modeling situations, however, such a global view is not possible, and it becomes necessary to study local systems influenced by external dynamics, possibly on different spatial or temporal scales. To better understand these phenomena, the PI will study open systems in which mass or energy may enter or exit through deterministic or random mechanisms, as well as large-scale systems of smaller interacting components that exchange mass or energy. These problems are strongly motivated by connections with statistical mechanics and seek to advance our understanding of fundamental questions related to energy transport and diffusion. This award will also support the involvement of undergraduates in mathematics research. The highly visual nature and physical motivation of the problems will enable the investigator to recruit undergraduate students to participate in related research projects. Special emphasis will be given to recruiting students from underrepresented groups in research mathematics. Students will disseminate results of their research via poster sessions, conference presentations and publications in peer-reviewed journals. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, this award will contribute to the important goal of integrating research and education.

The funded research includes three specific projects. The first project investigates the statistical and thermodynamic properties of both classical and non-equilibrium particle systems with collision interactions, an important class of models from statistical mechanics. The second concerns open systems, which relate on the one hand to physical systems in which mass or energy is allowed to escape, and on the other to the study of metastable states. The third project generalizes open systems to include linked and extended dynamical systems comprised of two or more components that exchange mass or energy through deterministic or random mechanisms. Important examples include the aperiodic Lorentz gas and mechanical models of heat conduction. The investigator will bring to bear several analytical techniques that he has been instrumental in developing for these classes of systems, including his recent work concerning the spectral decomposition of transfer operators for dispersing particle systems, contractions in projective cones due to Birkhoff, and the construction of Markov extensions adapted to open systems. None of these techniques require Markovian assumptions on the dynamics, making them widely applicable to a wide variety of nonuniformly hyperbolic and physically important systems. The application of these techniques to central models from equilibrium and non-equilibrium statistical mechanics will represent significant advances in the study of such systems. Efforts to understand these tools in one context strengthens them and aids in their application to other areas of mathematics. Their intellectual interest is enhanced by the application of these ideas to resolve problems posed and approached formally in the physics literature.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348384","Geometric Harmonic Analysis: Advances in Radon-like Transforms and Related Topics","DMS","ANALYSIS PROGRAM","07/01/2024","04/05/2024","Philip Gressman","PA","University of Pennsylvania","Standard Grant","Wing Suet Li","06/30/2027","$239,068.00","","gressman@math.upenn.edu","3451 WALNUT ST STE 440A","PHILADELPHIA","PA","191046205","2158987293","MPS","128100","","$0.00","The mathematics of geometric averages known as Radon-like operators is of fundamental importance in a host of technological applications related to imaging and data analysis: CT, SPECT, and NMR, as well as RADAR and SONAR applications, all depend on a deep understanding of the Radon transform, and related ideas appear in optical-acoustic tomography, scattering theory, and even some motion-detection algorithms. Somewhat surprisingly, there are many basic theoretical problems in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. This project studies a family of questions in the area of geometric averages which, for example, correspond to quantifying the relationship between small changes in the imaged objects and the expected changes in measured data (which in practice would be processed computationally to recover an approximate picture of the original object). The theoretical challenge in a problem such as this is to precisely quantify the notion of change and to establish essentially exact relationships between the magnitude of input and output changes. Thanks to recent advances in the PI's work to understand these objects, the project is well-positioned to yield important results. Achieving the main goals of this project would lead to advances in a number of related areas of mathematics and may influence future imaging technologies. The project furthermore provides unique opportunities for the advanced mathematical training of both undergraduate and PhD students, who can transfer these skills to other areas of critical need once in the workforce.

The PI studies topics in mathematical analysis related to the development of new geometric approaches to Radon-like transforms, oscillatory integrals, and Fourier restriction problems. This work includes various special cases of both sublevel set and oscillatory integral problems. Major special cases deserving mention include multiparameter sublevel set estimates, maximal curvature for Radon-like transforms of intermediate dimension, degenerate Radon transforms in low codimension, Fourier restriction and related generalized determinant functionals, and multilinear oscillatory integrals of convolution and related types. The PI's approach to these involves a variety of new tools developed within the last 5 years which incorporate techniques from Geometric Invariant Theory, geometric measure theory, decoupling theory, and other areas. Among these new tools is a recent result of the PI which provides an entirely new way to estimate norms of Radon-Brascamp-Lieb inequalities in terms of geometric quantities which can be understood as analogous to Lieb's formula for the Brascamp-Lieb constant. A major goal of this project is to understand the local geometric criteria which implicitly govern the finiteness of the nonlocal integrals appearing in the Radon-Brascamp-Lieb condition. The project has numerous potential applications to other problems of interest at the intersection of harmonic analysis, geometric measure theory, and incidence geometry.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2400191","Group Actions, Rigidity, and Invariant Measures","DMS","ANALYSIS PROGRAM","06/01/2024","04/05/2024","Aaron Brown","IL","Northwestern University","Standard Grant","Jan Cameron","05/31/2027","$353,236.00","","awb@northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","128100","","$0.00","This project focuses on questions at the interface of dynamical systems and rigidity of group actions. Many mathematical objects admit large groups of symmetries. The structure of such groups may highly constrain the underlying object or properties of the action. Questions across fields of mathematics can often be reformulated as questions about the (non-)fractal nature of invariant geometric structures (particularly sets and measures) for certain group actions. The project will employ tools from the field of dynamical systems to study group actions, with broad aims of classifying actions and the objects on which groups act, classifying certain invariant geometric structures, and showing certain actions do not admit fractal invariant structures. The project will also support the training of PhD students.

The project will focus on actions of groups, including higher-rank abelian groups and higher-rank lattices, with an emphasis on classifying actions with certain dynamical properties, classifying the underlying spaces on the group acts, or classifying invariant measures and orbit closures. The project will employ tools from hyperbolic dynamical systems (dynamical systems with positive Lyapunov exponents) with a common theme of studying invariant measures for the action (or certain subgroups). Classifying or ruling out fractal properties of certain invariant measures will produce further rigidity properties of the action including additional invariance of the measure, local homogeneous structures for the action, or dimension constraints on the space.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2348996","Variable Coefficient Fourier Analysis","DMS","ANALYSIS PROGRAM","07/01/2024","04/05/2024","Christopher Sogge","MD","Johns Hopkins University","Continuing Grant","Wing Suet Li","06/30/2027","$129,987.00","","sogge@jhu.edu","3400 N CHARLES ST","BALTIMORE","MD","212182608","4439971898","MPS","128100","","$0.00","The PI will study several problems in Geometric Harmonic Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated with a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits, as well as the sound volume. Similar phenomena arise for manifolds, and the PI will study precisely how their shapes, such as how they are curved, affect the properties properties of eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI will study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. This project provides research training opportunities for graduate students.

Among the specific problems the PI shall study, they wish to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, they will develop what is called ``global harmonic analysis??, which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates are Lp-estimates for eigenfunctions and quasimodes and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much-studied (but still not well-understood) questions of concentration, oscillation, and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation, and resolvent estimates coming from the metric Laplacian. High frequency solutions and obtaining sharp results under geometric assumptions are particularly interesting. They will also study functions that saturate the estimates in different ways depending on the sign of the sectional curvatures of the manifolds.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2409251","Conference: Maryland Dynamics Conference","DMS","ANALYSIS PROGRAM","04/15/2024","04/04/2024","Adam Kanigowski","MD","University of Maryland, College Park","Standard Grant","Jeremy Tyson","03/31/2027","$49,800.00","Giovanni Forni, Rodrigo Trevino, Bassam Fayad","adkanigowski@gmail.com","3112 LEE BUILDING","COLLEGE PARK","MD","207425100","3014056269","MPS","128100","7556","$0.00","This award provides funding for three years for an annual workshop, to be held in the spring, on dynamical systems and related topics. The workshop will take place on the campus of the University of Maryland at College Park. The event provides a forum for both early career and established researchers to exchange ideas with each other and with their counterparts from around the world. Conference proceedings will be produced at the conclusion of each workshop; these publications will help early career mathematicians to gain familiarity with the presented material. Funding from the award will be prioritized for the reimbursement of travel expenses incurred by junior participants and participants without access to other sources of support.

The goals of this workshop are to promote the dissemination of mathematical results; to facilitate interaction and research progress in dynamical systems and related fields; to nurture the sense of community and common mission in these fields; to promote the participation and visibility of women and under-represented groups in the field; and to contribute to the training of graduate students and recent Ph.D. recipients and to their integration into the dynamics community. Talks at the conference come from widely varying areas of dynamical systems, as well as related areas such as analysis, geometry, and topology. At the same time, each instance of the conference incorporates a particular thematic focus within the overall field of dynamical systems. Special effort will be taken to promote the involvement of early career researchers and individuals from groups under-represented in mathematics research. For instance, graduate students and postdocs in attendance at the conference will be invited to contribute to the creation of a post-conference booklet based on notes of the lectures, which will be made available on the conference?s website. (https://www-math.umd.edu/dynamics-conference.html)

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2402022","Conference: Dynamical Systems and Fractal Geometry","DMS","ANALYSIS PROGRAM","04/15/2024","04/03/2024","Pieter Allaart","TX","University of North Texas","Standard Grant","Jan Cameron","03/31/2025","$32,017.00","Kiko Kawamura, Kirill Lazebnik","allaart@unt.edu","1112 DALLAS DR STE 4000","DENTON","TX","762051132","9405653940","MPS","128100","7556","$0.00","This award provides support for participants to attend the conference ?Dynamical Systems and Fractal Geometry? to be held at the University of North Texas from May 14-17, 2024. The primary goal of the conference is to foster interaction and collaboration between researchers in several fields of mathematics: fractal geometry, complex dynamics, thermodynamic formalism, random dynamical systems, and open dynamical systems. These fields are interrelated through both the methods used and in the fundamental questions of their study. The conference will bring together mathematicians from these fields ranging from senior experts to graduate students; experts will give standard 45?50-minute plenary lectures, and students will have the opportunity to give 5-10 minute ?lightning talks?. The conference will also include a career panel. More information on the conference, including a list of speakers, can be found on the conference website: https://pcallaart3.wixsite.com/conference.

The fields represented in this conference have broad motivations and applications in several classical areas of mathematics and physics beyond dynamical systems and geometry, including number theory, probability theory, and statistical mechanics. Thermodynamic formalism is a framework for unifying many aspects of these fields, and its investigation triggers research and collaboration on the problem of the existence and uniqueness of equilibrium states of the various systems studied in these fields. Limit sets of conformal dynamical systems, and in particular Julia sets arising in complex dynamics, are typically of a fractal nature and understanding their fine fractal properties such as Hausdorff, packing, Assouad and Fourier dimensions provides a true challenge for fractal geometers. The conference aims to advance research in these directions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2400332","NSF-BSF: C*-algebras and Dynamics Beyond the Elliott Program","DMS","ANALYSIS PROGRAM","08/01/2024","04/08/2024","Norman Phillips","OR","University of Oregon Eugene","Standard Grant","Jan Cameron","07/31/2027","$343,286.00","","ncp@uoregon.edu","1776 E 13TH AVE","EUGENE","OR","974031905","5413465131","MPS","128100","","$0.00","A C*-algebra is a kind of mathematical object which, for example, appears in quantum mechanics. Simple C*-algebras are those that cannot be broken apart into smaller (""simpler"") C*-algebras. The largest part of this project is about when a simple C*-algebra is isomorphic to its opposite algebra, that is, mathematically the same as what might be thought of as its mirror image. For an example from everyday life, an ordinary sock is the same as its mirror image, since a sock which fits on a right foot will fit equally well on the left foot. A glove isn't like that: whatever one does, a right glove will not fit on a left hand. A nonsimple C*-algebra can be made of very elementary parts, but put together in a tricky way, so as to not be isomorphic to its opposite. Simple C*-algebras which are not separable or not nuclear (""too large"", but in different senses) can also fail to be isomorphic to their opposites. On the other hand, simple C*-algebras covered by the Elliott classification program are isomorphic to their opposites. A long-term goal of the project is to exhibit a simple separable nuclear C*-algebra which is not isomorphic to its opposite. Such an algebra could not be covered even by any proposed expansion of the Elliott program. The project will also contribute to US workforce development through the training of graduate and undergraduate students.

The intended example is a simple unital AH algebra with fast dimension growth. The intended proof that it is not isomorphic to its opposite depends on nonexistence theorems for certain homomorphisms from one matrix algebra over the algebra of continuous functions on a compact space to a different matrix algebra over the continuous functions on a different compact space. When the second matrix size is large enough, all homomorphisms not ruled out for fairly obvious reasons actually exist. When it is small, known obstructions rule out most homomorphisms. The application requires information about an intermediate range. Here, even the simplest case, asked by Blackadar over 30 years ago, remains open; understanding this case is a necessary preliminary step. This case can almost certainly be settled by computations in rational homotopy theory, a new use of algebraic topology in C*-algebras.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2401019","Conference: Supplementary funding for the BIRS-CMO workshop Optimal Transport and Dynamics (24s5198)","DMS","ANALYSIS PROGRAM","04/15/2024","04/05/2024","Jun Kitagawa","MI","Michigan State University","Standard Grant","Jan Cameron","03/31/2025","$14,420.00","","jun@msu.edu","426 AUDITORIUM RD RM 2","EAST LANSING","MI","488242600","5173555040","MPS","128100","7556","$0.00","The funds from this award will support local expenses for additional participants from US institutions to the Banff International Research Station-Casa Matemática Oaxaca workshop 24w5198, ?Optimal Transport and Dynamics? which will be held August 11 to August 16, 2024, in Oaxaca, Mexico. This workshop will focus on applications of the optimal transport problem, a mathematical problem where the goal is to minimize the total cost of transporting mass from one location to another, to problems involving physical processes that change with time. Such processes include interface motion (such as how water spreads on a surface), models for tumor growth, modeling fluid flows, multi-species population dynamics, and reconstruction of the state of the early universe. The workshop will provide a unique opportunity for early-career researchers to develop connections with and be exposed to the cutting-edge research of well-established leaders in the field. Additionally, the workshop will establish connections between mathematicians and cosmologists, to further accelerate development of the tools and theory behind computation in early universe reconstruction. More information on the workshop may be found at https://www.birs.ca/events/2024/5-day-workshops/24w5198.


The workshop will bring together experts working in optimal transport (Monge-Kantorovich) theory with connections to dynamics interpreted in a broad sense. This includes using optimal transport and related tools to analyze and model fluid flows, interface motion in evolutionary PDE, and also the use of dynamical techniques such as the theory of the parabolic Monge-Ampčre PDE for computational and theoretical analysis of optimal transport itself. Optimal transport theory has also been used as a computational model for early universe reconstruction that is consistent with the Zel?dovich approximation, by cosmologists with great success. With recent developments in cosmological surveying and the availability of new data, this area is currently experiencing a revival and is a particularly timely topic. The workshop will consist of a combination of short and long talks solicited from participants, with priority given to presentations by early-career researchers (i.e., graduate students, postdoctoral researchers, and pre-tenure faculty). To take advantage of the international diversity present in the participant list, there will also be a panel discussion on differences in academic job search procedures in different countries.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350530","Analysis and Geometry of Conformal and Quasiconformal Mappings","DMS","ANALYSIS PROGRAM","06/01/2024","04/02/2024","Malik Younsi","HI","University of Hawaii","Standard Grant","Jeremy Tyson","05/31/2027","$211,262.00","","malik.younsi@gmail.com","2425 CAMPUS RD SINCLAIR RM 1","HONOLULU","HI","968222247","8089567800","MPS","128100","9150","$0.00","This project aims to better understand the analytic and geometric properties of conformal and quasiconformal mappings. Conformal mappings are planar transformations which locally preserve angles. An important example is the Mercator projection in cartography, used to project the surface of the Earth to a two-dimensional map. More recently, much attention has been devoted to the study of quasiconformal mappings, a generalization of conformal mappings where a controlled amount of angle distortion is permitted. Because of this additional flexibility, quasiconformal mappings have proven over the years to be of fundamental importance in a wide variety of areas of mathematics and applications. Many of these applications involve planar transformations that are quasiconformal inside a given region except possibly for some exceptional set of points inside the region. The study of this exceptional set leads to the notion of removability, central to this research project and closely related to fundamental questions in complex analysis, dynamical systems, probability and related areas. Another focus of this project is on the study of certain families of quasiconformal mappings called holomorphic motions. The principal investigator will study how quantities such as dimension and area change under holomorphic motions, leading to a better understanding of the geometric properties of quasiconformal mappings. The project also provides opportunities for the training and mentoring of early career researchers, including graduate students. In addition, the principal investigator will continue to be involved in a science and mathematics outreach program for local high school students.

Two strands of research comprise the planned work. The first component involves the study of conformal removability. Motivated by the long-standing Koebe uniformization conjecture, the principal investigator will investigate the relationship between removability and the rigidity of circle domains. This part of the project also involves the study of conformal welding, a correspondence between planar Jordan curves and functions on the circle. Recent years have witnessed a renewal of interest in conformal welding along with new generalizations and variants, notably in the theory of random surfaces and in connection with applications to computer vision and numerical pattern recognition. The second component of the project concerns holomorphic motions. The principal investigator will study the variation of several notions of dimension under holomorphic motions. A new approach to this topic by the principal investigator and his collaborators using inf-harmonic functions has already yielded a unified treatment of several celebrated theorems about quasiconformal mappings, and many more fruitful connections are anticipated as progress continues to be made towards a better understanding of holomorphic motions. This part of the project also involves the relationship between global quasiconformal dimension and conformal dimension.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350356","Dynamics of Nonlinear and Disordered Systems","DMS","ANALYSIS PROGRAM","06/01/2024","04/02/2024","Wilhelm Schlag","CT","Yale University","Continuing Grant","Marian Bocea","05/31/2027","$149,265.00","","wilhelm.schlag@yale.edu","150 MUNSON ST","NEW HAVEN","CT","065113572","2037854689","MPS","128100","","$0.00","Observations of solitary waves that maintain their shape and velocity during their propagation were recorded around 200 years ago. First by Bidone in Turin in 1826, and then famously by Russell in 1834 who followed a hump of water moving at constant speed along a channel for several miles. Today these objects are known as solitons. Lying at the intersection of mathematics and physics, they have been studied rigorously since the 1960s. For completely integrable wave equations, many properties of solitons are known, such as their elastic collisions, their stability properties, as well as their role as building blocks in the long-time description of waves. The latter is particularly important, as it for example predicts how waves carrying information decompose into quantifiable units. In quantum physics, quantum chemistry, and material science, these mathematical tools allow for a better understanding of the movement of electrons in various media. This project aims to develop the mathematical foundations which support these areas in applied science, which are of great importance to industry and society at large. The project provides research training opportunities for graduate students.

The project?s goal is to establish both new results and new techniques in nonlinear evolution partial differential equations on the one hand, and the spectral theory of disordered systems on the other hand. The long-range scattering theory developed by Luhrmann and the Principal Investigator (PI) achieved the first results on potentials which exhibit a threshold resonance in the context of topological solitons. This work is motivated by the fundamental question about asymptotic kink stability for the phi-4 model. Asymptotic stability of Ginzburg-Landau vortices in their own equivariance class is not understood. The linearized problem involves a non-selfadjoint matrix operator, and the PI has begun to work on its spectral theory. With collaborators, the PI will engage on research on bubbling for the harmonic map heat flow and attempt to combine the recent paper on continuous-in-time bubbling with a suitable modulation theory. The third area relevant to this project is the spectral theory of disordered systems. More specifically, the PI will continue his work on quasiperiodic symplectic cocycles which arise in several models in condensed matter physics such as in graphene and on non-perturbative methods to analyze them.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348908","Low Regularity and Long Time Dynamics in Nonlinear Dispersive Flows","DMS","ANALYSIS PROGRAM","08/01/2024","04/02/2024","Mihaela Ifrim","WI","University of Wisconsin-Madison","Standard Grant","Marian Bocea","07/31/2027","$343,401.00","","ifrim@wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","128100","","$0.00","The primary objective of this project is to examine solutions to a broad class of equations that can be described as nonlinear waves. These mathematical equations model a wide range of physical phenomena arising in fluid dynamics (oceanography), quantum mechanics, plasma physics, nonlinear optics, and general relativity. The equations being studied range from semilinear to fully nonlinear, and from local to nonlocal equations, and we aim to investigate them in an optimal fashion both locally and globally in time. This research develops and connects ideas and methods in partial differential equations, and in some cases also draws a clear path towards other problems in fields such as geometry, harmonic analysis, complex analysis, and microlocal analysis. The project provides research training opportunities for graduate students.

The strength of the nonlinear wave interactions is the common feature in the models considered in this proposal, and it significantly impacts both their short-time and their long-time behavior. The project addresses a series of very interesting questions concerning several classes of nonlinear dispersive equations: (i) short-time existence theory in a low regularity setting; (ii) breakdown of waves, and here a particular class of equations is provided by the water wave models; and (iii) long-time persistence and/or dispersion and decay of waves, which would involve either a qualitative aspect attached to it, that is, an asymptotic description of the nonlinear solution, or a quantitative description of it, for instance nontraditional scattering statements providing global in time dispersive bounds. All of this also depends strongly on the initial data properties, such as size, regularity and localization.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2347850","Structure theory for measure-preserving systems, additive combinatorics, and correlations of multiplicative functions","DMS","ALGEBRA,NUMBER THEORY,AND COM, ANALYSIS PROGRAM","07/01/2024","04/02/2024","Terence Tao","CA","University of California-Los Angeles","Continuing Grant","Wing Suet Li","06/30/2027","$244,222.00","","tao@math.ucla.edu","10889 WILSHIRE BLVD STE 700","LOS ANGELES","CA","900244200","3107940102","MPS","126400, 128100","","$0.00","Consider a stream of digital data - a sequence of zeroes and ones. This sequence could be highly structured - for instance, it could alternate periodically between 0 and 1. Or it could be completely random, with the value of each member of the sequence having no relation whatsoever to the next. It could also be ""pseudorandom"" - described by a deterministic algorithm, but yet statistically indistinguishable from a genuinely random sequence. Or it could be some complex mixture of structure and (pseudo)randomness. Can one define precisely what structure and randomness mean and describe arbitrary data as combinations of these two different components? Such questions are of importance in cryptography, computer science, combinatorics, dynamics, and number theory, as they allow one to mathematically determine whether certain patterns in arbitrary streams of data are guaranteed to occur or not. For instance, in 2004, Ben Green and the PI were able to settle a long-standing conjecture in number theory that the prime numbers contained arbitrarily long arithmetic progressions, with the key idea being to break up the prime numbers into structured and random components and study the contribution of each component. In computer science, this theory has led, for instance, to efficient ways to generate pseudorandom bits for several types of applications. In the subsequent twenty years, much progress has been made in quantifying more precisely what structure and randomness mean, particularly in the area of mathematics now known as higher-order Fourier analysis. More understanding has been gained on the precise way in which number-theoretic structures, such as the primes, exhibit (pseudo-)random behavior at both large and small scales. There has been steady progress in this direction in recent years, in which the scale on which one is able to definitively demonstrate various types of pseudorandomness has narrowed over time, and further work will be carried out in this project, in particular, it is tantalizingly near to resolve (a version) of a well-known conjecture in number theory - the Chowla conjecture - which could be in turn a stepping stone to even more famous conjectures such as the twin prime conjecture. This project provides research training opportunities for graduate students.


In this project, the PI (in conjunction with collaborators) plans to work on two related projects. Firstly, the PI will continue recent work on developing general inverse theorems for the Gowers uniformity norms in additive combinatorics on one hand and the Host--Kra uniformity seminorms in ergodic theory on the other. Secondly, the PI will continue building upon recent breakthroughs in the understanding of multiplicative functions, to make further progress towards the (logarithmically averaged) Chowla and Elliott conjectures for such functions, and to apply these results to related problems in analytic number theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400008","A description of surface dynamics","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Enrique Pujals","NY","CUNY Graduate School University Center","Standard Grant","Jeremy Tyson","06/30/2026","$249,103.00","","epujals@gc.cuny.edu","365 5TH AVE STE 8113","NEW YORK","NY","100164309","2128177526","MPS","128100","5913, 5918","$0.00","This project seeks to understand the mechanisms that underlie the transition of a dynamical system from an ordered state to a random (chaotic) state. In other words, the aim is to understand the processes through which a system's behavior evolves from periodicity toward chaos, as one or more governing parameters are varied. A related goal is to identify the primary bifurcation responsible for qualitative changes exhibited by a dynamical system. While such comprehension has previously been attained for low-dimensional dynamical systems, this project introduces a novel approach to transcend the low-dimensional limitation. The project will offer new conceptual ideas and approaches to provide fresh perspectives on advances in mathematics and science. Additionally, the project will facilitate the training of graduate students directly engaged in the research, and will afford educational opportunities to undergraduate students through the organization of a summer school presenting topics in mathematics, including topics related to dynamical systems.

The theory of one-dimensional dynamical systems successfully explains the depth and complexity of chaotic phenomena in concert with a description of the dynamics of typical orbits for typical maps. Its remarkable universality properties supplement this understanding with powerful geometric tools. In the two-dimensional setting, the range of possible dynamical scenarios that can emerge is at present only partially understood, and a general framework for those new phenomena that do not occur for one-dimensional dynamics remains to be developed. In prior work supported by the NSF, the principal investigator introduced a large open class of two-dimensional dynamical systems, including the classical Henon family without the restriction of large area contraction, that is amenable to obtaining results as in the one-dimensional case. Moreover, major progress was reached to understand the transition from zero entropy to positive entropy using renormalization schemes. The present project has several components. First, existing renormalization schemes will be adapted to the positive entropy realm. Next, initial steps towards a characterization of dissipative diffeomorphisms in more general contexts will be addressed. Finally, the principal investigator will seek to develop the theory of differentiable renormalization without an a priori assumption of proximity to the one-dimensional setting. These results will open the door to a global description of dissipative diffeomorphisms and their behavior under perturbation, bringing both new tools and new perspectives to smooth dynamical systems theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2350128","Conference: PDE in Moab: Advances in Theory and Application","DMS","ANALYSIS PROGRAM","04/15/2024","04/05/2024","Mark Allen","UT","Brigham Young University","Standard Grant","Jan Cameron","03/31/2025","$35,875.00","Blair Davey, Mariana Smit Vega Garcia","mkallen2@gmail.com","A-153 ASB","PROVO","UT","846021128","8014223360","MPS","128100","7556","$0.00","The purpose of this award is to fund a research conference on Partial Differential Equations (PDE) to take place on June 3-7, 2024, at the Utah State University (USU) building located in Moab, Utah. The conference, called ""PDE in Moab: Advances in Theory and Application"" will feature 14 invited talks, along with 9 contributed talks from early career mathematicians, with a total of approximately 40 participants. Funding attached to this grant will be used to support travel and lodging expenses for participants in the conference, with priority for junior participants who do not have access to other sources of travel funding. The conference website is https://pdemoab.byu.edu

This conference aims to explore the tools and methods of partial differential equations (PDE), and their applications in related fields such as geometric measure theory (GMT), harmonic analysis, and free boundary problems. Historically, these areas of mathematics have benefited from many fruitful interconnections. Indeed, pioneering advancements in free boundary problems adapted techniques from regularity theory in both PDE and GMT. Moreover, recent advances in both nonlinear and nonlocal PDE have enlarged the intersection of the aforementioned fields, thereby increasing interactions, collaborations, and the overall advancement of these areas. This conference will bring together experts from the areas of PDE, GMT, harmonic analysis, and free boundary problems to explore and build on recent progress. The list of speakers is comprised of a dynamic group of mathematicians specializing in complementary fields, many of whom already have intersecting interests. It is expected that by bringing these researchers together, there will be further interaction between research areas, leading to the cross-pollination of techniques and novel research results.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350481","Complex Analysis and Random Geometry","DMS","ANALYSIS PROGRAM","06/15/2024","04/01/2024","Steffen Rohde","WA","University of Washington","Standard Grant","Jeremy Tyson","05/31/2027","$299,834.00","","rohde@math.washington.edu","4333 BROOKLYN AVE NE","SEATTLE","WA","981951016","2065434043","MPS","128100","5918, 5946, 5955","$0.00","The project explores probabilistic and deterministic aspects of self-similar geometry. Self-similar sets are characterized by the property that they look the same at different scales. Such sets arise in the study of dynamical systems, for instance, in complex dynamics and the study of the Mandelbrot set. On the other hand, in probability theory and statistical physics one often encounters stochastically self-similar sets. Such objects only have the same statistical properties at different scales. There are surprising analogies between the probabilistic theory and its deterministic counterpart. The research supported by this award explores these analogies and addresses foundational questions regarding self-similar objects, using methods from complex analysis. The project also provides opportunities for the training and mentoring of junior researchers, including graduate students and postdoctoral researchers. The PI will contribute to the dissemination of mathematical knowledge through the organization of conferences, workshops, and summer schools.

Research to be conducted under this award involves the geometry of conformally self-similar structures, both in stochastic and deterministic settings. Julia sets for the iteration of complex mappings illustrate the latter setting, while the former includes topics such as Schramm-Loewner evolution. The project aims to answer fundamental regularity questions for conformally self-similar objects, including Jordan curves of finite Loewner energy. A new parametrization of the Teichmueller spaces of punctured spheres will also be studied. Additional motivation for the project arises from the interaction between the deterministic and stochastic frameworks, notably, the transfer of methods and results between these two areas. For instance, the concept of conformal mating of polynomials in complex dynamics bears close similarity to Sheffield's mating of trees construction for random spheres. The PI?s research uses methods developed in complex dynamics to provide analytic constructions for random structures. Conversely, insights from the probabilistic theory translate to new research avenues in complex dynamics. Conformal welding is a tool of central importance in both theories, and the proposal aims to resolve several fundamental questions regarding Weil-Petersson curves, welding of Liouville Quantum Gravity discs, and Werner's conformal restriction measure on Jordan curves.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349846","Parabolic and elliptic boundary value and free boundary problems","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Steven Hofmann","MO","University of Missouri-Columbia","Standard Grant","Wing Suet Li","06/30/2027","$247,227.00","","hofmanns@missouri.edu","121 UNIVERSITY HALL","COLUMBIA","MO","652113020","5738827560","MPS","128100","","$0.00","This project is concerned with the theory of boundary value problems and free boundary problems for elliptic and parabolic partial differential equations. Such equations arise, for example, in the mathematical theory of heat conduction: an equation of elliptic type describes steady state (equilibrium) temperature distributions and a related parabolic equation governs heat conduction in the time-evolutive case. In a boundary value problem, one uses mathematical knowledge of either 1) the temperature distribution on the boundary (i.e., perimeter) of some region in space (or of some evolving region in space-time) or 2) the heat flux (the rate at which heat flows across the boundary), to deduce information about the internal temperature distribution inside the region. In free boundary problems, one uses simultaneous knowledge of both the boundary temperature distribution and the heat flux to deduce information about the geometry of the region and its boundary. A central goal of this project is to understand the interplay between analytic information and geometry. This project provides research training opportunities for graduate students.

The project has three main areas of focus: 1) to find a geometric characterization of the space-time domains for which the Dirichlet (or initial-Dirichlet) problem is solvable for the heat equation with singular (p-integrable) data, and to study related free boundary problems. The PI and coauthors have previously treated such problems in the steady state (elliptic) setting; in the present project, the PI seeks to treat the more difficult time-evolutive case. 2) to solve the Kato square root problem for elliptic equations in non-divergence form. The solution of the Kato problem for divergence form elliptic operators has led to significant progress in the theory of boundary value problems for divergence form equations. As a first step towards opening up the analogous theory in the nondivergence setting, the PI plans to treat the Kato problem for non-divergence elliptic operators. 3) to solve the Dirichlet problem in Lipschitz domains for non-symmetric divergence from elliptic equations with periodic coefficients. A primary motivation for the study of operators with periodic coefficients is their applicability to the theory of homogenization, which in turn provides a mathematical model for materials with periodic microstructure.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2400040","Approximation properties in von Neumann algebras","DMS","ANALYSIS PROGRAM","06/01/2024","03/27/2024","Jesse Peterson","TN","Vanderbilt University","Standard Grant","Wing Suet Li","05/31/2027","$291,569.00","","jesse.d.peterson@vanderbilt.edu","110 21ST AVE S","NASHVILLE","TN","372032416","6153222631","MPS","128100","","$0.00","Von Neumann algebras were introduced in the 1930's and 40's to study representation theory of groups, and to use as a tool for developing a mathematical foundation for quantum physics. They have since developed into a full area of study as a natural noncommutative notion of measure theory. The noncommutative setting of topology (C*-algebras) emerged shortly after, and the two subjects have historically been closely connected. This project explores these connections to develop new ideas, to reach a broad mathematical community and providing engagement and support for new students in the field. The investigator is actively participating in the training of students and postdocs in von Neumann algebras and the research from this project will directly impact these students and postdocs.

The project investigator is studying approximation properties (or the lack thereof) in von Neumann algebras and C*-algebras, especially relating to group von Neumann algebras and group measure space constructions. This has historically been a significant area of study in the classification of operator algebras, with amenability/injectivity playing a major role in the development of von Neumann algebras, and nuclearity playing a major corresponding role in the theory of C*-algebras. The emergence of Popa's deformation/rigidity theory has led to numerous breakthroughs in the classification of von Neumann algebras beyond the amenability setting, and approximation properties, such as Ozawa's notion of a biexact group, have created new opportunities to study approximation properties in the setting of von Neumann algebras. The research developed in this project investigates these approximation properties, creating new connections between C* and von Neumann algebras. This allows new C*-algebraic tools to be used in the setting of von Neumann algebras, leading to new structural results for group and group measure space von Neumann algebras, and giving a deeper insight into interactions between operator algebras, ergodic theory, and geometric group theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2337630","CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis","DMS","GEOMETRIC ANALYSIS, ANALYSIS PROGRAM","06/01/2024","12/28/2023","Yiming Zhao","NY","Syracuse University","Continuing Grant","Eriko Hironaka","05/31/2029","$69,246.00","","yzhao197@syr.edu","900 S CROUSE AVE","SYRACUSE","NY","132444407","3154432807","MPS","126500, 128100","1045","$0.00","Isoperimetric problems and Minkowski problems are two central ingredients in Convex Geometric Analysis. The former compares geometric measurements (such as volume and surface area) while the latter recovers the shape of geometric figures using local versions of these measurements. The two types of problems are inherently connected. This project will exploit this connection to seek answers to either isoperimetric problems or Minkowski problems in various settings when answers to one exist while answers to the other remain elusive. Although these problems originate from a geometric background, their applications extend beyond mathematics into engineering and design, including areas like antenna reflector design and urban planning. The principal investigator will organize a series of events and workshops at local science museums, community centers, and schools, involving high school teachers and students as well as undergraduate and graduate students. These events and workshops aim to expose the fun and exploratory side of the principal investigator?s research and mathematics in general to students early in their educational careers, raise society's awareness and interest in mathematics, and promote mathematics among historically underrepresented populations.

The existence of solutions to the dual Minkowski problem (that characterizes dual curvature measures) in the original symmetric case has been largely settled (by the principal investigator and his collaborators) through techniques from geometry and analysis. This naturally leads to conjectures involving isoperimetric problems connected to the dual Minkowski problem. Such conjectured isoperimetric inequalities are also connected to an intriguing question behind many other conjectures in convexity: how does certain symmetry improve estimates? The principal investigator will also study Minkowski problems and isoperimetric inequalities coming from affine geometry. Special cases of these isoperimetric inequalities are connected to an affine version of the sharp fractional Sobolev inequalities of Almgren-Lieb. The techniques involved in studying these questions are from Convex Geometric Analysis and PDE. In the last few decades (particularly the last two), there has been a community-wide effort to extend results in the theory of convex bodies to their counterparts in the space of log-concave functions. In this project, the principal investigator will also continue his past work to extend dual curvature measures, their Minkowski problems, and associated isoperimetric inequality to the space of log-concave functions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2400191","Group Actions, Rigidity, and Invariant Measures","DMS","ANALYSIS PROGRAM","06/01/2024","04/05/2024","Aaron Brown","IL","Northwestern University","Standard Grant","Jan Cameron","05/31/2027","$353,236.00","","awb@northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","128100","","$0.00","This project focuses on questions at the interface of dynamical systems and rigidity of group actions. Many mathematical objects admit large groups of symmetries. The structure of such groups may highly constrain the underlying object or properties of the action. Questions across fields of mathematics can often be reformulated as questions about the (non-)fractal nature of invariant geometric structures (particularly sets and measures) for certain group actions. The project will employ tools from the field of dynamical systems to study group actions, with broad aims of classifying actions and the objects on which groups act, classifying certain invariant geometric structures, and showing certain actions do not admit fractal invariant structures. The project will also support the training of PhD students.

The project will focus on actions of groups, including higher-rank abelian groups and higher-rank lattices, with an emphasis on classifying actions with certain dynamical properties, classifying the underlying spaces on the group acts, or classifying invariant measures and orbit closures. The project will employ tools from hyperbolic dynamical systems (dynamical systems with positive Lyapunov exponents) with a common theme of studying invariant measures for the action (or certain subgroups). Classifying or ruling out fractal properties of certain invariant measures will produce further rigidity properties of the action including additional invariance of the measure, local homogeneous structures for the action, or dimension constraints on the space.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2348996","Variable Coefficient Fourier Analysis","DMS","ANALYSIS PROGRAM","07/01/2024","04/05/2024","Christopher Sogge","MD","Johns Hopkins University","Continuing Grant","Wing Suet Li","06/30/2027","$129,987.00","","sogge@jhu.edu","3400 N CHARLES ST","BALTIMORE","MD","212182608","4439971898","MPS","128100","","$0.00","The PI will study several problems in Geometric Harmonic Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated with a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits, as well as the sound volume. Similar phenomena arise for manifolds, and the PI will study precisely how their shapes, such as how they are curved, affect the properties properties of eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI will study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. This project provides research training opportunities for graduate students.

Among the specific problems the PI shall study, they wish to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, they will develop what is called ``global harmonic analysis??, which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates are Lp-estimates for eigenfunctions and quasimodes and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much-studied (but still not well-understood) questions of concentration, oscillation, and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation, and resolvent estimates coming from the metric Laplacian. High frequency solutions and obtaining sharp results under geometric assumptions are particularly interesting. They will also study functions that saturate the estimates in different ways depending on the sign of the sectional curvatures of the manifolds.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2409251","Conference: Maryland Dynamics Conference","DMS","ANALYSIS PROGRAM","04/15/2024","04/04/2024","Adam Kanigowski","MD","University of Maryland, College Park","Standard Grant","Jeremy Tyson","03/31/2027","$49,800.00","Bassam Fayad, Rodrigo Trevino, Giovanni Forni","adkanigowski@gmail.com","3112 LEE BUILDING","COLLEGE PARK","MD","207425100","3014056269","MPS","128100","7556","$0.00","This award provides funding for three years for an annual workshop, to be held in the spring, on dynamical systems and related topics. The workshop will take place on the campus of the University of Maryland at College Park. The event provides a forum for both early career and established researchers to exchange ideas with each other and with their counterparts from around the world. Conference proceedings will be produced at the conclusion of each workshop; these publications will help early career mathematicians to gain familiarity with the presented material. Funding from the award will be prioritized for the reimbursement of travel expenses incurred by junior participants and participants without access to other sources of support.

The goals of this workshop are to promote the dissemination of mathematical results; to facilitate interaction and research progress in dynamical systems and related fields; to nurture the sense of community and common mission in these fields; to promote the participation and visibility of women and under-represented groups in the field; and to contribute to the training of graduate students and recent Ph.D. recipients and to their integration into the dynamics community. Talks at the conference come from widely varying areas of dynamical systems, as well as related areas such as analysis, geometry, and topology. At the same time, each instance of the conference incorporates a particular thematic focus within the overall field of dynamical systems. Special effort will be taken to promote the involvement of early career researchers and individuals from groups under-represented in mathematics research. For instance, graduate students and postdocs in attendance at the conference will be invited to contribute to the creation of a post-conference booklet based on notes of the lectures, which will be made available on the conference?s website. (https://www-math.umd.edu/dynamics-conference.html)

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2401019","Conference: Supplementary funding for the BIRS-CMO workshop Optimal Transport and Dynamics (24s5198)","DMS","ANALYSIS PROGRAM","04/15/2024","04/05/2024","Jun Kitagawa","MI","Michigan State University","Standard Grant","Jan Cameron","03/31/2025","$14,420.00","","jun@msu.edu","426 AUDITORIUM RD RM 2","EAST LANSING","MI","488242600","5173555040","MPS","128100","7556","$0.00","The funds from this award will support local expenses for additional participants from US institutions to the Banff International Research Station-Casa Matemática Oaxaca workshop 24w5198, ?Optimal Transport and Dynamics? which will be held August 11 to August 16, 2024, in Oaxaca, Mexico. This workshop will focus on applications of the optimal transport problem, a mathematical problem where the goal is to minimize the total cost of transporting mass from one location to another, to problems involving physical processes that change with time. Such processes include interface motion (such as how water spreads on a surface), models for tumor growth, modeling fluid flows, multi-species population dynamics, and reconstruction of the state of the early universe. The workshop will provide a unique opportunity for early-career researchers to develop connections with and be exposed to the cutting-edge research of well-established leaders in the field. Additionally, the workshop will establish connections between mathematicians and cosmologists, to further accelerate development of the tools and theory behind computation in early universe reconstruction. More information on the workshop may be found at https://www.birs.ca/events/2024/5-day-workshops/24w5198.


The workshop will bring together experts working in optimal transport (Monge-Kantorovich) theory with connections to dynamics interpreted in a broad sense. This includes using optimal transport and related tools to analyze and model fluid flows, interface motion in evolutionary PDE, and also the use of dynamical techniques such as the theory of the parabolic Monge-Ampčre PDE for computational and theoretical analysis of optimal transport itself. Optimal transport theory has also been used as a computational model for early universe reconstruction that is consistent with the Zel?dovich approximation, by cosmologists with great success. With recent developments in cosmological surveying and the availability of new data, this area is currently experiencing a revival and is a particularly timely topic. The workshop will consist of a combination of short and long talks solicited from participants, with priority given to presentations by early-career researchers (i.e., graduate students, postdoctoral researchers, and pre-tenure faculty). To take advantage of the international diversity present in the participant list, there will also be a panel discussion on differences in academic job search procedures in different countries.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2350128","Conference: PDE in Moab: Advances in Theory and Application","DMS","ANALYSIS PROGRAM","04/15/2024","04/05/2024","Mark Allen","UT","Brigham Young University","Standard Grant","Jan Cameron","03/31/2025","$35,875.00","Blair Davey, Mariana Smit Vega Garcia","mkallen2@gmail.com","A-153 ASB","PROVO","UT","846021128","8014223360","MPS","128100","7556","$0.00","The purpose of this award is to fund a research conference on Partial Differential Equations (PDE) to take place on June 3-7, 2024, at the Utah State University (USU) building located in Moab, Utah. The conference, called ""PDE in Moab: Advances in Theory and Application"" will feature 14 invited talks, along with 9 contributed talks from early career mathematicians, with a total of approximately 40 participants. Funding attached to this grant will be used to support travel and lodging expenses for participants in the conference, with priority for junior participants who do not have access to other sources of travel funding. The conference website is https://pdemoab.byu.edu

This conference aims to explore the tools and methods of partial differential equations (PDE), and their applications in related fields such as geometric measure theory (GMT), harmonic analysis, and free boundary problems. Historically, these areas of mathematics have benefited from many fruitful interconnections. Indeed, pioneering advancements in free boundary problems adapted techniques from regularity theory in both PDE and GMT. Moreover, recent advances in both nonlinear and nonlocal PDE have enlarged the intersection of the aforementioned fields, thereby increasing interactions, collaborations, and the overall advancement of these areas. This conference will bring together experts from the areas of PDE, GMT, harmonic analysis, and free boundary problems to explore and build on recent progress. The list of speakers is comprised of a dynamic group of mathematicians specializing in complementary fields, many of whom already have intersecting interests. It is expected that by bringing these researchers together, there will be further interaction between research areas, leading to the cross-pollination of techniques and novel research results.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350129","Problems in Regularity Theory of Partial Differential Equations","DMS","ANALYSIS PROGRAM, EPSCoR Co-Funding","07/01/2024","04/02/2024","Hongjie Dong","RI","Brown University","Standard Grant","Marian Bocea","06/30/2027","$351,235.00","","Hongjie_Dong@brown.edu","1 PROSPECT ST","PROVIDENCE","RI","029129100","4018632777","MPS","128100, 915000","9150","$0.00","This project focuses on understanding certain types of partial differential equations (PDE) commonly encountered in physics and engineering, such as those governing elasticity and conductivity. When we study how materials deform under stress or conduct electricity, we often use equations to describe these phenomena. However, some equations don't behave smoothly, especially when dealing with high contrast materials or complex shapes. These situations can lead to equations that are much harder to analyze, and traditional methods may not work. Another area of study is equations from fluid dynamics. Understanding these questions is crucial for practical applications like designing airplanes or predicting weather patterns, and it also inspires new ideas in mathematics and statistics. Finally, the Principal Investigator (PI) is interested in kinetic equations, which describe how particles move and interact in systems like nuclear fusion experiments. By studying these equations, scientists hope to improve our understanding of how plasmas behave in extreme conditions, such as inside a tokamak. The project provides research training opportunities for graduate students.

As part of this project, the PI will carry out research closely related to the aforementioned topics and will attempt to address some of the open problems in these areas. The focus will be on several projects that can be gathered into three main topical areas. First, the project will develop new methods to study elliptic equations arising in composite materials (e.g., elasticity problems, conductivity problems). The PI is particularly interested in the blowup behaviors of solutions to PDE in domains with Lipschitz inclusions, equations involving the p-Laplacian, and the insulated problem for the Lamé system. Second, the project will explore the free boundary problem involving an incompressible fluid permeating a porous medium, often referred to as the one-phase Muskat problem. The focus will be on investigating the regularity of solutions to the two- and three-dimensional one-phase Muskat problem in the whole space, as well as on exploring the short-term and long-term smoothing effects of these solutions. Finally, the project will investigate boundary regularity of linear kinetic equations as well as the stability and global well-posedness of nonlinear kinetic equations, including the relativistic Vlasov-Maxwell-Landau system and the spatially inhomogeneous Boltzmann equations in general domains.

This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences (DMS) and the Established Program to Stimulate Competitive Research (EPSCoR).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350101","Some problems in harmonic analysis","DMS","ANALYSIS PROGRAM","07/01/2024","04/02/2024","Xiaochun Li","IL","University of Illinois at Urbana-Champaign","Standard Grant","Wing Suet Li","06/30/2027","$327,441.00","","xcli@math.uiuc.edu","506 S WRIGHT ST","URBANA","IL","618013620","2173332187","MPS","128100","","$0.00","The principal investigator (PI) intends to delve into challenges situated at the junction of harmonic analysis, number theory, and dispersive equations. In addition to focusing on classical Fourier analysis, the PI aims to establish connections with diverse fields, including number theory, combinatorics, dispersive equations on tori, and Ergodic theory. Furthermore, the PI plans to mentor students, disseminate their findings through talks, and foster collaborations, thereby generating broader impacts.


The PI plans to continue the research efforts in several areas. Firstly, the PI and his collaborators will delve into the rapidly advancing field of modern mathematics, particularly focusing on additive combinatorics alongside Fourier analysis. Within this realm, they aim to further explore Roth's theorem, a fundamental result that determines the minimum subset size required for the existence of arithmetic progressions within {1, ..., N}. Their work will extend their previous investigations into the polynomial Roth theorem on rings and/or finite fields. Secondly, in classical harmonic analysis, the PI is dedicated to investigating the conjectured pointwise convergence of the Bochner-Riesz mean on the plane, as proposed by Sogge and Tao. Thirdly, in collaboration with Yang, the PI has made strides in improving both Gauss's circle problem and Dirichlet's divisor problem. They believe there is still room for additional progress in these areas. Finally, the PI will continue his study of the Waring problem, which can be approached as a decoupling problem for a function whose Fourier transform is confined to a broken line.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349794","Regularity Problems in Free Boundaries and Degenerate Elliptic Partial Differential Equations","DMS","ANALYSIS PROGRAM","07/01/2024","04/02/2024","Ovidiu Savin","NY","Columbia University","Standard Grant","Marian Bocea","06/30/2027","$273,927.00","","os2161@columbia.edu","615 W 131ST ST","NEW YORK","NY","100277922","2128546851","MPS","128100","","$0.00","The goal of this project is to develop new methods for the mathematical theory in several problems of interest involving partial differential equations (PDE). The problems share some common features and are motivated by various physical phenomena such as the interaction of elastic membranes in contact with one another, jet flows of fluids, surfaces of minimal area, and optimal transportation between the elements of two regions. Advancement in the theoretical knowledge about these problems would be beneficial to the scientific community in general and possibly have applications to more concrete computational aspects of solving these equations numerically. The outcomes of the project will be disseminated at a variety of seminars and conferences.

The project focuses on the regularity theory of some specific free boundary problems and nonlinear PDE. The first part is concerned with singularity formation in the Special Lagrangian equation. The equation appears in the context of calibrated geometries and minimal submanifolds. The Principal Investigator (PI) studies the stability of singular solutions under small perturbations together with certain degenerate Bellman equations that are relevant to their study. The second part of the project is devoted to free boundary problems. The PI investigates regularity questions that arise in the study of the two-phase Alt-Phillips family of free boundary problems. Some related questions concern rigidity of global solutions in low dimensions in the spirit of the De Giorgi conjecture. A second problem of interest involves coupled systems of interacting free boundaries. They arise in physical models that describe the configuration of multiple elastic membranes that are interacting with each other according to some specific potential. Another part of the project is concerned with the regularity of nonlocal minimal graphs and some related questions about the boundary Harnack principle for nonlocal operators.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348748","Exploring Large-Scale Geometry via Local and Nonlocal Potential Theory","DMS","ANALYSIS PROGRAM","09/01/2024","04/02/2024","Nageswari Shanmugalingam","OH","University of Cincinnati Main Campus","Standard Grant","Jan Cameron","08/31/2027","$343,850.00","","shanmun@uc.edu","2600 CLIFTON AVE","CINCINNATI","OH","452202872","5135564358","MPS","128100","","$0.00","This project will develop new mathematical tools for the analysis of metric measure spaces ? that is, spaces equipped (like Euclidean space) with notions of distance and volume ? with a focus on metric measure spaces that (unlike Euclidean space) lack smooth structure. The analysis of non-smooth spaces is a vital area of research with diverse applications across the mathematical and physical sciences, including fluid mechanics, neurophysiology, and fractal geometry. The PI will investigate the large-scale geometric behavior of objects in these spaces using the mathematical tools of local and nonlocal energies. Given a function measuring a physical phenomenon, such as temperature or momentum, local energies measure the function?s nearby or small-scale oscillations, while nonlocal energies measure its variations over long distances. A primary goal of this work is to develop much-needed mathematical tools for analyzing nonlocal energies. The project will also enhance the professional training of graduate students and postdoctoral scholars, through collaborative research projects, instruction in effective mathematical communication, and opportunities for research interactions with undergraduate students.


The primary objects of study in this project are represented as metric measure spaces that lack smooth structure. The finite dimensionality of the ambient space is represented by the property of supporting a doubling Radon measure. In such a setting, nearby or asymptotic oscillation of a function is measured using upper gradients, which are viable substitutes for the derivative of a function, and the local energy is associated with the collection of functions on the object, called Sobolev functions. The large-scale variation energy is associated with the collection called Besov space of functions. Recent research has uncovered a connection between local energies on a region in a metric measure space and nonlocal energies on the boundary of the region. The project will leverage this connection to explore the large-scale geometry of nonlocal energies on the boundary of the region by linking them with small-scale behavior of local energies on the region itself. In particular, connections between Dirichlet-type boundary value problems and Neumann-type boundary value problems will be investigated.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348315","Structural Properties of Measurable and Topological Dynamical Systems","DMS","ANALYSIS PROGRAM","06/01/2024","04/02/2024","Bryna Kra","IL","Northwestern University","Standard Grant","Jan Cameron","05/31/2027","$349,929.00","","kra@math.northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","128100","","$0.00","The investigator proposes a program of research, education, and outreach in dynamics. The research is focused on structural questions in dynamics that lie at the heart of understanding many different types of abstract systems, focused on gaining a deeper understanding of connections among different dynamical properties. The education and outreach portions of the project aim to broaden the cohort of researchers working in these areas.

The proposed research problems have a combinatorial flavor, and the methods used to approach them are dynamical. Using structural results in ergodic theory and topological dynamics, the PI proposes studying the types of infinite configurations that must arise in any sufficiently large set of integers. To gain a better understanding of the interactions of topological measurable structures, the PI proposes studying a newly defined class of systems that gives rise to a type of rigidity. A symbolic viewpoint underlines the approaches, and the PI plans to study the interactions of algebraic, measurable, and dynamical properties of these systems. This includes gaining a better understanding of how the automorphism group of the system reflects the underlying dynamics and the system supports a measure invariant under this group of symmetries.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2350242","Hilbert's Sixth Problem: From Particles to Waves","DMS","ANALYSIS PROGRAM","06/01/2024","04/03/2024","Zaher Hani","MI","Regents of the University of Michigan - Ann Arbor","Continuing Grant","Marian Bocea","05/31/2027","$130,000.00","","zhani@umich.edu","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","128100","","$0.00","Hilbert?s sixth problem, posed in 1900, asks for a rigorous mathematical derivation of the macroscopic laws of statistical physics, formulated by Maxwell and Boltzmann in the nineteenth century, starting from the microscopic laws of dynamics (aka first principles). The classical setting of this problem pertains to particle systems which collide according to the laws of classical mechanics. The same problem emerges in more modern theories of statistical physics, where particles are replaced by waves that interact according to some Hamiltonian wave-type partial differential equation. Such theories of statistical physics for waves often go by the name of ?wave turbulence theory?, because they play a central role in understanding turbulent behaviors in wave systems. This has applications in many areas of science such as quantum mechanics, oceanography, climate science, etc. Broadly speaking, the goal of this project is to advance the mathematical, and hence scientific, understanding of such turbulence theories, and settle some longstanding conjectures in mathematical physics on the foundations of statistical mechanics. The project provides research training opportunities for graduate students.

Even in its classical setting, Hilbert?s sixth problem remains a formidable task, that has only been resolved for short times. The project seeks to provide its long-time resolution, thus giving a final answer to this longstanding open problem. This amounts to giving the rigorous derivation of Boltzmann?s kinetic equation starting from Newton?s laws, followed by the derivation of the macroscopic fluid models (Euler?s and Navier-Stokes equations). In parallel, the project proposes similar justifications in the setting of wave turbulence theory. There too, the Principal Investigator (PI) seeks to provide the long-time derivation of the corresponding ?wave kinetic equations? for various wave systems of scientific interest. Starting with the nonlinear Schrödinger equation as a prime model for nonlinear wave systems, this will be followed by similar investigations for other wave systems, like many-particle quantum systems and some models coming from ocean and climate science. Finally, the project will investigate mathematical problems related to the turbulence aspects of wave turbulence theory. There, the PI intends to use the above rigorous derivation of the wave kinetic equations, combined with an analysis of solutions to those equations, to understand turbulence phenomena for wave systems, such as energy cascades and growth of Sobolev norms.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350049","Entropy and Boundary Methods in von Neumann Algebras","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Srivatsav Kunnawalkam Elayavalli","CA","University of California-San Diego","Continuing Grant","Jan Cameron","06/30/2027","$47,255.00","","srivatsavke@gmail.com","9500 GILMAN DR","LA JOLLA","CA","920930021","8585344896","MPS","128100","","$0.00","The theory of von Neumann algebras, originating in the 1930's as a mathematical foundation for quantum physics, has since evolved into a beautifully rich subfield of modern functional analysis. Studying the precise structure of von Neumann algebras is rewarding for many reasons, as they appear naturally in diverse areas of modern mathematics such as dynamical systems, ergodic theory, analytic and geometric group theory, continuous model theory, topology, and knot theory. They also continue to be intimately involved in a variety of fields across science and engineering, including quantum physics, quantum computation, cryptography, and algorithmic complexity. The PI will focus on developing a new horizon for research on structural properties of von Neumann algebras, by combining entropy (quantitative) and boundary (qualitative) methods, with applications to various fundamental open questions. This project will also contribute to US workforce development through diversity initiatives and mentoring of graduate students and early career researchers.

In this project, the PI will develop two new research directions in the classification theory of finite von Neumann algebras: applications of Voiculescu's free entropy theory to the structure of free products and of ultrapowers of von Neumann algebras; the small at infinity compactification and structure of von Neumann algebras arising from relatively properly proximal groups. This will involve a delicate study of structure, rigidity and indecomposability properties via innovative interplays between three distinct successful approaches: Voiculescu's free entropy theory, Popa's deformation rigidity theory, Ozawa's theory of small at infinity boundaries and amenable actions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348720","Advances in rational operations in free analysis","DMS","ANALYSIS PROGRAM","06/01/2024","04/01/2024","Jurij Volcic","PA","Drexel University","Standard Grant","Wing Suet Li","05/31/2027","$155,009.00","","jurij.volcic@drexel.edu","3141 CHESTNUT ST","PHILADELPHIA","PA","191042875","2158956342","MPS","128100","","$0.00","The order of actions or operations typically matters; for example, one should first wash the clothes and then dry them, not the other way around. In other words, operations typically do not commute; this is why matrices, which encode noncommutativity in mathematics, are omnipresent in science. While matrix and operator theory has been profoundly developed in the past, the fast-evolving technological advances raise new challenges that have to be addressed. Concretely, expanding quantum technologies, complex control systems, and new resources in optimization and computability pose questions about ensembles of matrices and their features that are independent of the matrix size. The common framework for studying such problems is provided by free analysis (""free"" as in size-free), which investigates functions in matrix and operator variables. This project focuses on such functions that are built only using variables and arithmetic operations, and are therefore called noncommutative polynomials and rational functions. While these are more tangible and computationally accessible than general noncommutative functions, most of their fundamental features are yet to be explored. The scope of the project is to investigate noncommutative rational functions and their variations, develop a theory that allows resolving open problems about them, and finally apply these resolutions to tackle emerging challenges in optimization, control systems, and quantum information. This project provides research training opportunities for graduate students.

The scope of this project is twofold. Firstly, the project aims to answer several function-theoretic open problems on rational operations in noncommuting variables. Among these are singularities and vanishing of rational expressions in bounded operator variables, geometric and structural detection of composition in noncommutative rational functions using control-theoretic tools, noncommutative tensor-rational functions and their role in computational complexity, and existence of low-rank values of noncommutative polynomials with a view towards noncommutative algebraic geometry and approximate zero sets. These fundamental problems call for new synergistic methods that combine complex analysis, representation theory, algebraic geometry and operator theory. Secondly, the project aims to advance the framework of positivity and optimization in several operator variables without dimension restrictions, where the objective functions and constraints are noncommutative polynomials and their variations. The approach to this goal leads through functional analysis, real algebraic geometry and operator algebras. Moreover, the project seeks to apply these new optimization algorithms in quantum information theory, to study nonlinear Bell inequalities in complex quantum networks and the self-testing phenomenon in device-independent certification and cryptographic security.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2334874","Conference: Pittsburgh Links among Analysis and Number Theory (PLANT)","DMS","ALGEBRA,NUMBER THEORY,AND COM, ANALYSIS PROGRAM","02/01/2024","01/19/2024","Carl Wang Erickson","PA","University of Pittsburgh","Standard Grant","James Matthew Douglass","01/31/2025","$20,000.00","Theresa Anderson, Armin Schikorra","carl.wang-erickson@pitt.edu","4200 FIFTH AVENUE","PITTSBURGH","PA","152600001","4126247400","MPS","126400, 128100","7556","$0.00","This award will support the four-day conference ""Pittsburgh Links among Analysis and Number Theory (PLANT)"" that will take place March 4-7, 2024 in Pittsburgh, PA. The purpose of the conference is to bring together representatives of two disciplines with a shared interface: number theory and analysis. There is a large potential for deeper collaboration between these fields resulting in new and transformative mathematical perspectives, and this conference aims at fostering such an interchange. In particular, the conference is designed to attract PhD students and post-doctoral scholars into working on innovations at this interface.

To encourage the development of new ideas, the conference speakers, collectively, represent many subfields that have developed their own distinctive blend of analysis and number theory, such as analytic number theory, arithmetic statistics, analytic theory of modular and automorphic forms, additive combinatorics, discrete harmonic analysis, and decoupling. While there have been a wide variety of conferences featuring these subfields in relative isolation, the PIs are excited at PLANT's potential for sparking links among all of these subfields and giving early-career participants the opportunity to be part of this exchange. The conference website is https://sites.google.com/view/plant24/home.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2400036","New Challenges in the Study of Propagation of Randomness for Nonlinear Evolution Equations","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Andrea Nahmod","MA","University of Massachusetts Amherst","Standard Grant","Marian Bocea","06/30/2027","$388,536.00","","nahmod@math.umass.edu","101 COMMONWEALTH AVE","AMHERST","MA","010039252","4135450698","MPS","128100","","$0.00","Waves are everywhere in nature. We observe them when we look at the ripples that form when we throw a pebble in a lake, the expanding ring called a wave-packet; or when we look at a rainbow that is formed when light wave passes through a prism or water droplet and note the spatial separation of white light into different colors. Partial differential equations (PDE) modeling wave propagation phenomena have played a fundamental role in understanding such physical and natural events as well as quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other physical models. In these cases, wave phenomena are never too smooth or too simple, and in fact the byproduct of nonlinear wave interactions as they propagate in time. Being able to understand and describe the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagates, is fundamental to accurately predict wave phenomena when studying the natural world. This project is aimed at answering several central questions about long-time dynamics and the propagation of randomness in this context using analytical and probabilistic methodologies. The work of the project, and its connections to science, promotes interdisciplinary interactions and fosters the training of graduate students and junior researchers in the United States thus fundamentally contributing to its STEM workforce.

The interplay of deterministic methods in nonlinear PDE and probabilistic ones naturally feed off each other and when combined contribute to a deep understanding of wave phenomena, which opens the door to new paradigms that move research forward in various directions. The Principal Investigator studies several projects at the forefront of current research. The problems, grouped in two interrelated directions, aim broadly at: (1) studying the out of equilibrium long-time dynamics of dispersive flows from a probabilistic viewpoint in energy subcritical regimes by means of suitable quantitative quasi-invariance, modified energies and stability theory of random structures; (2) establishing the invariance of Gibbs measures for the probabilistically critical three-dimensional nonlinear Schrödinger equation (also known as a model in constructive quantum field theory) in the context of equilibrium statistical mechanics; (3) establishing a suitable probabilistic local theory of the hyperbolic sine-Gordon equation on 2D tori and the invariance of its associated Gibbs measure; and (4) the development of the random tensor theory for the nonlinear wave equations and for non-Gaussian data. The research bridges between the dispersive and wave equations communities that specialize in stochastic equations and contributes to understanding in a fundamental way propagation of randomness in nonlinear wave phenomena.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2349868","Testing Theorems in Analytic Function Theory, Harmonic Analysis and Operator Theory","DMS","ANALYSIS PROGRAM","07/01/2024","04/02/2024","Brett Wick","MO","Washington University","Standard Grant","Wing Suet Li","06/30/2027","$310,000.00","","wick@math.wustl.edu","ONE BROOKINGS DR","SAINT LOUIS","MO","63110","3147474134","MPS","128100","","$0.00","This proposal involves basic fundamental mathematical research at the intersection of analytic function theory, harmonic analysis, and operator theory. Motivation to study these questions can be found in partial differential equations, which are fundamental to the study of science and engineering. The solution to a partial differential equation is frequently given by an integral operator, a Calderon-Zygmund operator, whose related properties can be used to deduce related properties of these partial differential equations. In general, studying these Calderon-Zygmund operators is challenging and one seeks to study their action on certain spaces of functions, by checking the behavior only on a simpler class of test functions. In analogy, this can be seen as attempting to understand a complicated musical score by simply understanding a simpler finite collection of pure frequencies. The proposed research is based on recent contributions made by the PI, leveraging the skills and knowledge developed through prior National Science Foundation awards. Through this proposal the PI will address open and important questions at the interface of analytic function theory, harmonic analysis, and operator theory. Resolution of questions in these areas will provide for additional lines of inquiry. Funds from this award will support a diverse group of graduate students whom the PI advises; helping to increase the national pipeline of well-trained STEM students for careers in academia, government, or industry.

The research program of this proposal couples important open questions with the PI's past work. The general theme will be to use methods around ``testing theorems,'' called ``T1 theorems'' in harmonic analysis or the ``reproducing kernel thesis'' in analytic function theory and operator theory, to study questions that arise in analytic function theory, harmonic analysis, and operator theory. In particular, applications of the proof strategy of testing theorems will: (1) be used to characterize when Calderon-Zygmund operators are bounded between weighted spaces both for continuous and dyadic variants of these operators; (2) serve as motivation for a class of questions related to operators on the Fock space of analytic functions that are intimately connected to Calderon-Zygmund operators; and, (3) be leveraged to provide a method to study Carleson measures in reproducing kernel Hilbert spaces of analytic functions. Results obtained will open the door to other lines of investigation.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2400040","Approximation properties in von Neumann algebras","DMS","ANALYSIS PROGRAM","06/01/2024","03/27/2024","Jesse Peterson","TN","Vanderbilt University","Standard Grant","Wing Suet Li","05/31/2027","$291,569.00","","jesse.d.peterson@vanderbilt.edu","110 21ST AVE S","NASHVILLE","TN","372032416","6153222631","MPS","128100","","$0.00","Von Neumann algebras were introduced in the 1930's and 40's to study representation theory of groups, and to use as a tool for developing a mathematical foundation for quantum physics. They have since developed into a full area of study as a natural noncommutative notion of measure theory. The noncommutative setting of topology (C*-algebras) emerged shortly after, and the two subjects have historically been closely connected. This project explores these connections to develop new ideas, to reach a broad mathematical community and providing engagement and support for new students in the field. The investigator is actively participating in the training of students and postdocs in von Neumann algebras and the research from this project will directly impact these students and postdocs.

The project investigator is studying approximation properties (or the lack thereof) in von Neumann algebras and C*-algebras, especially relating to group von Neumann algebras and group measure space constructions. This has historically been a significant area of study in the classification of operator algebras, with amenability/injectivity playing a major role in the development of von Neumann algebras, and nuclearity playing a major corresponding role in the theory of C*-algebras. The emergence of Popa's deformation/rigidity theory has led to numerous breakthroughs in the classification of von Neumann algebras beyond the amenability setting, and approximation properties, such as Ozawa's notion of a biexact group, have created new opportunities to study approximation properties in the setting of von Neumann algebras. The research developed in this project investigates these approximation properties, creating new connections between C* and von Neumann algebras. This allows new C*-algebraic tools to be used in the setting of von Neumann algebras, leading to new structural results for group and group measure space von Neumann algebras, and giving a deeper insight into interactions between operator algebras, ergodic theory, and geometric group theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2350242","Hilbert's Sixth Problem: From Particles to Waves","DMS","ANALYSIS PROGRAM","06/01/2024","04/03/2024","Zaher Hani","MI","Regents of the University of Michigan - Ann Arbor","Continuing Grant","Marian Bocea","05/31/2027","$130,000.00","","zhani@umich.edu","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","128100","","$0.00","Hilbert?s sixth problem, posed in 1900, asks for a rigorous mathematical derivation of the macroscopic laws of statistical physics, formulated by Maxwell and Boltzmann in the nineteenth century, starting from the microscopic laws of dynamics (aka first principles). The classical setting of this problem pertains to particle systems which collide according to the laws of classical mechanics. The same problem emerges in more modern theories of statistical physics, where particles are replaced by waves that interact according to some Hamiltonian wave-type partial differential equation. Such theories of statistical physics for waves often go by the name of ?wave turbulence theory?, because they play a central role in understanding turbulent behaviors in wave systems. This has applications in many areas of science such as quantum mechanics, oceanography, climate science, etc. Broadly speaking, the goal of this project is to advance the mathematical, and hence scientific, understanding of such turbulence theories, and settle some longstanding conjectures in mathematical physics on the foundations of statistical mechanics. The project provides research training opportunities for graduate students.

Even in its classical setting, Hilbert?s sixth problem remains a formidable task, that has only been resolved for short times. The project seeks to provide its long-time resolution, thus giving a final answer to this longstanding open problem. This amounts to giving the rigorous derivation of Boltzmann?s kinetic equation starting from Newton?s laws, followed by the derivation of the macroscopic fluid models (Euler?s and Navier-Stokes equations). In parallel, the project proposes similar justifications in the setting of wave turbulence theory. There too, the Principal Investigator (PI) seeks to provide the long-time derivation of the corresponding ?wave kinetic equations? for various wave systems of scientific interest. Starting with the nonlinear Schrödinger equation as a prime model for nonlinear wave systems, this will be followed by similar investigations for other wave systems, like many-particle quantum systems and some models coming from ocean and climate science. Finally, the project will investigate mathematical problems related to the turbulence aspects of wave turbulence theory. There, the PI intends to use the above rigorous derivation of the wave kinetic equations, combined with an analysis of solutions to those equations, to understand turbulence phenomena for wave systems, such as energy cascades and growth of Sobolev norms.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350351","Unique continuation and the regularity of elliptic PDEs and generalized minimal submanifolds","DMS","GEOMETRIC ANALYSIS, ANALYSIS PROGRAM","06/01/2024","03/27/2024","Zihui Zhao","MD","Johns Hopkins University","Standard Grant","Jeremy Tyson","05/31/2027","$253,734.00","","zhaozh@jhu.edu","3400 N CHARLES ST","BALTIMORE","MD","212182608","4439971898","MPS","126500, 128100","5920, 5936, 5946, 5950","$0.00","This award supports research on the regularity of solutions to elliptic partial differential equations and regularity of generalized minimal submanifolds. Elliptic differential equations govern the equilibrium configurations of various physical phenomena, for instance, those arising from minimization problems for natural energy functionals. Examples include the shape of free-hanging bridges, the shape of soap bubbles, and the sound of drums. Elliptic differential equations are also used to quantify the degree to which physical objects are bent or distorted, with far-reaching implications and applications in geometry and topology. The proposed research focuses on the regularity of solutions to such equations. Questions to be addressed include the following: Do non-smooth points (singularities) exist? How large can the set of singularities be? What is the behavior of the solution near a singularity? Is it possible to perturb the underlying environment in order to eliminate the singularity? The project will also provide opportunities for the professional development of graduate students, both via individual mentoring and via the organization of a directed learning seminar on geometric analysis and geometric measure theory.

The mathematical objectives of the project are twofold. First, the principal investigator will study unique continuation for solutions to elliptic partial differential equations, with a focus on quantitative estimates on the size and structure of the singular set of these solutions. A second topic for consideration is the regularity theory for generalized minimal submanifolds (a generalized notion of smooth submanifolds which arise as critical points for the area functional under local deformations). In particular, the principal investigator will study branch singular points in the interior as well as at the boundary of a generalized minimal submanifold, under an area-minimizing or stability assumption. Research on the latter topic, which can be viewed as a non-linear analogue of quantitative unique continuation for elliptic equations, requires the integration of ideas from geometric measure theory, partial differential equations and geometric analysis.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350263","Diffusion in Kinetic Equations","DMS","ANALYSIS PROGRAM","07/01/2024","03/27/2024","Luis Silvestre","IL","University of Chicago","Standard Grant","Marian Bocea","06/30/2027","$363,887.00","","luis@math.uchicago.edu","5801 S ELLIS AVE","CHICAGO","IL","606375418","7737028669","MPS","128100","","$0.00","Kinetic equations model the evolution of densities of a large system of interactive particles. They may be used, for example, to study the evolution of a gas or a plasma. The Principal Investigator (PI) is interested in the study of the Boltzmann and Landau equations, for systems of particles that repel each other by power-law potentials. These equations exhibit a regularization effect. An outstanding open problem is to understand if a singularity could emerge from the natural flow of the equation, or if the regularization effects actually dominate the evolution and keep the solutions smooth. The PI mentors graduate students and postdocs in research on the topics of this project.

This project aims at developing tools in the analysis of nonlocal equations, parabolic equations and hypoelliptic theory targeted to their applications in kinetic equations. The Boltzmann collision operator acts as a nonlinear diffusive operator of fractional order. It can be studied in the framework of parabolic integro-differential equations. The Landau equation is a model from statistical mechanics used to describe the dynamics of plasma. It can be obtained as a limit case of the Boltzmann equation when grazing collisions prevail. It is a second order, nonlinear, parabolic equation. The project connects different areas of mathematics and mathematical physics, relating recent progress in nonlinear integro-differential equations with the classical Boltzmann equation from statistical mechanics. Kinetic equations involve a nonlinear diffusive operator with respect to velocity, combined with a transport equation with respect to space. The regularization effect in all variables requires ideas from hypoelliptic theory. For the Boltzmann equation in the case of very soft potentials, as well as for the Landau equation with Coulomb potentials, the diffusive part of the equations is not strong enough to prevent the solution from blowing up in theory. In that case, new ideas are needed to properly understand the regularization effects of the equation.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348633","Free Information Theory Techniques in von Neumann Algebras","DMS","ANALYSIS PROGRAM","07/01/2024","03/27/2024","Dimitri Shlyakhtenko","CA","University of California-Los Angeles","Standard Grant","Jan Cameron","06/30/2027","$421,000.00","","shlyakht@ipam.ucla.edu","10889 WILSHIRE BLVD STE 700","LOS ANGELES","CA","900244200","3107940102","MPS","128100","","$0.00","Von Neumann algebras arose in the 1930s as a mathematical framework for quantum mechanics. In classical mechanics it is possible to simultaneously observe and measure various properties of a physical system ? for example, the locations and velocities of all of its components. Such properties are often called observables. Observables be viewed as functions of the underlying system and form an algebra ? they can be added and multiplied. In quantum mechanics, simultaneous measurements are no longer possible. Mathematically this is reflected by the non-commutativity of the algebra of observables for quantum systems. Nonetheless, many of the operations that can be done with ordinary functions have quantum analogs. The current proposal studies such non-commutative algebras of observables from the angle of Voiculescu?s free probability theory, which treats observables as random variables. This results in an extremely rich theory that leads to free probability generalizations of classical objects such as partial differential equations and Brownian motion, amenable to analysis by techniques inspired by classical information theory. This project will promote human resource development through graduate and undergraduate research opportunities and will support students under the auspices of the UCLA Olga Radko Endowed Math Circle.

The proposed research deals with several questions in von Neumann algebras which are approached by free probability and free information methods, including free entropy theory. This includes further developing PDE based methods in the non-commutative context and strengthening the connection between free probability and random matrix theory. Among the research directions is a notion of dimension that is based on the behavior of optimal transportation distance, as well as applications of free information theory techniques to von Neumann algebra theory. The project includes a mixture of problems, some coming from existing research directions and some exploring new lines of inquiry.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2349828","Spatial restriction of exponential sums to thin sets and beyond","DMS","ANALYSIS PROGRAM","06/01/2024","04/01/2024","Ciprian Demeter","IN","Indiana University","Standard Grant","Wing Suet Li","05/31/2027","$299,999.00","","demeterc@indiana.edu","107 S INDIANA AVE","BLOOMINGTON","IN","474057000","3172783473","MPS","128100","","$0.00","In recent years, the PI has developed a new tool called decoupling that measures the extent to which waves traveling in different directions interact with each other. While this tool was initially intended to analyze differential equations that describe wave cancellations, it has also led to important breakthroughs in number theory. For example, Diophantine equations are potentially complicated systems of equations involving whole numbers. They are used to generate scrambling and diffusion keys, which are instrumental in encrypting data. Mathematicians are interested in counting the number of solutions to such systems. Unlike waves, numbers do not oscillate, at least not in an obvious manner. But we can think of numbers as frequencies and thus associate them with waves. In this way, problems related to counting the number of solutions to Diophantine systems can be rephrased in the language of quantifying wave interferences. This was the case with PI's breakthrough resolution of the Main Conjecture in Vinogradov's Mean Value Theorem. The PI plans to further extend the scope of decoupling toward the resolution of fundamental problems in harmonic analysis, geometric measure theory, and number theory. He will seek to make the new tools accessible and useful to a large part of the mathematical community. This project provides research training opportunities for graduate students.


Part of this project is aimed at developing the methodology to analyze the Schrödinger maximal function in the periodic setting. Building on his recent progress, the PI aims to incorporate Fourier analysis and more delicate number theory into the existing combinatorial framework. Decouplings have proved successful in addressing a wide range of problems in such diverse areas as number theory, partial differential equations, and harmonic analysis. The current project seeks to further expand the applicability of this method in new directions. One of them is concerned with finding sharp estimates for the Fourier transforms of fractal measures supported on curved manifolds. The PI seeks to combine decoupling with sharp estimates for incidences between balls and tubes. In yet another direction, he plans to further investigate the newly introduced tight decoupling phenomenon. This has deep connections to both number theory and the Lambda(p) estimates.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2348846","Interfaces, Degenerate Partial Differential Equations, and Convexity","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Benjamin Weinkove","IL","Northwestern University","Standard Grant","Marian Bocea","06/30/2027","$145,617.00","","weinkove@math.northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","128100","9251","$0.00","Partial differential equations (PDE) are essential mathematical objects for modeling physical processes. This project aims to understand the properties of some fundamental PDE models for the diffusion of gas, the shape of liquid droplets, and electric transmission in composite materials. Many such processes exhibit an interface, where the equation becomes degenerate or singular. In the case of a model of gas diffusion through a porous medium, the interface is the set which separates the region where there is gas from the region where there is no gas. The boundary of a liquid droplet is another example of an interface. The project will investigate qualitative and quantitative properties of these interfaces, including convexity and smoothness. For a composite material, consisting of two materials with different conductivity properties, the interface is where these materials meet. The Principal Investigator (PI) will study the behavior of the electric field when one of the materials has a very thin part. This project has possible implications for material failure, an important question in Engineering. Students and postdoctoral scholars will be trained on the techniques and theory of these PDE models.

This project centers on four topics, connected by the themes of interfaces, degeneracies, and convexity/concavity. The porous medium equation is a nonlinear degenerate parabolic equation used to model the diffusion of gas. The PI will investigate questions of concavity and convexity of solutions and finding global optimal regularity estimates. Secondly, the PI will study linear PDE whose coefficients are discontinuous along two almost touching interfaces, a model for transmission problems and composite materials. In this setting, the PI will investigate new approaches to obtaining optimal gradient estimates in the thin region between the interfaces. A third project is to study linear equations which are parabolic on the interior on a fixed domain but are degenerate at the boundary. These equations arise as linearizations of the porous medium equation and the Gauss curvature flow. The PI will investigate optimal conditions for existence and uniqueness of smooth solutions. Finally, questions of concavity of solutions to the elliptic torsion problem with Dirichlet boundary conditions and the dynamic version of this equation, known as the quasi-static droplet model, will be studied. In addition, the PI will carry out summer research projects with undergraduates, exploring explicit solutions to these equations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2400036","New Challenges in the Study of Propagation of Randomness for Nonlinear Evolution Equations","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Andrea Nahmod","MA","University of Massachusetts Amherst","Standard Grant","Marian Bocea","06/30/2027","$388,536.00","","nahmod@math.umass.edu","101 COMMONWEALTH AVE","AMHERST","MA","010039252","4135450698","MPS","128100","","$0.00","Waves are everywhere in nature. We observe them when we look at the ripples that form when we throw a pebble in a lake, the expanding ring called a wave-packet; or when we look at a rainbow that is formed when light wave passes through a prism or water droplet and note the spatial separation of white light into different colors. Partial differential equations (PDE) modeling wave propagation phenomena have played a fundamental role in understanding such physical and natural events as well as quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other physical models. In these cases, wave phenomena are never too smooth or too simple, and in fact the byproduct of nonlinear wave interactions as they propagate in time. Being able to understand and describe the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagates, is fundamental to accurately predict wave phenomena when studying the natural world. This project is aimed at answering several central questions about long-time dynamics and the propagation of randomness in this context using analytical and probabilistic methodologies. The work of the project, and its connections to science, promotes interdisciplinary interactions and fosters the training of graduate students and junior researchers in the United States thus fundamentally contributing to its STEM workforce.

The interplay of deterministic methods in nonlinear PDE and probabilistic ones naturally feed off each other and when combined contribute to a deep understanding of wave phenomena, which opens the door to new paradigms that move research forward in various directions. The Principal Investigator studies several projects at the forefront of current research. The problems, grouped in two interrelated directions, aim broadly at: (1) studying the out of equilibrium long-time dynamics of dispersive flows from a probabilistic viewpoint in energy subcritical regimes by means of suitable quantitative quasi-invariance, modified energies and stability theory of random structures; (2) establishing the invariance of Gibbs measures for the probabilistically critical three-dimensional nonlinear Schrödinger equation (also known as a model in constructive quantum field theory) in the context of equilibrium statistical mechanics; (3) establishing a suitable probabilistic local theory of the hyperbolic sine-Gordon equation on 2D tori and the invariance of its associated Gibbs measure; and (4) the development of the random tensor theory for the nonlinear wave equations and for non-Gaussian data. The research bridges between the dispersive and wave equations communities that specialize in stochastic equations and contributes to understanding in a fundamental way propagation of randomness in nonlinear wave phenomena.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400115","Collaborative Research: Conference: Brazos Analysis Seminar","DMS","ANALYSIS PROGRAM","04/01/2024","03/25/2024","Kate Juschenko","TX","University of Texas at Austin","Standard Grant","Wing Suet Li","03/31/2027","$16,000.00","","kate.juschenko@gmail.com","110 INNER CAMPUS DR","AUSTIN","TX","787121139","5124716424","MPS","128100","7556","$0.00","This award provides three years of funding to help defray the expenses of participants in the semi-annual conference series ""Brazos Analysis Seminar"" 2024-2026, the first meeting of which will be held in Spring 2024 at Texas Christian University. Subsequent meetings will rotate among the University of Texas at Austin, University of Houston, Texas A&M University, and Baylor University. The Brazos Analysis Seminar will bring together analysts at academic institutions within the South-Central region of the United States on a regular basis to communicate their research, with a particular emphasis on providing an opportunity for young researchers and graduate students to meet, collaborate and disseminate their work on a regular basis during the academic year. The format for the seminar provides ample opportunity for graduate students, postdocs, and junior investigators to present their work, start new collaborations, learn about the latest developments in modern analysis, and to advance their careers.

The scientific topics of this conference series will focus on the analytic theory of operator algebras and operator space theories and their connections to harmonic analysis, ergodic theory, dynamic systems, and the quantum information theory. These include free probability method in the study of quantum groups, Fourier multipliers theory on noncommutative Lp spaces, dynamical system, and K-theory of C*-algebras and von Neumann algebras. In each meeting, there will be 3 plenary talks given by prominent experts and 6 contributed talks presented by 3 experts from the region, and 3 postdoctoral or upper level PhD students. The goal is to keep both junior and senior researchers in the south-central institutions exposed and informed of the latest major mathematical developments in noncommutative Analysis, and to enhance and advance the research on the related topics. Additional information is available on the seminar website https://sites.google.com/site/brazosanalysisseminar.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400112","Collaborative Research: Conference: Brazos Analysis Seminar","DMS","ANALYSIS PROGRAM","04/01/2024","03/25/2024","Zhizhang Xie","TX","Texas A&M University","Standard Grant","Wing Suet Li","03/31/2027","$16,400.00","","xie@math.tamu.edu","400 HARVEY MITCHELL PKY S STE 30","COLLEGE STATION","TX","778454375","9798626777","MPS","128100","7556","$0.00","This award provides three years of funding to help defray the expenses of participants in the semi-annual conference series ""Brazos Analysis Seminar"" 2024-2026, the first meeting of which will be held in Spring 2024 at Texas Christian University. Subsequent meetings will rotate among the University of Texas at Austin, University of Houston, Texas A&M University, and Baylor University. The Brazos Analysis Seminar will bring together analysts at academic institutions within the South-Central region of the United States on a regular basis to communicate their research, with a particular emphasis on providing an opportunity for young researchers and graduate students to meet, collaborate and disseminate their work on a regular basis during the academic year. The format for the seminar provides ample opportunity for graduate students, postdocs, and junior investigators to present their work, start new collaborations, learn about the latest developments in modern analysis, and to advance their careers.

The scientific topics of this conference series will focus on the analytic theory of operator algebras and operator space theories and their connections to harmonic analysis, ergodic theory, dynamic systems, and the quantum information theory. These include free probability method in the study of quantum groups, Fourier multipliers theory on noncommutative Lp spaces, dynamical system, and K-theory of C*-algebras and von Neumann algebras. In each meeting, there will be 3 plenary talks given by prominent experts and 6 contributed talks presented by 3 experts from the region, and 3 postdoctoral or upper level PhD students. The goal is to keep both junior and senior researchers in the south-central institutions exposed and informed of the latest major mathematical developments in noncommutative Analysis, and to enhance and advance the research on the related topics. Additional information is available on the seminar website https://sites.google.com/site/brazosanalysisseminar.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2337630","CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis","DMS","GEOMETRIC ANALYSIS, ANALYSIS PROGRAM","06/01/2024","12/28/2023","Yiming Zhao","NY","Syracuse University","Continuing Grant","Eriko Hironaka","05/31/2029","$69,246.00","","yzhao197@syr.edu","900 S CROUSE AVE","SYRACUSE","NY","132444407","3154432807","MPS","126500, 128100","1045","$0.00","Isoperimetric problems and Minkowski problems are two central ingredients in Convex Geometric Analysis. The former compares geometric measurements (such as volume and surface area) while the latter recovers the shape of geometric figures using local versions of these measurements. The two types of problems are inherently connected. This project will exploit this connection to seek answers to either isoperimetric problems or Minkowski problems in various settings when answers to one exist while answers to the other remain elusive. Although these problems originate from a geometric background, their applications extend beyond mathematics into engineering and design, including areas like antenna reflector design and urban planning. The principal investigator will organize a series of events and workshops at local science museums, community centers, and schools, involving high school teachers and students as well as undergraduate and graduate students. These events and workshops aim to expose the fun and exploratory side of the principal investigator?s research and mathematics in general to students early in their educational careers, raise society's awareness and interest in mathematics, and promote mathematics among historically underrepresented populations.

The existence of solutions to the dual Minkowski problem (that characterizes dual curvature measures) in the original symmetric case has been largely settled (by the principal investigator and his collaborators) through techniques from geometry and analysis. This naturally leads to conjectures involving isoperimetric problems connected to the dual Minkowski problem. Such conjectured isoperimetric inequalities are also connected to an intriguing question behind many other conjectures in convexity: how does certain symmetry improve estimates? The principal investigator will also study Minkowski problems and isoperimetric inequalities coming from affine geometry. Special cases of these isoperimetric inequalities are connected to an affine version of the sharp fractional Sobolev inequalities of Almgren-Lieb. The techniques involved in studying these questions are from Convex Geometric Analysis and PDE. In the last few decades (particularly the last two), there has been a community-wide effort to extend results in the theory of convex bodies to their counterparts in the space of log-concave functions. In this project, the principal investigator will also continue his past work to extend dual curvature measures, their Minkowski problems, and associated isoperimetric inequality to the space of log-concave functions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2349868","Testing Theorems in Analytic Function Theory, Harmonic Analysis and Operator Theory","DMS","ANALYSIS PROGRAM","07/01/2024","04/02/2024","Brett Wick","MO","Washington University","Standard Grant","Wing Suet Li","06/30/2027","$310,000.00","","wick@math.wustl.edu","ONE BROOKINGS DR","SAINT LOUIS","MO","63110","3147474134","MPS","128100","","$0.00","This proposal involves basic fundamental mathematical research at the intersection of analytic function theory, harmonic analysis, and operator theory. Motivation to study these questions can be found in partial differential equations, which are fundamental to the study of science and engineering. The solution to a partial differential equation is frequently given by an integral operator, a Calderon-Zygmund operator, whose related properties can be used to deduce related properties of these partial differential equations. In general, studying these Calderon-Zygmund operators is challenging and one seeks to study their action on certain spaces of functions, by checking the behavior only on a simpler class of test functions. In analogy, this can be seen as attempting to understand a complicated musical score by simply understanding a simpler finite collection of pure frequencies. The proposed research is based on recent contributions made by the PI, leveraging the skills and knowledge developed through prior National Science Foundation awards. Through this proposal the PI will address open and important questions at the interface of analytic function theory, harmonic analysis, and operator theory. Resolution of questions in these areas will provide for additional lines of inquiry. Funds from this award will support a diverse group of graduate students whom the PI advises; helping to increase the national pipeline of well-trained STEM students for careers in academia, government, or industry.

The research program of this proposal couples important open questions with the PI's past work. The general theme will be to use methods around ``testing theorems,'' called ``T1 theorems'' in harmonic analysis or the ``reproducing kernel thesis'' in analytic function theory and operator theory, to study questions that arise in analytic function theory, harmonic analysis, and operator theory. In particular, applications of the proof strategy of testing theorems will: (1) be used to characterize when Calderon-Zygmund operators are bounded between weighted spaces both for continuous and dyadic variants of these operators; (2) serve as motivation for a class of questions related to operators on the Fock space of analytic functions that are intimately connected to Calderon-Zygmund operators; and, (3) be leveraged to provide a method to study Carleson measures in reproducing kernel Hilbert spaces of analytic functions. Results obtained will open the door to other lines of investigation.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2349828","Spatial restriction of exponential sums to thin sets and beyond","DMS","ANALYSIS PROGRAM","06/01/2024","04/01/2024","Ciprian Demeter","IN","Indiana University","Standard Grant","Wing Suet Li","05/31/2027","$299,999.00","","demeterc@indiana.edu","107 S INDIANA AVE","BLOOMINGTON","IN","474057000","3172783473","MPS","128100","","$0.00","In recent years, the PI has developed a new tool called decoupling that measures the extent to which waves traveling in different directions interact with each other. While this tool was initially intended to analyze differential equations that describe wave cancellations, it has also led to important breakthroughs in number theory. For example, Diophantine equations are potentially complicated systems of equations involving whole numbers. They are used to generate scrambling and diffusion keys, which are instrumental in encrypting data. Mathematicians are interested in counting the number of solutions to such systems. Unlike waves, numbers do not oscillate, at least not in an obvious manner. But we can think of numbers as frequencies and thus associate them with waves. In this way, problems related to counting the number of solutions to Diophantine systems can be rephrased in the language of quantifying wave interferences. This was the case with PI's breakthrough resolution of the Main Conjecture in Vinogradov's Mean Value Theorem. The PI plans to further extend the scope of decoupling toward the resolution of fundamental problems in harmonic analysis, geometric measure theory, and number theory. He will seek to make the new tools accessible and useful to a large part of the mathematical community. This project provides research training opportunities for graduate students.


Part of this project is aimed at developing the methodology to analyze the Schrödinger maximal function in the periodic setting. Building on his recent progress, the PI aims to incorporate Fourier analysis and more delicate number theory into the existing combinatorial framework. Decouplings have proved successful in addressing a wide range of problems in such diverse areas as number theory, partial differential equations, and harmonic analysis. The current project seeks to further expand the applicability of this method in new directions. One of them is concerned with finding sharp estimates for the Fourier transforms of fractal measures supported on curved manifolds. The PI seeks to combine decoupling with sharp estimates for incidences between balls and tubes. In yet another direction, he plans to further investigate the newly introduced tight decoupling phenomenon. This has deep connections to both number theory and the Lambda(p) estimates.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2348846","Interfaces, Degenerate Partial Differential Equations, and Convexity","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Benjamin Weinkove","IL","Northwestern University","Standard Grant","Marian Bocea","06/30/2027","$145,617.00","","weinkove@math.northwestern.edu","633 CLARK ST","EVANSTON","IL","602080001","3125037955","MPS","128100","9251","$0.00","Partial differential equations (PDE) are essential mathematical objects for modeling physical processes. This project aims to understand the properties of some fundamental PDE models for the diffusion of gas, the shape of liquid droplets, and electric transmission in composite materials. Many such processes exhibit an interface, where the equation becomes degenerate or singular. In the case of a model of gas diffusion through a porous medium, the interface is the set which separates the region where there is gas from the region where there is no gas. The boundary of a liquid droplet is another example of an interface. The project will investigate qualitative and quantitative properties of these interfaces, including convexity and smoothness. For a composite material, consisting of two materials with different conductivity properties, the interface is where these materials meet. The Principal Investigator (PI) will study the behavior of the electric field when one of the materials has a very thin part. This project has possible implications for material failure, an important question in Engineering. Students and postdoctoral scholars will be trained on the techniques and theory of these PDE models.

This project centers on four topics, connected by the themes of interfaces, degeneracies, and convexity/concavity. The porous medium equation is a nonlinear degenerate parabolic equation used to model the diffusion of gas. The PI will investigate questions of concavity and convexity of solutions and finding global optimal regularity estimates. Secondly, the PI will study linear PDE whose coefficients are discontinuous along two almost touching interfaces, a model for transmission problems and composite materials. In this setting, the PI will investigate new approaches to obtaining optimal gradient estimates in the thin region between the interfaces. A third project is to study linear equations which are parabolic on the interior on a fixed domain but are degenerate at the boundary. These equations arise as linearizations of the porous medium equation and the Gauss curvature flow. The PI will investigate optimal conditions for existence and uniqueness of smooth solutions. Finally, questions of concavity of solutions to the elliptic torsion problem with Dirichlet boundary conditions and the dynamic version of this equation, known as the quasi-static droplet model, will be studied. In addition, the PI will carry out summer research projects with undergraduates, exploring explicit solutions to these equations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2350340","Coarse-graining, Renormalization, and Fractal Homogenization","DMS","ANALYSIS PROGRAM","07/01/2024","04/02/2024","Scott Armstrong","NY","New York University","Continuing Grant","Marian Bocea","06/30/2027","$142,013.00","","sa3602@nyu.edu","70 WASHINGTON SQ S","NEW YORK","NY","100121019","2129982121","MPS","128100","","$0.00","This project is focused on the development of new mathematics for analyzing the statistical behavior of physical systems which exhibit complex behavior across a large number of length scales. A typical examples include turbulent fluids, such as the earth's atmosphere, which have fluctuations on the human scale (a gust of wind) and on the continental scale (weather patterns), and every scale in between. Other examples include important models in statistical mechanics and quantum field theory. Such chaotic physical systems have interesting behaviors which emerge through the interaction of these very different length scales, often called ""critical phenomena"" by physicists. Physicists have developed heuristic, non-rigorous ways of understanding and analyzing many such physical systems, some of which are called ""renormalization group"" arguments. One of the main goals of this project is to develop precise versions of these informal arguments which are mathematically rigorous. In the past decade, the work of the Principal Investigator (PI) and other mathematicians have led to a rigorous theory of ""quantitative homogenization"" of certain partial differential equations. These equations have some of the properties of the complex physical systems mentioned above, and the homogenization theory resembles renormalization group-type arguments in important ways. However, it currently works well only for problems with a small number of length scales. The project proposes to increase the level of sophistication of the homogenization methods until the theory can be deployed more flexibly on physical systems exhibiting critical behavior. This requires the development of new mathematical ideas and concepts and will require input from analysis, probability theory, partial differential equations and mathematical physics. The project provides research training opportunities for graduate students.

The project has two main goals. The first one concerns improving the quantitative homogenization theory, so that it is more explicit in its dependence on important parameters in the equation (like the ellipticity ratio) and allows for degenerate and possibly unbounded coefficient fields. This is a well-known open problem in the subfield, but the PI and his collaborator Kuusi have made recent progress on this question, and this project will continue to develop these new ideas. A second focus of the project is to use these analytic methods developed for homogenization as means of formalizing heuristic renormalization group arguments in physics. Such methods arise in a wide variety of contexts, but the project has a few specific problems in mind. One arises in fluid turbulence, and concerns proving the anomalous diffusion of a passive scalar advected by a rough vector field. The PI and his collaborator Vicol have made recent progress on this question by using homogenization to formalize a renormalization group argument. This points the way to further possibilities, including the construction of more physically realistic examples of anomalous diffusion. Another potential application lies in Euclidean field theory, following a stochastic quantization approach to study Gibbs measures.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2349981","Survival Threshold for Collective Plasma Oscillations","DMS","ANALYSIS PROGRAM","06/01/2024","04/01/2024","Toan Nguyen","PA","Pennsylvania State Univ University Park","Continuing Grant","Marian Bocea","05/31/2027","$127,987.00","","nguyen@math.psu.edu","201 OLD MAIN","UNIVERSITY PARK","PA","168021503","8148651372","MPS","128100","","$0.00","The main objective of this project is to investigate questions about the final states of matter consisting of a sufficiently large number of interacting particles such as plasmas in plasma physics and condensates in quantum mechanics. The research will advance the understanding of turbulence in plasma physics and quantum mechanics, provide foundational mathematics to tackle unsolved problems in physics, and push the boundaries of current mathematical techniques. The research will contribute new techniques to the theory of partial differential equations, mathematical physics, dynamical systems, and applied mathematics. The project includes activities aimed at training graduate students and young researchers.

The project will prove longstanding conjectures concerning the large time behavior of solutions to the mathematical mean field models that are used in plasma physics and quantum mechanics. The primary mathematical models under investigation include the relativistic Vlasov-Maxwell system and the Hartree equations used to model the nonlinear collective effects of infinitely many interacting particles. The research will rigorously validate nonlinear physical phenomena including plasma oscillations, phase transition, phase mixing, Landau damping, and the formation of coherent structures. The scattering theory as well as the formation of periodic structures for the Vlasov and Hartree equations near nontrivial translation-invariant equilibria will be established. The work of the project involves mathematical techniques from spectral theory, resolvent analysis, Fourier analysis, dispersive PDEs, probability, and statistical physics.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2348018","Well-posedness and Long-time Behavior of Dispersive Integrable Systems","DMS","ANALYSIS PROGRAM","07/01/2024","04/01/2024","Monica Visan","CA","University of California-Los Angeles","Continuing Grant","Marian Bocea","06/30/2027","$130,679.00","","visan@math.ucla.edu","10889 WILSHIRE BLVD STE 700","LOS ANGELES","CA","900244200","3107940102","MPS","128100","","$0.00","Integrable systems have long served as guides in the study of Hamiltonian partial differential equations. They arise as effective models of real physical systems, including in optics and many-body quantum mechanics. It is in the setting of completely integrable systems that solitons and multisolitons were first discovered. These structures have since found numerous applications in the applied sciences: for example, in fiber optics, solitons have been employed in the transmission of digital signals over long distances, while in biology, they are used to describe signal propagation in the nervous system and low-frequency collective motion in proteins. This project seeks to investigate both longstanding and newly introduced integrable models. Specifically, we seek to find the minimal conditions on the initial state under which one can construct global-in-time dynamics, investigate the (in)stability of special structures (such as solitons and multisolitons), and elucidate the long-time behavior of general solutions. The project provides significant research training opportunities for graduate students, who are integrated into the main objectives of the project.

The project investigates the following specific questions for the newly introduced continuum Calogero-Moser equations: (1) large data well-posedness in the scaling-invariant space, (2) scattering for both the defocusing model and the focusing equation for initial data with mass less than that of the ground state soliton, and (3) the determination of the blowup threshold in the focusing case. Further objectives include orbital and asymptotic stability of multisoliton solutions to the Benjamin-Ono equation in optimal well-posedness spaces, dispersive decay away from the soliton component for large solutions to this equation, and the construction of Gibbs dynamics for the Landau-Lifshitz model.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." @@ -43,8 +45,8 @@ "2400114","Collaborative Research: Conference: Brazos Analysis Seminar","DMS","ANALYSIS PROGRAM","04/01/2024","03/25/2024","Jose Carrion Muniz","TX","Texas Christian University","Standard Grant","Wing Suet Li","03/31/2027","$17,250.00","Travis Russell","j.carrion@tcu.edu","3101 BELLAIRE DRIVE NORTH","FORT WORTH","TX","761290001","8172577516","MPS","128100","7556","$0.00","This award provides three years of funding to help defray the expenses of participants in the semi-annual conference series ""Brazos Analysis Seminar"" 2024-2026, the first meeting of which will be held in Spring 2024 at Texas Christian University. Subsequent meetings will rotate among the University of Texas at Austin, University of Houston, Texas A&M University, and Baylor University. The Brazos Analysis Seminar will bring together analysts at academic institutions within the South-Central region of the United States on a regular basis to communicate their research, with a particular emphasis on providing an opportunity for young researchers and graduate students to meet, collaborate and disseminate their work on a regular basis during the academic year. The format for the seminar provides ample opportunity for graduate students, postdocs, and junior investigators to present their work, start new collaborations, learn about the latest developments in modern analysis, and to advance their careers.

The scientific topics of this conference series will focus on the analytic theory of operator algebras and operator space theories and their connections to harmonic analysis, ergodic theory, dynamic systems, and the quantum information theory. These include free probability method in the study of quantum groups, Fourier multipliers theory on noncommutative Lp spaces, dynamical system, and K-theory of C*-algebras and von Neumann algebras. In each meeting, there will be 3 plenary talks given by prominent experts and 6 contributed talks presented by 3 experts from the region, and 3 postdoctoral or upper level PhD students. The goal is to keep both junior and senior researchers in the south-central institutions exposed and informed of the latest major mathematical developments in noncommutative Analysis, and to enhance and advance the research on the related topics. Additional information is available on the seminar website https://sites.google.com/site/brazosanalysisseminar.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400113","Collaborative Research: Conference: Brazos Analysis Seminar","DMS","ANALYSIS PROGRAM","04/01/2024","03/25/2024","Tao Mei","TX","Baylor University","Standard Grant","Wing Suet Li","03/31/2027","$16,000.00","Sheng Yin","tao_mei@baylor.edu","700 S UNIVERSITY PARKS DR","WACO","TX","767061003","2547103817","MPS","128100","7556","$0.00","This award provides three years of funding to help defray the expenses of participants in the semi-annual conference series ""Brazos Analysis Seminar"" 2024-2026, the first meeting of which will be held in Spring 2024 at Texas Christian University. Subsequent meetings will rotate among the University of Texas at Austin, University of Houston, Texas A&M University, and Baylor University. The Brazos Analysis Seminar will bring together analysts at academic institutions within the South-Central region of the United States on a regular basis to communicate their research, with a particular emphasis on providing an opportunity for young researchers and graduate students to meet, collaborate and disseminate their work on a regular basis during the academic year. The format for the seminar provides ample opportunity for graduate students, postdocs, and junior investigators to present their work, start new collaborations, learn about the latest developments in modern analysis, and to advance their careers.

The scientific topics of this conference series will focus on the analytic theory of operator algebras and operator space theories and their connections to harmonic analysis, ergodic theory, dynamic systems, and the quantum information theory. These include free probability method in the study of quantum groups, Fourier multipliers theory on noncommutative Lp spaces, dynamical system, and K-theory of C*-algebras and von Neumann algebras. In each meeting, there will be 3 plenary talks given by prominent experts and 6 contributed talks presented by 3 experts from the region, and 3 postdoctoral or upper level PhD students. The goal is to keep both junior and senior researchers in the south-central institutions exposed and informed of the latest major mathematical developments in noncommutative Analysis, and to enhance and advance the research on the related topics. Additional information is available on the seminar website https://sites.google.com/site/brazosanalysisseminar.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400111","Collaborative Research: Conference: Brazos Analysis Seminar","DMS","ANALYSIS PROGRAM","04/01/2024","03/25/2024","Mehrdad Kalantar","TX","University of Houston","Standard Grant","Wing Suet Li","03/31/2027","$16,000.00","","kalantar@math.uh.edu","4300 MARTIN LUTHER KING BLVD","HOUSTON","TX","772043067","7137435773","MPS","128100","7556","$0.00","This award provides three years of funding to help defray the expenses of participants in the semi-annual conference series ""Brazos Analysis Seminar"" 2024-2026, the first meeting of which will be held in Spring 2024 at Texas Christian University. Subsequent meetings will rotate among the University of Texas at Austin, University of Houston, Texas A&M University, and Baylor University. The Brazos Analysis Seminar will bring together analysts at academic institutions within the South-Central region of the United States on a regular basis to communicate their research, with a particular emphasis on providing an opportunity for young researchers and graduate students to meet, collaborate and disseminate their work on a regular basis during the academic year. The format for the seminar provides ample opportunity for graduate students, postdocs, and junior investigators to present their work, start new collaborations, learn about the latest developments in modern analysis, and to advance their careers.

The scientific topics of this conference series will focus on the analytic theory of operator algebras and operator space theories and their connections to harmonic analysis, ergodic theory, dynamic systems, and the quantum information theory. These include free probability method in the study of quantum groups, Fourier multipliers theory on noncommutative Lp spaces, dynamical system, and K-theory of C*-algebras and von Neumann algebras. In each meeting, there will be 3 plenary talks given by prominent experts and 6 contributed talks presented by 3 experts from the region, and 3 postdoctoral or upper level PhD students. The goal is to keep both junior and senior researchers in the south-central institutions exposed and informed of the latest major mathematical developments in noncommutative Analysis, and to enhance and advance the research on the related topics. Additional information is available on the seminar website https://sites.google.com/site/brazosanalysisseminar.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." -"2349865","Analysis and Dynamics in Several Complex Variables","DMS","ANALYSIS PROGRAM","06/01/2024","03/21/2024","Xianghong Gong","WI","University of Wisconsin-Madison","Standard Grant","Jeremy Tyson","05/31/2027","$333,182.00","","gong@math.wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","128100","","$0.00","This award supports research at the interface of several complex variables, differential geometry, and dynamical systems. Complex analysis studies the behavior and regularity of functions defined on and taking values in spaces of complex numbers. It remains an indispensable tool across many domains in the sciences, engineering, and economics. This project considers the smoothness of transformations on a domain defined by complex valued functions when the domain is deformed. Using integral formulas, the PI will study how invariants of a domain vary when the underlying structure of the domain changes. Another component of the project involves the study of resonance. The PI will use small divisors that measure non-resonance to classify singularities of the complex structure arising in linear approximations of curved manifolds. The project will involve collaboration with researchers in an early career stage and will support the training of graduate students.

Motivated by recent counterexamples showing that smooth families of domains may be equivalent by a discontinuous family of biholomorphisms, the PI will study the existence of families of biholomorphisms between families of domains using biholomorphism groups and other analytic tools such as Bergman metrics. The PI will construct a global homotopy formula with good estimates for suitable domains in a complex manifold. One of the goals is to construct a global formula in cases when a local homotopy formula fails to exist. The PI will use such global homotopy formulas to investigate the stability of holomorphic embeddings of domains with strongly pseudoconvex or concave boundary in a complex manifold, when the complex structure on the domains is deformed. The PI will use this approach to investigate stability of global Cauchy-Riemann structures on Cauchy-Riemann manifolds of higher codimension. The project seeks a holomorphic classification of neighborhoods of embeddings of a compact complex manifold in complex manifolds via the Levi-form and curvature of the normal bundle. In addition, the PI will study the classification of Cauchy-Riemann singularities for real manifolds using methods from several complex variables and small-divisor conditions in dynamical systems.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2346736","Conference: CIRM 2024: Operators on analytic function spaces","DMS","ANALYSIS PROGRAM","08/01/2024","03/22/2024","Pamela Gorkin","PA","Bucknell University","Standard Grant","Wing Suet Li","07/31/2025","$40,200.00","Kelly Bickel, Elodie Pozzi","pgorkin@bucknell.edu","1 DENT DR","LEWISBURG","PA","178372005","5705773510","MPS","128100","7556","$0.00","The conference ""Operators on analytic function spaces"" will take place at the Centre International de Rencontres Mathematiques (CIRM) in Marseille, France from December 2 - 6, 2024. The goal is to create a diverse group of mathematicians poised to solve an important set of problems in function and operator theory, and to allow attendees to develop new directions and partnerships. Funding will be used for US participant support, with priority going to members of underrepresented groups and early career researchers. CIRM provides facilities and equipment as well as an excellent library and serves as a place for collaborative work.

The focus of the conference is on recent progress on Hilbert and Banach spaces of holomorphic functions and the operators acting on them. During the week at CIRM participants will discuss important open questions in function theory and operator theory, including operators on model spaces, Toeplitz and Hankel operators, cyclic vectors, sampling, frames, interpolation and reproducing kernels, and the Crouzeix conjecture. In addition to the talks, the conference will offer activities for attendees to interact and discuss future directions for research. More information may be found at the conference webpage, https://conferences.cirm-math.fr/3085.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." +"2349865","Analysis and Dynamics in Several Complex Variables","DMS","ANALYSIS PROGRAM","06/01/2024","03/21/2024","Xianghong Gong","WI","University of Wisconsin-Madison","Standard Grant","Jeremy Tyson","05/31/2027","$333,182.00","","gong@math.wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","128100","","$0.00","This award supports research at the interface of several complex variables, differential geometry, and dynamical systems. Complex analysis studies the behavior and regularity of functions defined on and taking values in spaces of complex numbers. It remains an indispensable tool across many domains in the sciences, engineering, and economics. This project considers the smoothness of transformations on a domain defined by complex valued functions when the domain is deformed. Using integral formulas, the PI will study how invariants of a domain vary when the underlying structure of the domain changes. Another component of the project involves the study of resonance. The PI will use small divisors that measure non-resonance to classify singularities of the complex structure arising in linear approximations of curved manifolds. The project will involve collaboration with researchers in an early career stage and will support the training of graduate students.

Motivated by recent counterexamples showing that smooth families of domains may be equivalent by a discontinuous family of biholomorphisms, the PI will study the existence of families of biholomorphisms between families of domains using biholomorphism groups and other analytic tools such as Bergman metrics. The PI will construct a global homotopy formula with good estimates for suitable domains in a complex manifold. One of the goals is to construct a global formula in cases when a local homotopy formula fails to exist. The PI will use such global homotopy formulas to investigate the stability of holomorphic embeddings of domains with strongly pseudoconvex or concave boundary in a complex manifold, when the complex structure on the domains is deformed. The PI will use this approach to investigate stability of global Cauchy-Riemann structures on Cauchy-Riemann manifolds of higher codimension. The project seeks a holomorphic classification of neighborhoods of embeddings of a compact complex manifold in complex manifolds via the Levi-form and curvature of the normal bundle. In addition, the PI will study the classification of Cauchy-Riemann singularities for real manifolds using methods from several complex variables and small-divisor conditions in dynamical systems.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2424015","Oscillatory Integrals and Falconer's Conjecture","DMS","ANALYSIS PROGRAM","03/01/2024","03/12/2024","Hong Wang","NY","New York University","Standard Grant","Marian Bocea","08/31/2024","$72,178.00","","hongwang@math.ucla.edu","70 WASHINGTON SQ S","NEW YORK","NY","100121019","2129982121","MPS","128100","","$0.00","The project is on the restriction theory in Fourier analysis. This field is concerns functions with Fourier transform (frequencies) supported (non-zero at most) on some curved objects such as a sphere or a cone. Such functions appear naturally in several areas of science and mathematics: in the study of Schrödinger equations, wave equations and number theory. For instance, a solution to the linear wave equation can be represented as a function with Fourier transform supported on a cone. Investigating these functions allows one to understand how waves evolve in time. In number theory, one can count the number of integer solutions to some Diophantine equations (polynomial equations with integer coefficients) by estimating such functions. Namely, if the corresponding functions are concentrated, then one expects the Diophantine equation to have many integer solutions. And an upper bound on the number of solutions can be given in terms of how spread out the functions are. This project will be focused on how the curvature of the Fourier support prevents the functions from being concentrated.

The work will be concentrated on oscillatory integrals and related to Falconer's conjecture. The latter is an unsolved question concerning the sets of Euclidean distances between points in compact d-dimensional spaces. The projects on oscillatory integrals concern the restriction conjecture, the Hormander operator, and decoupling questions. For the restriction conjecture, Stein's restriction conjecture will be studied in higher dimensions and in dimension three. For the Hörmander operator the Bochner-Riesz conjecture will be investigated by considering it as a Hörmander operator not satisfying Bourgain's ""generic failure"" condition. Work will be done on the dimension of radial projections with applications surrounding Falconer's conjecture.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2337344","Conference: Madison Lectures in Harmonic Analysis","DMS","ANALYSIS PROGRAM","02/01/2024","01/23/2024","Brian Street","WI","University of Wisconsin-Madison","Standard Grant","Wing Suet Li","01/31/2025","$50,000.00","Andreas Seeger, Betsy Stovall","street@math.wisc.edu","21 N PARK ST STE 6301","MADISON","WI","537151218","6082623822","MPS","128100","7556","$0.00","The Madison Lectures in Harmonic Analysis, a two-week program encompassing a week-long international conference, followed by a week-long summer school on Fourier analysis and related topics, will take place on the campus of the University of Wisconsin-Madison, May 13-23, 2024.

The conference, to take place during the first week, May 13-17, 2024, is devoted to recent developments in the field of harmonic analysis, and will also encompass its many applications to other mathematical areas. The conference will bring 26 internationally-recognized mathematicians to each give a 50-minute talk and participate throughout the week. It will also support the participation of a number of early career researchers. All participants will have the opportunity to present their research in a poster session running throughout the week, and an open problem session will focus on future directions of research. The Spring School will take place during the second week, May 19-23, 2024. It will be led by two young experts, Drs. Durcik and Roos, with 10 to 15 participants at the graduate student or postdoctoral level. The participants will prepare presentations from a list of important papers in the field in advance and during the Spring School, they will study the papers together under the guidance of Drs. Durcik and Roos which will introduce these junior researchers to important recent developments in the field and gain a broad understanding of this every active area of mathematics. These two events are connected for synergistic purposes; junior researchers will be invited to the conference to learn more from more senior researchers, where they can present their own work in a poster session, and they can stay for the summer school for a more in-depth study. Thus, organizing the two events consecutively will provide enhanced opportunities for junior researchers to network, share their results, and learn from their peers and more senior researchers. The organizers have dedicated a substantial portion of the conference budget to the participation of early-career mathematicians. They will make special efforts to invite and encourage the participation of women and members of other underrepresented groups.

The topics to be discussed at the conference will include, but will not be limited to, the following important developments in harmonic analysis in recent years: the Fourier restriction and Bochner?Riesz problems; decoupling inequalities and local smoothing problems in Fourier analysis and their applications, applications to geometric measure theory; inequalities of Brascamp-Lieb type; sparse domination inequalities and their application to weighted norm theory; analysis problems for the Hamming cube; subelliptic partial differential equations and generalized pseudo-differential operators. The spring school will focus on the exciting and active area of multilinear singular integrals with applications to ergodic theory.

More information may be found at the conference webpage, https://sites.google.com/wisc.edu/mlha-2024/home.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria." "2400046","Conference: 2024 Great Plains Operator Theory Symposium","DMS","ANALYSIS PROGRAM","05/15/2024","01/23/2024","David Pitts","NE","University of Nebraska-Lincoln","Standard Grant","Wing Suet Li","12/31/2024","$50,000.00","Allan Donsig, Christopher Schafhauser","dpitts2@math.unl.edu","2200 VINE ST","LINCOLN","NE","685032427","4024723171","MPS","128100","7556, 9150","$0.00","This grant will provide partial participants support for the 2024 Great Plains Operator Theory Symposium (GPOTS) conference at the University of Nebraska-Lincoln, June 3-7, 2024. GPOTS is the largest annual national conference in operator algebras and operator theory held in the United States. Operator algebras and operator theory are major research areas in mathematics, with many connections to other branches of mathematics and with applications across the sciences, particularly to quantum phenomena. Since its beginning in 1981, GPOTS has grown into a major international conference with attendance approximately between 100-150 each year. Researchers will learn about the latest developments and early-career participants will have the opportunity to share their work and connect with other researchers in this part of Mathematics.

The themes of the conference have connections both to many areas of mathematics, including dynamical systems, logic, group theory, ring theory, and to a range of applications, including in quantum phenomena, such quantum computing, and in mathematical physics. Topics which will be featured at GPOTS 2024 include: C*-algebras, operator spaces, operator theory, non-commutative geometry, von Neumann algebras, and quantum information theory. Aside from plenary lectures, GPOTS 2024 will have contributed talks, with many given by early-career researchers, graduate students and postdocs. A significant portion of the proposed funding will be used to support participation by early-career researchers. More information is available at https://math.unl.edu/events/special/gpots2024

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria."