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Geometric-Analysis/Awards-Geometric-Analysis-2024.csv

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 "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"
-"2411029","Curvature, Metric Geometry and Topology","DMS","GEOMETRIC ANALYSIS","07/01/2024","06/03/2024","Krishnan Shankar","VA","James Madison University","Standard Grant","Qun Li","06/30/2027","$142,565.00","","[email protected]","800 S MAIN ST","HARRISONBURG","VA","228013104","5405686872","MPS","126500","","$0.00","The PI?s focus is the study of objects in dimensions higher than three that admit positive or non-negative curvature.  Intuitively one may think of positive curvature in the following manner: On the surface of Earth any two longitudes from the North pole appear to bend towards each other and indeed they meet at the South pole. This is true of all points on Earth if we imagine longitudes emanating from each point. Because of this, we say that the surface of Earth has positive curvature everywhere. By the same token, a saddle has negative curvature at the point where the rider sits while a flat table has zero curvature. In higher dimensions, matters are far less visually apparent. One deals almost exclusively with equations and sophisticated geometrical techniques that describe the curvature of manifolds, a term that refers to objects that, roughly speaking, have no sharp edges. Manifolds of bounded size are called compact manifolds.  One of the great mysteries in the study of positive or non-negative curvature is the dearth of examples. The techniques at hand are few and the number of known examples remains relatively small.  In this project the PI aims to study positively and non-negatively curved manifolds with proposed new methods of construction. The project also presents several broader impact activities including outreach, inclusivity, and undergraduate research.<br/> <br/> <br/>The study of manifolds with positive or non-negative sectional curvature has a long history with roots as far back as Felix Klein in the late nineteenth century.  The PI's work for many years has been to try and construct new examples while attempting to prove rigidity theorems in the presence of additional hypotheses.  Complete manifolds of non-negative sectional curvature fall into two broad categories: compact and non-compact. In the non-compact case we have the beautiful Soul theorem of Cheeger and Gromoll:  A complete, non-compact manifold with non-negative sectional curvature is diffeomorphic to a vector bundle over a closed, totally convex submanifold.  In the compact case, Gromov's theorem restricts the topology sharply:  The total Betti number is bounded by a constant depending only on the dimension.  Beyond these theorems the question remains: which closed manifolds admit metrics of non-negative sectional curvature?  In this project the PI continues their recently successful research program on constructing new manifolds with non-negative curvature as well as a new proposed method for constructing positive curvature.<br/><br/>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."
+"2405266","RUI:  Quotient Spaces and the Double Soul Conjecture","DMS","GEOMETRIC ANALYSIS","07/01/2024","06/07/2024","Jason DeVito","TN","University of Tennessee Martin","Standard Grant","Qun Li","06/30/2027","$136,007.00","","[email protected]","304 ADMINISTRATION BLDG","MARTIN","TN","382380001","7318817015","MPS","126500","","$0.00","The Principal Investigator will investigate shapes called manifolds which are of particular importance due to the critical role they play in Einstein's general theory of relativity.  His main focus will be on a special class of manifolds which are called double disk bundles.  The PI intends to increase our understanding of the relationships which exist between double disk bundles and other important classes of manifolds.   The PI will accomplish this via projects ranging in scope from undergraduate research experiences to international collaborations.<br/><br/>The PI's work, motivated by the Double Soul Conjecture and the recent introduction of codimension one biquotient foliations, seeks to improve our understanding of double disk bundles both topologically and geometrically.  From the topological side, he intends to increase our understanding of when certain spaces of geometric interest, such as homogeneous spaces and biquotients, carry a double disk bundle structure.  On the geometric side, he intends to use double disk bundle structures in the construction of new examples of manifolds with interesting Riemannian metrics.<br/><br/>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."
 "2405440","Rigidity and Flexibility through Group Actions","DMS","GEOMETRIC ANALYSIS","07/01/2024","06/03/2024","Kurt Vinhage","UT","University of Utah","Standard Grant","Eriko Hironaka","06/30/2027","$207,485.00","","[email protected]","201 PRESIDENTS CIR","SALT LAKE CITY","UT","841129049","8015816903","MPS","126500","","$0.00","A traditional dynamical system is a time lapse of a space that describes the motion of points. The time lapse can be in a single, discrete time step or a continuous time flow. One natural way in which dynamical systems can be considered the same, called conjugacy, is through change of coordinates. That is, two dynamical systems are conjugate if there is an equivalence between the spaces which connects the way in which time steps are made. One of the central classification questions in dynamics is to classify dynamical systems up to conjugacy. This question has variations based on what it means for two systems to be equivalent, usually taking the forms of measurable, continuous and smooth equivalences. The goal of the proposal is to study the classification question from various perspectives, including generalizing the notion of a dynamical system to a group action, understanding possible values for conjugacy invariants and relaxing the notion of conjugacy to allow for time reparameterization. The proposal also includes work with students at various levels to deepen the collective understanding.<br/><br/>The proposal aims to capitalize on momentum in 3 key areas: smooth rigidity for actions of abelian groups and higher-rank semisimple Lie groups, Kakutani equivalence for flows and group actions, and flexibility for conjugacy invariants. Each of these questions is related to a classification question, the first working toward the Katok-Spatzier conjecture and Zimmer program, the second being an extension of results about Kakutani equivalence of parabolic flows to the setting of abelian group actions, and the third being a natural extension of the seminal work of Erchonko-Katok describing the possible values for topological and metric entropy for geodesic flows on surfaces.<br/><br/>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."
+"2411029","Curvature, Metric Geometry and Topology","DMS","GEOMETRIC ANALYSIS","07/01/2024","06/03/2024","Krishnan Shankar","VA","James Madison University","Standard Grant","Qun Li","06/30/2027","$142,565.00","","[email protected]","800 S MAIN ST","HARRISONBURG","VA","228013104","5405686872","MPS","126500","","$0.00","The PI?s focus is the study of objects in dimensions higher than three that admit positive or non-negative curvature.  Intuitively one may think of positive curvature in the following manner: On the surface of Earth any two longitudes from the North pole appear to bend towards each other and indeed they meet at the South pole. This is true of all points on Earth if we imagine longitudes emanating from each point. Because of this, we say that the surface of Earth has positive curvature everywhere. By the same token, a saddle has negative curvature at the point where the rider sits while a flat table has zero curvature. In higher dimensions, matters are far less visually apparent. One deals almost exclusively with equations and sophisticated geometrical techniques that describe the curvature of manifolds, a term that refers to objects that, roughly speaking, have no sharp edges. Manifolds of bounded size are called compact manifolds.  One of the great mysteries in the study of positive or non-negative curvature is the dearth of examples. The techniques at hand are few and the number of known examples remains relatively small.  In this project the PI aims to study positively and non-negatively curved manifolds with proposed new methods of construction. The project also presents several broader impact activities including outreach, inclusivity, and undergraduate research.<br/> <br/> <br/>The study of manifolds with positive or non-negative sectional curvature has a long history with roots as far back as Felix Klein in the late nineteenth century.  The PI's work for many years has been to try and construct new examples while attempting to prove rigidity theorems in the presence of additional hypotheses.  Complete manifolds of non-negative sectional curvature fall into two broad categories: compact and non-compact. In the non-compact case we have the beautiful Soul theorem of Cheeger and Gromoll:  A complete, non-compact manifold with non-negative sectional curvature is diffeomorphic to a vector bundle over a closed, totally convex submanifold.  In the compact case, Gromov's theorem restricts the topology sharply:  The total Betti number is bounded by a constant depending only on the dimension.  Beyond these theorems the question remains: which closed manifolds admit metrics of non-negative sectional curvature?  In this project the PI continues their recently successful research program on constructing new manifolds with non-negative curvature as well as a new proposed method for constructing positive curvature.<br/><br/>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."
 "2428771","RUI: Configuration Spaces of Rigid Origami","DMS","GEOMETRIC ANALYSIS","04/15/2024","04/22/2024","Thomas Hull","PA","Franklin and Marshall College","Continuing Grant","Christopher Stark","08/31/2025","$59,029.00","","[email protected]","415 HARRISBURG AVE","LANCASTER","PA","176032827","7173584517","MPS","126500","9229","$0.00","Origami, the art of paper folding, has been practiced for centuries. The mathematics behind origami, however, is not yet fully understood. In particular, some origami models can be folded and unfolded in such a way that we could make the crease lines be hinges and the paper between them stiff like sheet metal. Such models are called rigidly flexible origami and have applications that span the physical and biological sciences, ranging from unfolding solar sails to collapsible heart stents. This project will add mathematical tools that allow industrial applications to employ cutting-edge research, from large-scale architectural structures to nano-scale robotics driven by origami mechanics. The tools from this project will help design self-foldable structures. Currently self-folding designs in engineering, architecture, and the biological sciences involve building physical models in a trial-and-error approach, wasting time and resources. The self-folding research provided by this project will allow designers to avoid pitfalls and tighten the design-to-realization process significantly. In addition to the research component, the PI shall organize a diverse range of educational activities including in-service teacher training and education, undergraduate mentoring and preparation for graduate school; high-school and undergraduate classes on the mathematics of folding; for the public, general-audience articles, lectures, and exhibitions. This will increase interest in STEM fields through the fun, hands-on nature of origami while simultaneously disseminating project results.<br/><br/>The methods of this project involve a blend of practical experimentation with theory. Programmed self-foldability of structures will be achieved by trimming away undesired paths from the configuration space of all possible rigid foldings. One approach is to transform a given rigid folding of a crease pattern into a kinematically equivalent rigid folding with fewer degrees of freedom. The PI has proposed such a transform and will develop others. Key to all of this, however, is gaining a better understanding of rigid origami configuration spaces, which are algebraically complicated and not well understood. The project seeks to understand, and exploit, local-to-global behavior that is present in many known examples of rigid origami. In these examples approximating the configuration space near the origin (the unfolded state) leads to exact equations for the global configuration space. Formulating rigid origami configuration spaces in this way will add insight into the general field of flexible polyhedral surfaces, as well as provide the data needed to prove the feasibility of origami crease pattern transforms and design reliably self-foldable origami mechanisms.<br/><br/>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."
 "2404529","Semi-global Kuranishi Structures in Symplectic Field Theory","DMS","GEOMETRIC ANALYSIS","07/01/2024","05/22/2024","Erkao Bao","MN","University of Minnesota-Twin Cities","Continuing Grant","Swatee Naik","06/30/2027","$110,904.00","","[email protected]","200 OAK ST SE","MINNEAPOLIS","MN","554552009","6126245599","MPS","126500","9251","$0.00","Contact manifolds are a special type of space that naturally emerges in various contexts. For instance, they are used to describe the orbital paths of satellites. Taking the satellite scenario, for example, a natural question to ask is whether there are recurring orbits that a satellite can traverse. Contact homology offers a systematic way to explore the geometric features of these contact manifolds. In the satellite scenario, one can gain deep insights into all possible orbital paths of satellites by analyzing recurring satellite orbits. Contact homology has been highly successful in distinguishing different contact manifolds. This project aims to refine our understanding of contact manifolds using an enhanced approach based on contact homology. Furthermore, the PI intends to apply the techniques developed in this process to study other types of spaces beyond contact manifolds, such as spaces with symmetries. Symmetries play a crucial role in many aspects of daily life. For instance, ensuring that machine learning models treat people fairly, irrespective of their gender or race, reflects a key symmetry requirement. To address these issues, the PI will conduct an REU (Research Experience for Undergraduates) program focused on developing symmetrical neural networks to mitigate gender and racial biases.<br/><br/>Symplectic Field Theory, introduced two decades ago, aims to provide invariants for symplectic and contact manifolds. It encompasses essential concepts such as cylindrical contact homology, contact homology, chain homotopy types of contact differential graded algebras (dga), and linearized contact homology. The projects presented here revolve around the foundational aspects of Symplectic Field Theory. The primary challenge addressed by the Principal Investigator (PI) concerns achieving transversality while preserving symmetries to derive the desired algebraic formula. Various tools and techniques, including obstruction bundle gluing and evaluation maps for cylindrical contact homology, and semi-global Kuranishi structures for contact homology, have been introduced or employed by the PI. The project applies these tools to investigate two specific invariants: the chain homotopy type of contact dga and linearized contact homology. Furthermore, the PI has developed a new tool, the semi-global Kuranishi structure for clean intersections, which lies between obstruction bundle gluing and semi-global Kuranishi structures. This tool offers increased computational efficiency and bridges seemingly unrelated techniques. The project also aims to establish a Smith-type rank inequality related to the Floer homology of the fixed point set, contributing to the understanding of the L-space conjecture.<br/><br/>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."
 "2350423","Conference: Moving to higher rank: from hyperbolic to Anosov","DMS","GEOMETRIC ANALYSIS","07/01/2024","01/30/2024","Ilesanmi Adeboye","CT","Wesleyan University","Standard Grant","Eriko Hironaka","06/30/2025","$40,000.00","Sara Maloni","[email protected]","237 HIGH ST","MIDDLETOWN","CT","064593208","8606853683","MPS","126500","7556","$0.00","This award supports participation of US based mathematicians in the conference entitled ""Moving to higher rank: from hyperbolic to Anosov,"" which will take place in Centraro, Italy, from July 15- 19, 2024. The conference will bring together researchers and students from the classical field of hyperbolic geometry and the more recent area of higher Teichmuller theory to explore and further develop the rich connection between them. The conference will facilitate the exchange of ideas, and promote collaboration between experts in both fields, while reinforcing cooperation between the US and European mathematical communities. The organizing committee will encourage and support broad and diverse participation, and the training of the new generation of researchers.<br/><br/>In recent decades, the areas of hyperbolic geometry and Higher Teichmuller theory have undergone a dynamic convergence of concepts, attracting numerous scholars from hyperbolic geometry who have shifted their focus toward higher rank phenomena. Concurrently, a new generation of researchers has emerged, working at the juncture of these two domains. The conference will focus on how phenomena from hyperbolic geometry generalize to higher Teichmuller theory. Past success along these lines includes generalizations of Fenchel-Nielssen coordinates, Weil- Petersson geometry, Collar Lemmas, Length rigidity, and Patterson-Sullivan Theory. The conference will feature 18 research talks, and two lightening talk sessions for junior researchers. The URL for the conference website is https://tinyurl.com/hyp2anosov.<br/><br/>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."