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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" | ||
"2403557","Fundamental Gap Estimates and Geometry /Topology of Ricci Limit Spaces","DMS","GEOMETRIC ANALYSIS","07/01/2024","05/01/2024","Guofang Wei","CA","University of California-Santa Barbara","Continuing Grant","Qun Li","06/30/2027","$181,166.00","","[email protected]","3227 CHEADLE HALL","SANTA BARBARA","CA","931060001","8058934188","MPS","126500","","$0.00","Various problems of mathematical physics can be modeled by the Laplacian or more general Schrodinger equations. The difference of the first two eigenvalues of the Laplacian is referred to as the fundamental gap, which represents the energy needed to excite a particle from ground level to the next level in quantum mechanics. The principal investigator will estimate the fundamental gap for various spaces. The proposed activities are related to optimal transport, information geometry and discrete geometry. The project will also support educational activities and diversity through mentoring undergraduate and graduate students as well as postdocs; recruiting women and other underrepresented groups; organizing seminars, workshops and research programs promoting young scholars.<br/><br/>The project is centered around Riemannian geometry and geometric analysis with three parts. The first is about the fundamental gap estimates of the Laplacian with Dirichlet boundary conditions on a horoconvex domain in the hyperbolic space and convex domain in locally symmetric spaces by comparison with some suitable 1-dim model. The second concerns geometry and topology of spaces with Ricci curvature lower bound, especially the fundamental group of noncompact manifolds with nonnegative Ricci curvature; minimal volume entropy rigidity for metric measure spaces with curvature lower bounds. The last is to study integral curvature for the critical power.<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." | ||
"2405361","Variational Problems In The Theory of Minimal Surfaces","DMS","GEOMETRIC ANALYSIS","06/01/2024","05/02/2024","Giada Franz","MA","Massachusetts Institute of Technology","Standard Grant","Qun Li","05/31/2027","$194,991.00","","[email protected]","77 MASSACHUSETTS AVE","CAMBRIDGE","MA","021394301","6172531000","MPS","126500","","$0.00","A submanifold is called minimal if it is a critical point of the area functional. Minimal submanifolds are of central importance in differential geometry and arise naturally in mathematical physics, as soap films and black hole horizons, for example. Therefore, understanding their behavior is of great interest from the mathematical point of view but also for applications. The objective of this project is to take steps towards a full description of all minimal submanifolds in a given ambient manifold, inspired by the variational nature of these objects. The investigator will also conduct educational activities and practice community building, with particular attention to students and junior researchers.<br/> <br/>The project consists of three interwoven research lines. The first seeks new insights into the topological and analytical properties of minimal surfaces obtained via min-max constructions. The second line focuses on minimal surfaces with free boundary in the three-dimensional ball, with a focus on existence theorems and global properties. Finally, the project will investigate rigidity results for minimal submanifolds of higher codimension in ambient manifolds with 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." | ||
"2404992","Minimal Surfaces and Harmonic Maps in Differential Geometry","DMS","GEOMETRIC ANALYSIS","07/01/2024","05/02/2024","Daniel Stern","NY","Cornell University","Continuing Grant","Qun Li","06/30/2027","$76,032.00","","[email protected]","341 PINE TREE RD","ITHACA","NY","148502820","6072555014","MPS","126500","","$0.00","Solutions of geometric variational problems--objects which (locally) minimize natural notions of energy--play a central role in modern geometry and analysis, as well as physics, materials science, and engineering, where they characterize equilibrium states for various systems. Among the most important examples are harmonic maps, which arise in computer graphics and the study of liquid crystals, and minimal surfaces, which model soap films and the boundaries of black holes. The central goal of this project is to advance our understanding of the existence and structure of these objects, with an emphasis on connections to spectral geometry and certain geometric equations arising in particle physics. This project also involves the training of graduate students and postdocs, the organization of seminars and workshops on related topics, and dissemination of ideas to non-expert audiences through public lectures and survey articles.<br/><br/>This project concerns the existence and geometric structure of harmonic maps, minimal surfaces and minimal submanifolds of codimension 2 and 3, in relation to isoperimetric problems in spectral geometry and singularity formation for gauge-theoretic PDEs. With his collaborators, the PI will exploit new techniques for constructing extremal metrics for Laplacian and Steklov eigenvalues--developed in recent work of the PI with Karpukhin, Kusner, and McGrath--to produce many new minimal surfaces of prescribed topology in low-dimensional spheres and balls, and study related constructions of minimal surfaces in generic ambient spaces via mapping methods. Building on prior work with Pigati and Parise-Pigati, the PI will continue to investigate the relationship between the abelian Higgs model and minimal submanifolds of codimension two, and explore an analogous correspondence between the SU(2)-Yang-Mills-Higgs equations and minimal varieties of codimension 3. In another direction, the PI will further develop the existence and regularity theory for harmonic maps on manifolds of supercritical dimension n ? 3, combining variational methods with new analytic techniques to study the existence and compactness theory for harmonic maps with bounded Morse index into general targets.<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." | ||
"2404309","Rigidity Properties in Dynamics and Geometry","DMS","GEOMETRIC ANALYSIS","06/01/2024","05/01/2024","Ralf Spatzier","MI","Regents of the University of Michigan - Ann Arbor","Continuing Grant","Eriko Hironaka","05/31/2027","$204,000.00","","[email protected]","1109 GEDDES AVE, SUITE 3300","ANN ARBOR","MI","481091079","7347636438","MPS","126500","","$0.00","Dynamical systems and ergodic theory investigate the evolution of physical, biological or mathematical systems over time, such as turbulence in a fluid flow, the motions in planetary systems or the evolution of diseases. Fundamental ideas and concepts such as information, entropy, chaos and fractals have had a profound impact on our understanding of the world. Dynamical systems and ergodic theory have developed superb tools with applications to sciences and engineering. Symbolic dynamics, for example, has been instrumental in developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology. Geometry is a highly developed, ancient yet superbly active field in mathematics. It studies curves, surfaces and their higher dimensional analogs, their shapes, shortest paths, and maps between such spaces. Surveying the land for his principality, Gauss developed the fundamental notions of geodesics and curvature, laying the groundwork for modern differential geometry. It has close links with physics and applied areas like computer vision or the more current geometric and topological data analysis. Geometry and dynamics are closely connected. Indeed, important dynamical systems such as the geodesic flow come from geometry, and conversely one can use geometric tools to study dynamics. One main goal of this project is to study symmetries of dynamical systems, especially when one system is unaffected by the changes brought on by the other. The quest is to study these systems via unexpected symmetries. Important examples arise from geometry when the space contains many flat subspaces. Under additional assumptions, one can classify such spaces. Finally, group theory is introduced in both dynamics and geometry via the group of symmetries of a geometry or dynamical system. The principal investigator will continue training a new generation of researchers in mathematics, and students at all levels in their mathematical endeavors. This project includes support for research training opportunities for graduate students and summer research experiences for undergraduates.<br/><br/>This project will investigate rigidity phenomena in geometry and dynamics, especially actions of higher rank abelian and semi-simple Lie groups and their lattices. The latter is part of the Zimmer program. Particular emphasis will be put on hyperbolic actions of such groups. As higher-rank semisimple Lie groups and their lattices contain higher-rank abelian groups, the classification and rigidity problems for the abelian and semi-simple cases are closely related, with abundant cross-fertilization. The goal is the classification of such actions. Closely related are the study of automorphism groups of geometric structures. A further goal is to understand topological joinings of lattice actions on Furstenberg boundaries and the related problem of classifying discrete subgroups in semisimple ie groups with higher Prasad-Rapinchuk rank. Investigations in geometry will address higher-rank Riemannian manifolds and their classification, introducing novel methods. The dynamics of geodesic and frame flows will also be studied, with investigations of discrete subgroups of Lie groups for rank rigidity and measure properties. Besides establishing new results, the principal investigator also strives to find and introduce novel methods for investigating these problems which will lend themselves to applications in other areas.<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." | ||
"2403981","Scalar Curvature, Optimal Transport, and Geometric Inequalities","DMS","GEOMETRIC ANALYSIS","07/01/2024","05/01/2024","Simon Brendle","NY","Columbia University","Standard Grant","Qun Li","06/30/2027","$254,266.00","","[email protected]","615 W 131ST ST","NEW YORK","NY","100277922","2128546851","MPS","126500","","$0.00","This project focuses on questions at the intersection of differential geometry and the theory of partial differential equations. Differential geometry uses techniques from calculus to understand the shape and curvature of surfaces. These ideas can be generalized to higher-dimensional manifolds. In particular, they provide the mathematical framework for the Einstein equations in general relativity, which link the matter density to the curvature of space-time. A major theme in differential geometry has been to study the interplay between the curvature and the large-scale properties of a manifold. To study these questions, various techniques have been developed, many of them based on partial differential equations. Examples include the minimal surface equation and the partial differential equations governing optimal mass transport. This project is aimed at understanding these partial differential equations. This is of significance within mathematics. There are also connections with general relativity. Moreover, ideas from optimal transport have found important applications in statistics and computer science. The project also includes a variety of mentoring and outreach activities. <br/><br/>An important topic in geometry is to understand the geometric meaning of the scalar curvature. The PI recently obtained a new rigidity theorem for metrics with nonnegative scalar curvature on polytopes. The PI plans to extend that result to the more general setting of initial data sets satisfying the dominant energy condition. In another direction, the PI plans to work on geometric inequalities and optimal mass transport. On the one hand, the PI plans to use ideas from differential geometry and partial differential equations to study the behavior of optimal maps. On the other hand, the PI hopes to use ideas from optimal transport to prove new geometric inequalities. Ideas from optimal transport can be used to give elegant proofs of many classical inequalities, including the isoperimetric inequality and the sharp version of the Sobolev inequality. Moreover, the recent proof of the sharp isoperimetric inequality for minimal surfaces is inspired by optimal transport.<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." | ||
"2403728","Singularity, Rigidity, and Extremality Phenomena in Minimal Hypersurfaces","DMS","GEOMETRIC ANALYSIS","06/01/2024","05/01/2024","Christos Mantoulidis","TX","William Marsh Rice University","Standard Grant","Qun Li","05/31/2027","$215,057.00","","[email protected]","6100 MAIN ST","Houston","TX","770051827","7133484820","MPS","126500","","$0.00","Riemannian geometry is a modern version of geometry that studies shapes in any number of dimensions. Other than ""lengths"" and ""angles,"" its key notions also include ""minimal surfaces,"" which generalize the concept of a straight line, and ""curvature,"" which measures how a shape is bent. The principal investigator (PI) will study problems involving minimal surfaces and their curvature that arise from physical theories including Einstein?s general theory of relativity and the van der Waals?Cahn?Hilliard theory for phase transitions in multicomponent alloy systems. In addition to the research, this project will also support the PI's continued efforts to promote student learning and training through seminar organization, conferences, expository articles, and notes.<br/><br/>This project will specifically examine singularity, rigidity, and extremality phenomena in the theory of minimal surfaces. First, the PI will further investigate the structure of minimal surface singularities, meaning points of curvature blow-up, in area-minimization problems as well as their dynamic counterpart in mean curvature flow. Second, the PI will study enhanced rigidity properties of critical points in the van der Waals?Cahn?Hilliard phase transition theory, which can be thought of as diffuse variants of minimal surfaces. Third, the PI will study extremal behaviors of different quasi-local mass notions in general relativity, as seen through their interactions with scalar curvature and minimal surfaces, which correspond to energy density and boundaries of black hole regions.<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." | ||
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