Geometric Function Theory in Euclidean and Metric Spaces

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Calculus has a myriad of applications in science. In the nineteenth and twentieth centuries the notions of calculus were studied for functions defined on general spaces called smooth manifolds. The last thirty years have witnessed the emergence of new directions in mathematics—calculus, now called analysis, on more general spaces, called metric spaces—leading to new bridges between the distant worlds of non-smooth and smooth spaces. Analysis on metric spaces, together with quantitative topology, is nowadays an active and independent field bringing together researchers from disparate parts of the mathematical spectrum. It has far-reaching applications. The current project aims at investigating a broad spectrum of questions in analysis on metric spaces and quantitative topology while including other related topics in analysis. The work emphasizes how similar techniques can be successfully employed to answer seemingly unrelated questions in different areas of mathematics. The project also includes the training of graduate students.The common themes of the project are the analytic, geometric, and topological aspects of the theory of functions and mappings with low order of regularity (convex functions, Sobolev functions, Lipschitz and Hölder continuous mappings, mappings that are one time continuously differentiable, etc.). Such mappings appear in several areas of contemporary mathematics, and the project attempts to create bridges between different areas of analysis, geometry, and topology. More precisely the investigator will study topics in the following areas: (1) Approximation of convex functions; (2) Sign of the Jacobian of Sobolev homeomorphisms with connections to topology and the calculus of variations; (3) Homotopy groups of spheres and geometry of mappings whose derivatives have low rank, with connections to quantitative topology; (4) Approximation of mappings whose derivatives have low rank; (5) Area and coarea formulas in metric spaces; (6) Generalization of the implicit function theorem to metric spaces; (7) Quantitative implicit function theorem and factorization through trees; (8) Analytic properties of Hölder continuous mappings in the Heisenberg groups; (9) Lipschitz and Hölder homotopy groups of the Heisenberg groups; (10) Whitney extension theorem for the Heisenberg group with applications to approximation of contact mappings.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.