Piotr Hajlasz has recently been funded by the US National Science Foundation (award DMS 2055171) for the proposal “Geometric Function Theory in Euclidean and Metric Spaces” for the period of July 15, 2021 – June 30, 2024.
While classical topology aims to study spaces and functions that are continuous but not smooth, analysis often deals with smooth objects like manifolds and smooth functions. Nonetheless, non-smooth functions such as Sobolev functions have been studied in analysis, but they were defined on smooth objects: domains in the Euclidean space, or more generally, Riemannian manifolds. Both the theories of classical topology and analysis have been constantly developed since the nineteenth century, building the framework for contemporary mathematics and its applications. However, in the last thirty years, we have witnessed the emergence of new directions in mathematics--analysis on metric spaces and quantitative topology--leading to new bridges between the distant worlds of non-smooth topology and smooth analysis. Analysis on metric spaces and quantitative topology is nowadays an active and independent field bringing together researchers from disparate parts of the mathematical spectrum. It has far-reaching applications to areas as diverse as geometric group theory, nonlinear partial differential equations, and even theoretical computer science. Moreover, analysis on metric spaces was recognized in the 2010 MSC classification as a category (30L: Analysis on metric spaces). The current project aims at investigating the broad spectrum of problems in analysis on metric spaces and quantitative topology while including other related problems in analysis.
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 Holder 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. It is often emphasized in the project how similar techniques can be successfully employed to seemingly unrelated problems in different areas of mathematics. More precisely Dr. Hajlasz will study problems in the followings areas: (1) Approximation of convex functions; (2) Sign of the Jacobian of Sobolev homeomorphisms with connections to topology and problems in calculus of variations; (3) Homotopy groups of spheres and geometry of mappings whose derivatives have low rank, with connections to problems in 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 Holder continuous mappings in the Heisenberg groups; (9) Lipschitz and Holder homotopy groups of the Heisenberg groups; (10) Whitney extension theorem for the Heisenberg group with applications to approximation of contact mappings; (11) Holder extension problem for mappings between the Heisenberg groups; (12) Sobolev spaces on Euclidean domains and on metric-measure spaces. (13) Existence of measurable differentiable structures.