Weakly Differentiable Mappings and Functions: Analysis, Geometry, and Topology

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

In 1900, David Hilbert presented a famous list of 23 open questions that had fundamental influence in the mathematics of the 20th century. Hilbert's 20th problem was about existence of generalized solutions of partial differential equations and variational problems. This was a crucial question since the theory was at a dead end: the classical notion of a differentiable function was not sufficient for further development. This led to one of the greatest discoveries of mathematics of 20th century: the theory of Sobolev spaces. Without Sobolev spaces, further development (both theoretical and practical) of nonlinear partial differential equations would not be possible. Recent decades showed however, that the scope of applications of Sobolev spaces goes far beyond the theory of partial differential equations; the theory applies to differential geometry, geometric group theory, algebraic topology, sub-Riemannian geometry, and analysis on metric spaces, just to name a few. This is a very active area of contemporary mathematics with many unsolved problems and new emerging areas of research. This research project concerns analytic, geometric, and topological properties of Sobolev functions and mappings as well as other related classes of mappings that have low differentiability regularity. The focus of the principal investigator is on finding new bridges between seemingly unrelated aspects of the fields of analysis, geometry, and topology. Graduate students and young researchers will be trained through research involvement in the project. In more detail, the principal investigator plans to explore the following topics. (1) Approximation of convex functions (which have second order distributional derivatives being Radon measures and thus low regularity from the differentiable viewpoint). (2) General theory of Sobolev extension domains. (3) Existence of translation-invariant operators (a question that does not deal with weakly differentiable functions). (4) Continuity of Orlicz-Sobolev mappings of finite distortion. (5) Regularity of Sobolev isometric immersions. (6) Sign of the Jacobian of a Sobolev homeomorphism. (7) Topologically nontrivial Kaufman-type counterexamples to the Sard theorem (while the Sard theorem deals with sufficiently smooth mappings, the principal investigator will investigate the case of mappings with low regularity). (8) Sobolev spaces on metric spaces. (9) Implicit function theorem for Lipschitz mappings into metric spaces. (10) Lipschitz homotopy groups of the Heisenberg groups. (11) Whitney extension theorem for contact mappings into the Heisenberg group. (12) Calculus of differential forms and Hölder continuous mappings with applications to the Heisenberg groups. (13) Preparation of a monograph about the Heisenberg 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.