Collaborative Research: FRG: Geometric Function Theory: From Complex Functions to Quasiconformal Geometry and Nonlinear Analysis

University Of Washington, Seattle WA

Investigators

Abstract

FRG Geometric Function Theory is a broad area of mathematics that has its roots in the classical theory of analytic functions of one complex variable. From the very beginning this field has had connections to potential theory, partial differential equations, the calculus of variations, and geometric topology. The second half of the twentieth century brought about new areas like quasiconformal and quasiregular mappings, with links to nonlinear PDEs and harmonic analysis. The research group is planning to tackle some of the most important open problems in this broadly construed field by using our diverse strengths. Examples of the problems include understanding the integrability properties of derivatives of conformal mappings, finding criteria for recognizing metric spaces up to bi-Lipschitz or quasiconformal equivalence, further developing the theory of holomorphic curves and its quasiregular generalizations, and investigating algebraic conditions related to quasiconvexity of energy functionals. The intellectual merit of our activity will be found in a deepened understanding of fundamental questions in Geometric Function Theory, in an increase of the links to other fields of mathematics, and in a broader scope of possible applications. Core mathematics keeps reappearing outside its own realm with dramatic success and consequences. Recent examples range from cosmology (where deep topological issues arise regarding the proposed new dimensions for the universe) to material science (where deformation of elastic bodies are studied by methods of the Calculus of Variations) to engineering (where function theoretic methods have led to advances in control theory). The latter two examples are directly connected with the work of our research group, as are topological issues pertaining to the geometry of three dimensional spaces. Another new feature with possible far reaching reverberations is to use function theoretic methods in studying spaces that are not smooth in the classical sense; such spaces naturally occur when Riemannian structures degenerate and form singularities. The main strength of our group is that its members have common roots, but multifarious interests, so as to make advancement in and connections between separate fields. The broader impact of our activity will be the education of new scholars who understand the methods and techniques in the field, and who know how to find applications of their knowledge to other parts of mathematics and sciences. We will put great weight on passing on the important questions to the younger generation and on enabling them to perform independent research.