Topics in quasiconformal mappings and subelliptic PDE

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SubRiemannian geometry and subelliptic Partial Differential Equations are used to model real life systems where there is a constrained dynamics. Examples of such systems include the motion of robot arms, structural functions of the first layer of the mammalian visual cortex, the Black-Scholes model for financial markets and quantum computing. Geometric and analytic properties of such spaces are captured in a quantitative fashion by studying the behavior of certain families of transformations of the space into itself. This project aims at studying fine properties of such transformations. In particular, the proposed research will provide a theoretical basis for implementing numerical simulations of real-life systems. In terms of broader impacts, the PI will involve graduate and undergraduate students in several aspects of the research and design outreach activities to attract K12 students to mathematics.The technical focus of the proposed research addresses the Liouville Theorem for quasiconformal mappings with minimal distortion. The main goal is to prove smoothness and rigidity of such mappings. The proposed approach is based on the solution of longstanding open problems of regularity for a class of subelliptic quasilinear PDE. The study of such PDE is important on its own and has broader applications. This project encompasses regularity for subelliptic p-laplacian beyond the Heisenberg group and construction of p-harmonic coordinates for subRiemannian manifolds.