Applications of Quasiconformal Geometry and Partial Differential Equations

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Partial differential equations (PDE) are used to model important systems in various application areas; in particular, subelliptic PDE are helpful in settings 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 terms of broader impacts, the principal investigator will involve graduate and undergraduate students in several aspects of the research and will design outreach activities to attract K-12 students to mathematics.The technical focus of the proposed research addresses a curve-shrinking flow in Carnot groups, the study of harmonic extensions of quasiconformal mappings between boundaries of certain Gromov hyperbolic spaces, and regularity of certain nonlinear, degenerate parabolic PDE.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.