Mathematical Sciences: Quasiregular Mappings and the Heat Equation

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This project continues mathematical research on quasiregular mappings and the heat equation. Quasiregular mappings were introduced during the past quarter century to describe transformations of the plane and space whose distortion (dilatation) about any point is bounded. Quasiconformal maps have the same definition except that they are required to be univalent. Quasiregular maps are closely related to questions of uniqueness in the theory of partial differential equations. This project will focus on investigations into weak solutions of elliptic and parabolic differential equations and the relationship between the branch set of an entire quasiregular or quasimeromorphic function and its value distribution. Work will also continue on singular integrals and the heat equation for certain time varying domains in space. The one space variable case is now well understood. Current plans are to carry forward recent results into the multidimensional case. The goals are to study the Dirichlet and Neumann problems on these domains and seek to determine the mutual absolute continuity of parabolic measure with respect to a certain projective Lebesgue measure. This will use the method of layer potentials, the David buildup scheme and singular integral estimates. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is.