The Regularity of Cauchy-Riemann Mappings and Solutions of Systems of Nonlinear Partial Differential Equations

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The aim of this research project is to study the solutions of systems of partial differential equations that arise in several complex variables. Important mathematical applications of this research include establishing the regularity of solutions and the convergence of solutions for which only Taylor series coefficients are known. Important applications to physical systems include equations that arise in statistical mechanics and meteorology. For example, the planar dimer model is a statistical mechanical model of certain two-dimensional surfaces in space that is a higher dimensional generalization of the simple random walk on integers. The limit shape of such surfaces is modeled by a partial differential equation closely related to the complex Burgers' equation studied in this project. The same equation is also relevant in the study of the quasi-geostrophic equations that model the behavior of certain atmospheric phenomena. The main part of the project involves the problem of understanding geometric conditions that imply that a sufficiently smooth CR mapping between two Cauchy-Riemann (CR) manifolds is smooth when the manifolds are smooth, and real analytic when the manifolds are real analytic. Closely related problems that will be investigated include the local and microlocal regularity of CR functions on abstract CR manifolds. The tools that may be used include a new family of Fourier-Bros-Iagolnitzer transforms, analytic discs, and the methods developed in the theory of holomorphic extendability of CR functions on hypersurfaces.