Linear and nonlinear problems in CR manifolds

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This project focuses on the study of systems of linear first-order partial differential equations and nonlinear problems that exploit the results and methods of analysis on CR-manifolds. The systems of equations under investigation arise from locally integrable structures, a good model for which is a CR-manifold. The tools that are of potential use include the Baouendi-Treves approximation scheme, ideas from microlocal analysis (e.g., the FBI transform), analytic discs, the geometry of Sussmann's orbits, and the techniques developed in the theory of holomorphic extendability of CR-functions on semirigid CR-manifolds. Potential application areas for the research in this project include partial differential equations and geometry. Some of the nonlinear partial differential equations under consideration surface in meteorology, in attempts to model and explain atmospheric phenomena. The same equations describe the evolution of the amplitudes of nonlinear waves in elastic solids. Understanding the propagation and interaction of nonlinear waves is a very important and challenging problem in physics. In geometry and physics, these equations arise in the study of the shapes of surfaces in space that turn up as limit shapes in a class of random surface models, limit shapes that minimize surface tension. The research activity is also expected to generate interesting problems for graduate students.