Contact Geometry, Complex Analysis and Imaging

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A major current in analysis and topology for the last decade has been the generalization of constructions and results in complex algebraic geometry to the almost complex, symplectic category. In recent work, Dr. Epstein has shown that the dbar-Neumann boundary condition has a symplectic analogue for Spin-C manifolds with contact boundary. This work is based on the notion of tame Fredholm pairs of projectors. Dr. Epstein will further explore this category and these boundary value problems, to obtain explicit formulae for the index of the Spin-C Dirac operator and gluing formulae under various convexity conditions. He will also use this framework to investigate the unmodified dbar-Neumann problem on a strictly pseudoconvex symplectic manifold, to see if there is a reasonable analogue of the Bergman projection, and if its range defines a useful analogue of holomorphic functions in the non-compact symplectic context.In many mathematical and physical problems a question of principal interest is: how many solutions does a partial differential equation have? For certain classes of equations, a partial answer to this question, called the "index" of the equation, can be provided that does not depend on the details of equation. It is computed from the geometry of the space on which the solutions are defined. Dr. Epstein's work is broadly directly toward clarifying the relationship between this counting procedure and which aspects of the geometry of the underlying space are important determinants of the final result. One such approach to this problem involves cutting the space into parts and describing how the indices of the parts can be combined to compute the index of the whole.