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Prof. Leon Ehrenpreis Prof. Ehrenpreis is a renowned mathematician working in a number of different areas of mathematics including harmonic analysis, partial differential equations, complex analysis, number theory, group representations, and integral geometry. He is the author of over 70 peer-reviewed journal papers and books. “In his writing, one can find analytic ideas applied in the context of number theory, geometric thinking within analysis, transcendental number theory applied to partial differential equations, and so forth. He is a mathematician gifted with an uncommon ability to shed light upon one area of mathematics by bringing to bear concepts from another. Many basic notions, theorems and proofs in these subjects have been strongly shaped by Ehrenpreis” (from the volume: Analysis, Geometry, Number Theory: The mathematics of Leon Ehrenpreis, Contemporary Mathematics, 251. American Mathematical Society, 2000). |
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LECTURE Abstract: The first striking difference between holomorphic functions of 1 and several complex variables is Hatrogs' result that a functions defined and holomorphic in the neighborhood of the boundary of a ball in C^n (n>1) can be extended to be holomorphic in the interior. This is certainly not the case for functions of 1 complex variable. The difference between 1 and >1 can be attributed to the fact that holomorphicity for functions of 1 complex variable is defined by a single differential equation (Cauchy- Riemann) while a holomorphic function of several complex variables satisfies several Cauchy- Riemann equations. Thus Hartogs' theorem applies to solutions of certain systems of partial differential equations, namely, those that are overdetermined (more equations than functions). The main thrust of the talk is to sharpen Hartogs' theorem on various ways. For example, the ball can be replaced by certain noncompact regions. One of the most interesting regions is the union of the sets S+ and S- where S+ is the region for which the imaginary parts of the variables are all positive (and S- is defined similarly). These regions are called "wedges" and their intersection (real space) is called "edge". The resulting theorem is called the "edge of the wedge theorem". This will be extended to overdetermined systems of differential equations and some very sharp forms of the theorem will be introduced. . |
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