Mathematical Sciences: Envelopes of Holomorphy and Holomorphic Motions

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Professor Slodkowski will investigate the structure of envelopes of holomorphy where he is especially interested in problems related to their construction and univalence. Some questions of existence of analytic discs and varieties in envelopes of holomorphy and of the uniqueness of Levi-flat hypersurfaces will also be addressed. Finding answers to these questions should be very helpful in solving a second set of problems involving the extendibility of holomorphic motions and related questions concerning the holomorphic axiom of choice. The areas under investigation here involve the theory of functions of several complex variables. The structure imposed by requiring that a function of several complex variables be differentiable (as in elementary calculus) is a very rich one. In particular, given a set in which such a function is defined, it is important to know the largest set to which all such functions can be extended maintaining their differentiability. This "envelope of holomorphy" is not well understood and one of Professor Slodkowski's projects is to develop methods to find it.