Geometric and Analytic Problems in Several Complex Variables

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One of the basic geometric and analytic problems in several complex variables is to determine when two real submanifolds in multidimensional complex space are locally biholomorphically equivalent. That is, when is it possible to find a local invertible holomorphic transformation sending one submanifold onto the other? This problem has attracted the attention of many mathematicians since the beginning of the twentieth century, starting with the work of Poincare and continuing with the major contributions of E. Cartan, Tanaka, Chern, Moser and others. The principal investigators will continue their research on several aspects of this problem. In particular, they will focus on determining when it is possible to reduce the biholomorphic equivalence problem to solving systems of polynomial equations with complex coefficients. They also plan to determine when a formal mapping sending a real submanifold into another is necessarily convergent. In addition, they will attempt to categorize those submanifolds for which such mappings are determined by finitely many derivatives at a given point. They expect that this study will lead to the discovery of new geometric, analytic, as well as algebraic invariants of these submanifolds.The Principal Investigators will continue their study of fundamental properties of analytic and geometric objects, such as surfaces and curves, in multidimensional complex spaces. A complete classification of these mathematical objects can have important implications for a number of other questions in mathematical science. In fact, this study is motivated by the rich interplay between several areas of mathematics and physics, including control theory, string theory, and other areas of mathematical physics. Progress on the problems proposed by the Principal Investigators will likely have impact on the above-mentioned areas as well.