Mathematical Sciences: Geometric and Analytic Problems in Several Complex Variables and Partial Equations

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DMS-9501516 Rothschild/Baouendi The investigators will study the geometry of real submanifolds in complex space. In particular, they will focus on geometric and analytic invariants of these manifolds under holomorphic changes of coordinates. Special consideration will be given to those manifolds globally defined by real polynomial equations. For many manifolds defined by the vanishing of real polynomials, any component of a holomorphic transformation from one such manifold to another must also be a root of a complex polynomial. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.