Mathematical Sciences: Three Problems in Complex Analysis

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9401580 Catlin This award supports mathematical research directed at problems arising in the field of several complex variables. The work focuses on three problem areas. The first concerns Stein spaces of dimension n with isolated singularities. It is proposed to investigate whether or not any relatively compact subset can be mapped biholomorphically into an affine algebraic variety of the same dimension. This is not necessarily the case if the singularities are not isolated, thus the result would be best possible in this sense. The second goal of the research concerns points of finite type on smoothly bounded pseudoconvex domains. These are points on the boundary at which the tangent space acts reasonably like Euclidean space. It is planned to show that at each point a peaking function can be found. These are functions which achieve their maximum (modulus) at the given point and nowhere else. Finally, it is planned to examining smooth pseudoconvex compact CR-manifolds with strictly plurisubharmonic functions. Then, work will be done to show that the manifold can be extended to a smoothly bounded integrable almost complete manifold in which the original appears as the inside boundary. Several complex variables grew from the classical theory of functions at the turn of the century into a separate and profound discipline of its own. The problems of interest are significantly different from those which originally motivated the field: to find that part of the one-variable theory which generalized to several. Most of the generalizations turned out to be false. The subject now focuses on applications toward understanding the geometry of higher dimensional spaces, partial differential equations and certain parts of control theory. ***