Mathematical Sciences: Three Problems in Complex Analysis

9401580卡特林该奖项支持针对在几个复变量领域中出现的问题的数学研究。这项工作集中在三个问题领域。第一个是关于具有孤立奇点的n维Stein空间。研究是否有相对紧致的子集可以双全纯地映射到同一维的仿射代数簇。如果奇点不是孤立的，情况就不一定是这样，因此从这个意义上来说，结果可能是最好的。研究的第二个目标是关于光滑有界伪凸域上的有限型点。这些是边界上的点，切线空间在这些点上的行为与欧几里得空间相当。计划证明在每个点上都可以找到一个峰值函数。这些函数在给定点达到最大值(模数)，而不是在其他地方。最后，我们计划研究具有严格重次调和函数的光滑伪凸紧致CR-流形。然后，我们将证明流形可以被推广到光滑有界的几乎完全的可积流形，其中原始流形看起来是内边界。几个复杂的变量从世纪之交的经典函数理论发展成为一门独立而深刻的学科。感兴趣的问题与最初激励该领域的问题有很大不同：找到概括为几个变量的单变量理论的那一部分。事实证明，大多数结论都是错误的。这门学科现在集中在理解高维空间的几何、偏微分方程式和控制理论的某些部分方面的应用。***

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. ***