Mathematical Sciences: Three Problems in Complex Analysis

小行星9401580 该奖项支持针对多个复变量领域中出现的问题的数学研究。 这项工作侧重于三个问题领域。 第一个问题涉及Stein空间的n维孤立奇点。 研究了任意相对紧的子集是否可以双全纯映射到同一维数的仿射代数簇。 如果奇点不是孤立的，则不一定是这种情况，因此在这个意义上，结果将是最好的。 研究的第二个目标是光滑有界伪凸域上的有限型点。 这些是边界上的点，在这些点上切空间的行为与欧几里得空间相当相似。 我们计划证明在每一点上都可以找到一个峰值函数，这些函数在给定的点上达到最大值（模），而在其他任何地方都没有。 最后，我们研究了具有严格多重次调和函数的光滑伪凸紧CR-流形。 然后，将做的工作表明，该流形可以扩展到一个光滑有界可积几乎完全流形，其中原始出现的内部边界。 几个复杂的变量增长从古典理论的职能在世纪之交成为一个独立的和深刻的纪律本身。 感兴趣的问题是显着不同的那些最初的动机领域：找到那部分的一个变量的理论推广到几个。 大多数的概括都被证明是错误的。 该主题现在侧重于应用于理解高维空间的几何，偏微分方程和控制理论的某些部分。 ***

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. ***