Mathematical Sciences: Three Problems in Complex Analysis

数学科学：复分析中的三个问题

• 批准号: 9401580
• 负责人: David Catlin
• 金额: $ 15万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401580&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Three; Problems; Complex

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. ***

[401580]卡特林该奖项支持针对几个复杂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n维的Stein空间，它具有孤立的奇点。研究是否有相对紧的子集可以被生物全纯地映射为相同维数的仿射代数变体。如果奇点不是孤立的，则不一定是这种情况，因此在这种意义上，结果将是最佳可能的。研究的第二个目标涉及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非常相似。计划表明，在每一点上都可以找到一个峰值函数。这些函数在给定点上达到最大值（模数），而在其他点上没有。最后，计划研究具有严格多次调和函数的光滑伪凸紧cr -流形。然后，证明了该流形可以推广为光滑有界的可积几乎完全流形，其中原流形为内边界。几个复变量在世纪之交从经典的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推动该领域的问题有很大的不同：找到可以推广到几个变量的单变量理论的那一部分。大多数的归纳结果都是错误的。该学科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。* * *