Function Theory on Symmetric Spaces

对称空间函数论

• 批准号: 0100459
• 负责人: Adam Koranyi
• 金额: --
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100459&HistoricalAwards=false
• 关键词: Function; Theory; Symmetric; Spaces

AbstractKoranyiThe proposal is concerned with several lines of investigation, The first one of these is the mapping problem of domains in several complex variables with the aid of symplectic quasiconformal maps (with respect to the Bergman metric). It has already been proved that bounded simply connected smooth strongly pseudoconvex domains can always be mapped ontoeach other by such maps. The present project is concerned with developing geometric methods for computing or estimating the minimal qusiconformal distortion. The second line of investigation has to do with Poisson transforms of sections of vector bundles. Such Poisson transforms havebeen used to extend infinitesimally quasiconformal maps from the complex unit ball to the interior. They are now to be further investigated for their injectivity properties and for pairs of differential operators intertwined by them. Here the machinery of semisimple Lie groups will be used to its full capacity. A further subject to be studied is generalized conformal maps of generalized flag manifolds; it is expected that Liouville's classic theorem will extend to this situation. (This question is related to the preceding one through the notion of parabolic geometry.) In addition, three further, more or less related problems will be investigated; these concern analysis on two-step nilpotent groups, harmonic functions on spaces with negative curvature, and an applicationof the theory of reflection groups to statistics.The main goal of the proposal is to discover new mathematical facts in the field of analysis of functions. The theory of functions of one complex variable has been the most central field of mathematics in the last 150 years. It has innumerable applications to physics and othersciences. The theory of functions of several complex variables has had important successes but has not up to now matched in perfection and applicability the one-variable theory. The first goal of the proposal is to try to improve on this situation by studying the so-called mapping properties of functions of several complex variables. It will also be investigated to what extent the fundamental notions and facts of this theory can be extended to more general situations. The project also includes the investigation of some further related problems, one of these concerns an application to statistics. A secondary goal of the proposal is to understand more clearly the connections between certain already known facts in the main subject. This should make them easier to learn; in this way a contribution to mathematical education is expected to be made.

Korany这个建议涉及几条研究路线，其中第一条是借助于辛拟共形映射（关于Bergman度量）的多复变域的映射问题。 有界单连通光滑强伪凸域总是可以通过这类映射相互映射。 本项目涉及的是发展几何方法来计算或估计最小的拟共形失真。 第二条线的调查与泊松变换的部分向量丛。 这种Poisson变换已被用来将无穷小拟共形映射从复单位球扩展到内部。 现在，他们的内射性质和对他们交织的微分算子进行了进一步的研究。 在这里，半单李群的机制将被充分利用。 另一个要研究的课题是广义旗流形的广义共形映射;人们期望刘维尔的经典定理将扩展到这种情况。 (This这个问题通过抛物几何的概念与前一个问题相关。） 此外，三个进一步的，或多或少相关的问题将被调查;这些关注分析两步幂零群，调和函数的空间负曲率，和applicationof理论的反射群statistics.The主要目标的建议是发现新的数学事实，在该领域的分析职能。 在过去的150年里，一元复变函数理论一直是数学中最核心的领域。 它在物理学和其他科学中有无数的应用。 多复变函数理论已经取得了重要的成就，但迄今为止在完善性和适用性方面还不能与单变量理论相匹配。 该提案的第一个目标是试图通过研究多复变量函数的所谓映射特性来改善这种情况。 它也将调查到什么程度的基本概念和事实，这一理论可以扩展到更一般的情况。 该项目还包括调查一些进一步的相关问题，其中之一涉及统计应用。 该提案的第二个目标是更清楚地了解主要主题中某些已知事实之间的联系。 这应该使他们更容易学习;以这种方式作出贡献的数学教育预计。