Function Theory on Symmetric Spaces

[摘要]本文涉及几个研究方向，其中第一个是借助（关于Bergman度量的）辛拟共形映射在几个复变量中的域的映射问题。已经证明了有界单连通光滑强伪凸域总是可以被这样的映射映射到彼此之间。本项目关注于发展计算或估计最小拟共形畸变的几何方法。研究的第二条线与向量束截面的泊松变换有关。这样的泊松变换已经被用来将无限小拟共形映射从复单位球扩展到内部。它们现在将进一步研究它们的注入性质和由它们交织的微分算子对。在这里，半单李群的机制将被充分利用。进一步研究的是广义标志流形的广义共形映射；预计刘维尔的经典定理将推广到这种情况。（这个问题通过抛物线几何的概念与前一个问题相关。）此外，还将进一步探讨三个或多或少相关的问题；讨论了两步幂零群的分析，负曲率空间上的调和函数，以及反射群理论在统计学中的应用。该提案的主要目标是在函数分析领域发现新的数学事实。单复变函数理论是近150年来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。复变函数理论虽取得了重要的成就，但在完善性和适用性方面仍不及单变量函数理论。该提案的第一个目标是试图通过研究几个复变量函数的所谓映射性质来改善这种情况。我们还将研究这一理论的基本概念和事实在多大程度上可以推广到更一般的情况。该项目还包括对一些进一步相关问题的调查，其中一个问题涉及统计学的应用。该建议的第二个目标是更清楚地理解主要主题中某些已知事实之间的联系。这将使它们更容易学习；通过这种方式，期望对数学教育做出贡献。

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.