Function Theory on Symmetric Spaces

Koranyi Koranyi将继续他对黎曼对称空间的调和分析的研究，重点是调和函数及其与复分析的关系。特别有趣的是解决对称空间上调和函数的局部Fatou定理的长期存在的问题。这将通过离散建筑的理论来实现。第二个研究方向将是在多个复变量中使用拟共形映射。这里主要关注的是关于Bergman度量的关于Bergman度量的辛拟共形映射从严格伪凸域的边界到内部的拟共形映射的扩张。这项研究涉及的分析基于李群理论，该理论以挪威数学家索菲斯·李的名字命名，是20世纪数学的主要主题之一。作为利用系统固有对称性的数学工具，李群表示理论对数学本身产生了深远的影响，特别是在分析和数论方面，对理论物理特别是量子力学和基本粒子物理产生了深远的影响。

Abstract Koranyi Koranyi will continue his investigations of harmonic analysis on Riemannian symmetric spaces with an emphasis on harmonic functions and relations with complex analysis. Of particular interest will be solving the long-standing problem of a local Fatou theorem for harmonic functions on symmetric spaces. This will be approached through the theory of discrete buildings. A second direction of research will be the use of quasiconformal maps in several complex variables. A primary concern here is the extension of quasiconformal maps of the boundary of a strictly pseudoconvex domain to the interior as symplectic quasiconformal maps with respect to the Bergman metric. The analysis involved in this research rests on the theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, which has beenone of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics especially quantum mechanics and elementary particle physics.