Mathematical Sciences: Function Theory on Symmetric Spaces

DMS-9500794 PI：Koranyi Koranyi将在柯西-黎曼流形的背景下继续他对拟共形映射的研究。该项目的一部分是研究向量丛上的泊松变换及其与齐次微分算子的相互作用。第二个方向是研究调和函数的边界行为。第三个方向是利用正定矩阵对称空间的调和分析，在存在一些先验信息的情况下，得到高斯分布的最佳估计方法。这项研究涉及的分析基于李群理论，李群是以挪威数学家索菲斯·李的名字命名的，一直是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，李群的表示理论对数学本身，特别是在分析和数论方面，以及对理论物理，特别是量子力学和基本粒子物理，都产生了深远的影响。

DMS-9500794 PI: Koranyi Koranyi will continue his investigation of quasiconformal maps in the context of Cauchy-Riemann manifolds. Part of the project is the study of Poisson transforms on vector bundles and their interaction with homogeneous differential operators. A second direction will be the study of boundary behavior for harmonic functions. A third direction will be the use of the harmonic analysis of the symmetric space of positive definite matrices to obtain methods of best estimation of a Gaussain distribution in the presence of some prior information. The analysis involved in this research rests on the theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one 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.