Mathematical Sciences: Function Theory on Symmetric Spaces

Koranyi将继续他在Cauchy-Riemann流形背景下的拟共形映射的研究。项目的一部分是研究向量束上的泊松变换及其与齐次微分算子的相互作用。第二个方向将是研究调和函数的边界行为。第三个方向将是利用正定矩阵对称空间的调和分析来获得存在某些先验信息时高斯分布的最佳估计方法。本研究中涉及的分析依赖于李群理论，以挪威数学家索夫斯·李命名，一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李群表示理论对数学本身，特别是分析和数论，以及理论物理，特别是量子力学和基本粒子物理产生了深远的影响。

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.