Mathematical Sciences: Singular Unitary Representations and Cohomology of Arithmetic Manifolds

PI: Li Li将在Sarnak最近工作的基础上研究一种解决自同构形式耗尽问题的新方法。该方法不使用傅里叶系数，适用于小有理秩群，包括各向异性群。它涉及到矩阵系数的研究，p进群的小表示，和一些丢番图问题。李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是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-9501092 PI: Li Li will investigate a new approach to the problem of exhaustion of automorphic forms based on recent work by Sarnak. This approach does not make use of Fourier coefficients and is applicable to groups of small rational rank, including anisotropic groups. It involves the study of matrix coefficients, small representations of p-adic groups, and some Diophantine problems. 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.