Representation Theory of Reductive Groups over Local Fields

所提出的研究是关于局部域上的约化群的表示理论，即像实数域或p-进数域上的可逆矩阵群这样的群。所提出的问题在自同构形理论和朗兰兹程序中有直接的应用。它们也有解析和几何方面。问题分为三个主题：调和分析、群胚的表示理论和可积性定理。以下是对这些主题的简要描述：PI将球面空间上的调和分析视为表示理论的推广。他的目标是将某些基本结果从表示论转移到球面空间上的调和分析领域。群的概念是群概念的有趣推广。PI建议在群仿的情况下重新建立p-进群的表示理论。可积性定理是D-模理论中的一个定理，它在实流形上的不变分布理论中有很强的应用，而不变分布理论又是表示理论的重要组成部分。然而，D-模的理论并不适用于p-进的情况。基于前面的部分结果，PI提出了一个类似于p元情形的可积性定理。所提出的项目是关于表示理论和调和分析的。这个项目中的问题的一个模型例子可以是傅立叶级数。傅立叶级数是圆上函数的虚指数和(与三角函数正弦和余弦密切相关)的分解。当你旋转圆周时，这些指数会以一种非常简单的方式变化。在某种意义上，PI研究的问题是这种结构在高维情形下的推广。一般来说，群论可以被视为研究数学对象的对称性，表示论-作为向量空间的对称性的研究，以及调和分析-作为具有对称性的几何对象上的函数空间的研究。所研究的几何对象是实流形和p-进流形。实流形是几何对象，在局部上看起来像一条直线、一个平面、一个三维空间(就像我们生活的那个空间)或一个更高维度的空间。P-进流形是实流形的某些类比。实群的表示理论和实空间上的调和分析在几何、分析以及后来的物理、信号处理、图像处理和生物学中有着广泛的应用。对数和实数在数论中都有许多重要的应用。更具体地说，在自同构形式理论和朗兰兹计划中。

The proposed research is on representation theory of reductive groups over local fields, i.e. groups like the group of invertible matrices over fields like the field of real or p-adic numbers. The proposed problems have direct applications in the theory of automorphic forms and in the Langlands program. They also have analytic and geometric aspects. The problems are divided into three topics: Harmonic analysis, representation theory of groupoids and the integrability theorem. Here are brief descriptions of those topics: The PI views harmonic analysis on spherical spaces as a generalization of representation theory. His aim is to transfer certain fundamental results from representation theory to the realm of harmonic analysis on spherical spaces.The notion of a groupoid is an interesting generalization of the notion of group. The PI proposes to re-build the representation theory of p-adic groups for the case of groupoids. The integrability theorem is a theorem from the theory of D-modules which has powerful applications in the theory of invariant distributions on real manifolds, which in turn is an important ingredient of representation theory. The theory of D-modules, however, is not applicable to the p-adic case. Based on a previous partial result, the PI proposes to provide an analog of the integrability theorem for the p-adic case.The propose project is about representation theory and harmonic analysis. A model example of the problems in this project can be the Fourier series. The Fourier series is a decomposition of a function on the circle as a sum of imaginary exponent (which are closely related to the trigonometric functions sine and cosine). These exponents change in a very simple way when you rotate the circle. The problems that the PI studies are, in a sense, a generalization of this construction for higher dimensional cases. In general, group theory can be viewed as the study of symmetries of mathematical objects, representation theory - as the study of symmetries of vector spaces, and harmonic analysis - as the study of spaces of functions over geometric objects that possess symmetries. The geometric objects that are studied are real and p-adic manifolds. Real manifolds are geometric objects that locally look like a line, a plain, a three dimensional space (like the one we live in) or a higher dimensional space. p-adic manifolds are certain analogues of real manifolds. Representation theory of real groups and harmonic analysis on real spaces have various applications in geometry, analysis and subsequently in physics, signal processing, image processing and biology. Both the p-adic and the real case have many important applications in number theory. More specifically, in the theory of automorphic forms and in the Langlands program.