CAREER: Representation Theory on Curves

几何朗兰兹程序在曲线束的代数几何中提出了调和分析的非凡类比，其灵感来自朗兰兹哲学，该哲学将调和分析与伽罗瓦理论结合在数域上。在此几何调和分析中，将函数空间替换为轮轴的范畴，并利用几何傅里叶变换分析了这些轮轴上算子的谱理论。研究者正在将这些前沿思想应用于表征理论中的经典问题。特别是，在与D. Nadler合作中，他提出了对几何朗兰兹程序的改进，该程序对实李群的表示范畴进行谱分解。这大大增强了不可约表示的朗兰兹分类（经典朗兰兹程序的一部分），并将其与表示的几何（Borel-Weil）实现结合起来。研究者还带头努力将这些材料提供给更广泛的受众，特别是通过组织GRASP（几何表示和一些物理），以电子方式分发关于这个快速发展领域的基本思想和潜在趋势的说说性讲座系列。现代数学的一个基本主题是利用对称作为一种组织原则，在一个优雅的总体框架中将各种各样的、可能令人困惑的现象联系起来。也许这一趋势的主要例子是朗兰兹程序，它确定了代数和数论中对称性的一般模式，并将费马大定理的解列为其成功之一。近年来，出现了一种新的应用朗兰兹哲学的几何背景，它与新的对称原理，特别是那些在物理学中令人兴奋的弦理论发展基础上的原理联系在一起。我的CAREER方案旨在以两种方式推进这个几何朗兰兹项目。首先，GRASP（几何表示和一些物理）项目将为学生提供一个电子资源中心，这些学生对这个令人兴奋和多样但可能令人生畏和难以接近的领域感兴趣。GRASP的核心是一系列说明性讲座，通过网络向广大观众介绍朗兰兹计划的基本概念和相关概念。与此同时，提案的研究部分开发了一个新的程序，将前沿的几何朗兰兹技术应用于更经典的代数问题。特别是，我希望这些想法对实数线性代数中出现的对称性分类产生重大影响，这是一个在量子力学中具有杰出历史和起源的问题。

The geometric Langlands program proposes an extraordinary analog of harmonic analysis in the algebraic geometry of bundles on curves, inspired by the Langlands philosophy which binds harmonic analysis and Galois theory over number fields. In this geometric harmonic analysis, function spaces are replaced with categories of sheaves, and the spectral theory of operators on these sheaves is analyzed using geometric Fourier transforms. The investigator is developing applications of these cutting edge ideas to classical questions in representation theory. In particular, in collaboration with D. Nadler, he proposes an enhancement of the geometric Langlands program which performs a spectral decomposition of the categories of representations of real Lie groups. This significantly enhances the Langlands classification of irreducible representations (part of the classical Langlands program) and binds it to the geometric (Borel-Weil) realizations of representations. The investigator also spearheads efforts to make this material available to a far broader audience, in particular through the organization of GRASP (Geometry Representations and Some Physics), electronically distributed expository lecture series on the fundamental ideas and underlying currents in this rapidly developing area.A fundamental theme of modern mathematics is the exploitation of symmetry as an organizing principle, linking diverse and potentially baffling phenomena in an elegant overarching framework. Perhaps the prime example of this trend is the Langlands program, which identifies a general pattern in the appearance of symmetry in algebra and number theory, and counts among its successes the solution of Fermat's Last Theorem. In recent years, a new geometric setting for the applicationof the Langlands philosophy has emerged, which makes contact with new symmetry principles, in particular those underlying the exciting developments of string theory in physics. My CAREER proposal is aimed at advancing this geometric Langlands program in two ways. First, the GRASP (Geometry Representations And Some Physics) program will provide an electronic resource center for students interested in this exciting and varied but potentially intimidating and inaccessible area. At the center of GRASP is a series of expository lectures introducing the fundamental concepts underlying and relating to the Langlands program to a wide audience, via the web. In parallel, the research component of the proposal develops a novel program to apply the cutting edge geometric Langlands technology to more classical problems in algebra. In particular I expect these ideas to have a significant impact on the classification of symmetries arising in linear algebra with real numbers, a question with a distinguished history and origins in quantum mechanics.