CAREER: Representation Theory on Curves

几何朗兰兹纲领在曲线束的代数几何中提出了调和分析的非凡类比，其灵感来自于将调和分析和伽罗瓦理论结合在数域上的朗兰兹哲学。在该几何调和分析中，函数空间被滑轮类别所取代，并且使用几何傅里叶变换来分析这些滑轮上算子的谱理论。 研究人员正在开发这些前沿思想在表示论中经典问题中的应用。特别是，与 D. Nadler 合作，他提出了几何朗兰兹程序的增强，该程序对实李群的表示类别进行谱分解。这显着增强了不可约表示的朗兰兹分类（经典朗兰兹纲领的一部分），并将其与表示的几何（Borel-Weil）实现结合起来。 研究人员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），这是关于这个快速发展领域的基本思想和潜在趋势的电子分发说明性讲座系列。现代数学的一个基本主题是利用对称性作为一种组织原则，在一个优雅的总体框架中将各种可能令人困惑的现象联系起来。也许这种趋势的主要例子是朗兰兹纲领，它确定了代数和数论中对称性出现的一般模式，并将费马大定理的解决算作其成功之一。近年来，出现了朗兰兹哲学应用的新几何环境，它与新的对称原理相联系，特别是物理学中弦理论令人兴奋的发展背后的对称原理。 我的职业建议旨在以两种方式推进几何朗兰兹计划。 首先，GRASP（几何表示和一些物理）计划将为对这个令人兴奋且多样化但可能令人生畏且难以进入的领域感兴趣的学生提供一个电子资源中心。 GRASP 的核心是一系列说明性讲座，通过网络向广大受众介绍朗兰兹计划背后的基本概念以及与朗兰兹计划相关的基本概念。与此同时，该提案的研究部分开发了一个新颖的程序，将尖端的几何朗兰兹技术应用于代数中更经典的问题。我特别希望这些想法能够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悠久历史和起源的问题。

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.