CAREER: Representation Theory on Curves

几何朗兰兹计划在曲线上丛的代数几何中提出了一种非同寻常的调和分析模拟，灵感来自于朗兰兹哲学，该哲学将调和分析与数域上的伽罗瓦理论结合在一起。在这种几何调和分析中，用层范畴代替函数空间，并用几何傅立叶变换分析了这些层上算子的谱理论。研究人员正在开发这些前沿思想在表象理论中的经典问题上的应用。特别是，在与D.Nadler的合作中，他提出了对几何朗兰兹程序的增强，该程序执行实李群表示范畴的谱分解。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)，并将其绑定到表示的几何(Borel-Weil)实现。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GRAPH(几何表示和一些物理)，以电子方式分发的关于这个快速发展领域的基本思想和潜在潮流的说明性讲座系列。现代数学的一个基本主题是利用对称性作为组织原则，在一个优雅的总体框架中将各种和潜在令人费解的现象联系起来。也许这一趋势的主要例子是朗兰兹计划，该计划确定了代数和数论中对称性出现的一般模式，并将费马最后定理的解计算在其成功之列。近年来，朗兰兹哲学的应用出现了一个新的几何环境，它与新的对称性原理相联系，特别是那些在物理学中弦理论令人兴奋的发展背后的原理。我的职业建议旨在从两个方面推进这个几何兰兰兹项目。首先，GRAPH(几何表示和一些物理)课程将为对这一令人兴奋和多样但可能令人望而生畏和难以接近的领域感兴趣的学生提供一个电子资源中心。GRAPH的中心是一系列的说明性讲座，通过网络向广大观众介绍朗兰兹项目的基本概念和相关概念。同时，该提案的研究部分开发了一项新的计划，将尖端几何朗兰兹技术应用于更经典的代数问题。特别是，我预计这些想法将对实数线性代数中出现的对称性的分类产生重大影响，这是一个在量子力学中有着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.