CAREER: Representation Theory on Curves

几何朗兰兹计划提出了一个非凡的模拟调和分析的代数几何束的曲线，灵感来自朗兰兹哲学，其中结合调和分析和伽罗瓦理论的一些领域。在这个几何调和分析中，函数空间被替换为层的范畴，并且使用几何傅立叶变换分析这些层上的算子的谱理论。 研究人员正在开发应用这些前沿思想的经典问题表示论。特别是，与D。纳德勒，他提出了一个增强的几何朗兰兹计划，执行光谱分解的类别表示的真实的李群。这显著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）实现绑定。 调查员还带头努力，特别是通过大型类人猿生存项目的组织，向更广泛的受众提供这些材料（几何表示和一些物理学），电子分发的基本思想和基本电流在这个迅速发展的领域的临时讲座系列。现代数学的一个基本主题是利用对称性作为一个组织原则，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。也许这种趋势的最好例子是朗兰兹纲领，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算在其成功之列。近年来，朗兰兹哲学的应用出现了一种新的几何背景，它与新的对称性原理，特别是那些支撑物理学中令人兴奋的弦理论发展的对称性原理相联系。 我的职业建议旨在从两个方面推进几何朗兰兹纲领。 首先，GRASP（几何表示和一些物理）计划将为对这个令人兴奋和多样化但可能令人生畏和难以进入的领域感兴趣的学生提供电子资源中心。在GRASP的中心是一系列的讲座介绍基本概念的基础上，并与朗兰兹计划向广大观众，通过网络。与此同时，该提案的研究部分开发了一个新的程序，将前沿几何朗兰兹技术应用于代数中更经典的问题。特别是我希望这些想法有一个显着的影响，分类的对称性所产生的线性代数与真实的号码，一个问题与杰出的历史和起源，在量子力学。

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.