CAREER: Representation Theory on Curves

职业：曲线表示论

• 批准号: 0449830
• 负责人: David Ben-Zvi
• 金额: $ 40万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0449830&HistoricalAwards=false
• 关键词: CAREER; Representation; Theory; Curves

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.

几何兰兰兹计划在曲线上的代数几何形状中提出了一个谐波分析的非凡类似物，灵感来自兰兰兹哲学，该哲学结合了谐波分析和数字字段上的谐波分析和galois理论。在此几何谐波分析中，功能空间被带束带的类别取代，并使用几何傅立叶变换分析了这些系带带的算子光谱理论。 研究者正在将这些尖端思想的应用在表示理论中的经典问题中。特别是，他与D. Nadler合作，提出了对几何Langlands计划的增强，该计划对真实谎言组的表示类别进行了光谱分解。这显着增强了兰兰兹对不可约表示的分类（经典兰兰兹计划的一部分），并将其与表示形式的几何（Borel-Weil）结合。 研究者还率先努力使这种材料可为更广泛的受众提供，尤其是通过掌握（几何表示和某些物理学的组织），电子上分发了有关基本思想和基本潮流的电子传播性讲座系列，并且在这个快速发展的领域中，现代数学的基本数学主题是超出的构造，并构成了整体的范围。 框架。也许这种趋势的主要例子是兰兰兹计划，该计划在代数和数字理论中确定了对称性的一般模式，并且在其成功中算出了Fermat的最后一个定理的解决方案。近年来，出现了兰兰兹哲学应用的新几何环境，它与新的对称原理，尤其是那些在物理学中弦乐理论令人兴奋的发展的基础的几何环境。 我的职业建议旨在通过两种方式推进这一几何兰兰兹计划。 首先，GRASP（几何表示和某些物理学）计划将为对这个令人兴奋，多样但可能令人生畏且难以接近的领域感兴趣的学生提供电子资源中心。 Grasp的中心是一系列的说明性讲座，介绍了通过网络通过网络向广大受众的基本概念和与Langlands计划有关的基本概念。同时，该提案的研究组成部分开发了一个新颖的程序，将最先进的兰兰兹技术应用于代数的更古典问题。特别是我希望这些想法对具有实数的线性代数产生的对称性的分类产生重大影响，这是一个具有杰出历史和起源于量子力学的问题。