Geometric Harmonic Analysis and Applications

几何调和分析及应用

• 批准号: 1103525
• 负责人: David Ben-Zvi
• 金额: $ 44.44万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103525&HistoricalAwards=false
• 关键词: Geometric; Harmonic; Analysis; Applications

Geometric harmonic analysis is a new chapter in geometric representation theory, in which techniques from homotopical algebra and unifying structures from topological field theory are combined to describe the decomposition of categories in the presence of symmetries. The PI proposes a variety of projects (joint with David Nadler) that further the foundations of the subject and apply it to resolve classical problems in representation theory. The most far-reaching project aims to develop a geometric analog of the Arthur-Selberg trace formula, consistently enhancing vector spaces of functions to categories of sheaves (in particular providing a natural context for the work of Ngo). Other projects seek to prove general Langlands dualities (or "nonabelian Fourier transforms") for categories of representations of real groups and for cohomologies of character varieties (conjectures of Soergel and Hausel--Rodriguez-Villegas, respectively). The PI also proposes (with Jonathan Block and Nigel Higson) to bridge two disparate approaches to harmonic analysis of Lie groups --- noncommutative algebraic geometry (D-modules and Beilinson-Bernstein localization) and noncommutative topology (C*-algebras and the Baum-Connes conjecture). The PI is highly committed to mathematical exposition at a variety of levels, and endeavors to convey the intuitions behind sophisticated mathematical ideas to a broad range of audiences. The significant educational component of the proposal builds on the PI's experience and enthusiasm as an expositor.The Langlands program is one of the fundamental organizing principles in modern mathematics, predicting that diverse phenomena can be understood by the systematic exploitation of hidden symmetries. Among its successes in number theory are the solution of Fermat's Last Theorem, while in physics it underlies the symmetry between electricity and magnetism and its theoretical generalizations to other fundamental forces. Recently Ngo proved the Langlands Fundamental Lemma, one of Time Magazine's Top Ten Scientific Discoveries of 2009. This proof suggests a new link between the apparently unrelated number theoretic and physical settings for the Langlands program. The current proposal is aimed at developing these connections, finding physical analogues of classical results in number theory while using the rich geometric intuition behind the physics to suggest new patterns in the application of symmetries. An important component is the dissemination of the exciting but often inaccessible developments in this field through development of online resources, lecture series, books and courses.

几何谐波分析是几何表示理论的新章节，其中合并了来自同位代数和统一结构的技术，以描述在存在对称性中类别的分解。 PI提出了各种项目（与大卫·纳德勒（David Nadler）的联合），这些项目进一步提出了主题的基础，并将其应用于代表理论中的经典问题。最深远的项目旨在开发Arthur-Selberg Trace Formula的几何类似物，从而始终增强功能的向量空间到滑轮类别（特别是为NGO的工作提供了自然背景）。其他项目旨在证明一般的Langlands对偶性（或“非亚伯傅里叶变换”），以实现真实群体的表征和角色品种的共同体（分别是Soergel和Hausel-Rodriguez-Villegas的猜想）。 PI还建议（与Jonathan Block和Nigel Higson）桥接两种不同的方法来对谎言组进行谐波分析---非交通性代数几何（D-Modules和Beilinson-Berinson-Bernstein局部化）和非交换性拓扑（C* - Algebras and Baum-Connes Interization）。 PI高度致力于在各个层面上的数学博览会，并努力将精致的数学思想背后的直觉传达给广泛的受众。 该提案的重要教育组成部分是基于PI的经验和热情作为扩展人的。兰兰兹计划是现代数学中的基本组织原则之一，预测可以通过系统的隐藏对称性来理解多样化的现象。在数量理论中取得的成功之一是Fermat的最后一个定理的解决方案，而在物理学上，它是电力和磁性之间的对称性及其对其他基本力量的理论概括。最近，非政府组织证明了兰兰兹的基本引理，这是《时代》杂志2009年前十大科学发现之一。此证明暗示了兰格兰计划的明显不相关的数字理论和物理环境之间的新联系。当前的建议旨在建立这些联系，在数字理论中找到经典结果的物理类似物，同时使用物理背后的丰富几何直觉来提出对称性应用的新模式。一个重要的组成部分是通过开发在线资源，讲座系列，书籍和课程来传播该领域令人兴奋但常常无法访问的发展。