Geometric Harmonic Analysis and Applications

几何调和分析是几何表示理论的新篇章，它将同伦代数的技巧和拓扑场论的统一结构相结合来描述存在对称性的范畴的分解。PI提出了各种项目(与David Nadler联合)，进一步奠定了这门学科的基础，并将其应用于解决表示理论中的经典问题。影响最深远的项目旨在开发Arthur-Selberg迹公式的几何模拟，将函数的向量空间始终如一地增强为多个类别(特别是为NGO的工作提供一个自然的背景)。其他项目试图证明实群的表示范畴和特征标簇的上同调(分别是Soerel和Hausel-Rodriguez-Villegas的猜想)的一般朗兰兹对偶(或“非交换傅立叶变换”)。PI还建议(与Jonathan Block和Nigel Higson)架起李群调和分析的两种不同方法的桥梁-非交换代数几何(D-模和Beilinson-Bernstein局部化)和非交换拓扑(C*-代数和Baum-Connes猜想)。PI高度致力于不同层次的数学阐述，并努力将复杂的数学思想背后的直觉传达给广泛的受众。该提案的重要教育部分建立在PI作为展示者的经验和热情之上。朗兰兹计划是现代数学的基本组织原则之一，预测通过系统地利用隐藏的对称性可以理解各种现象。它在数论中的成功之一是费马大定理的解，而在物理学中，它是电和磁之间的对称性及其对其他基本力的理论推广。最近，NGO证明了朗兰兹基本引理，该引理被《时代》杂志评为2009年十大科学发现之一。这一证明表明，朗兰兹计划显然不相关的数论和物理环境之间存在着一种新的联系。目前的提议旨在发展这些联系，找到数论中经典结果的物理相似之处，同时利用物理学背后丰富的几何直觉来提出对称性应用的新模式。一个重要组成部分是通过开发在线资源、系列讲座、书籍和课程，传播这一领域令人兴奋但往往难以获得的发展。

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.