Geometric Harmonic Analysis and Applications

几何调和分析是几何表示论的一个新的篇章，其中同伦代数的技术和拓扑场论的统一结构相结合，描述了对称性存在时范畴的分解。PI提出了各种各样的项目（与大卫纳德勒联合），进一步巩固了这一学科的基础，并将其应用于解决表示论中的经典问题。最深远的项目旨在开发阿瑟-塞尔伯格迹公式的几何模拟，不断增强向量空间的功能，以类别的层（特别是提供一个自然的背景下，Ngo的工作）。其他项目试图证明一般朗兰兹对偶（或“非阿贝尔傅立叶变换”）的类别表示的真实的群体和上同调的字符品种（abutures的Soergel和Hausel-罗德里格斯-Villegas，分别）。PI还建议（与Jonathan Block和奈杰尔希格森）桥接李群调和分析的两种不同方法-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高度致力于各种层次的数学阐述，并努力将复杂的数学思想背后的直觉传达给广泛的受众。 该计划的重要教育部分建立在PI作为评审员的经验和热情之上。朗兰兹纲领是现代数学的基本组织原则之一，它预言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处在于解决了费马大定理，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概括的基础。最近Ngo证明了朗兰兹基本引理，这是时代杂志2009年十大科学发现之一。这一证明表明了朗兰兹纲领的数论和物理设置之间明显不相关的新联系。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类似物，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。一个重要的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无法获得的发展。

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.