Wall-Crossing and Dualities in Representation Theory

表示论是代数的一个分支，研究对称，特别是线性数学结构的对称性。这门学科令人难以置信的丰富部分是因为它在量子物理中扮演的基本角色，以及它进入朗兰兹对偶计划中所谓的数论的更高(非阿贝尔)互易定律的公式的方式，该计划正在搭建连接数论、分析和理论物理的桥梁。本项目涉及到这两个主题：其目的之一是利用无限维空间的几何来回答由朗兰兹对偶引发的问题；另一个目的是利用更多经典的有限维几何对象及其量子化来研究表示。该项目将通过应用镜像对称和朗兰兹对偶的思想来推进量子化辛解的表示理论。另一个目标是进一步发展L-进模束方法在p-进群调和分析中的应用。研究人员还将继续探索表示理论中的D-模方法，重点是Hodge D-模的应用以及导出的正特征局部化定理的推广。

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures. The incredible richness of the subject is partly due to the fundamental role it plays in quantum physics, as well as the way it enters formulations of the so-called higher (non-abelian) reciprocity laws of number theory in the Langlands duality program, which is building bridges connecting number theory, analysis, and theoretical physics. The present project is related to both these topics: one of its aims is to use geometry of infinite dimensional spaces to answer questions motivated by Langlands duality; another one is to exploit more classical, finite dimensional geometric objects and their quantization to study representations.The project will advance the representation theory of quantized symplectic resolutions by applying ideas of mirror symmetry and Langlands duality. Another goal is to further develop l-adic sheaf methods in harmonic analysis on p-adic groups. The investigator will also continue to explore D-module methods in representation theory, concentrating on applications of Hodge D-modules and generalizations of the derived localization theorem in positive characteristic.