Wall-Crossing and Dualities in Representation Theory

表征理论是研究对称性的代数分支，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由于它在量子物理学中所起的基本作用，以及它在朗兰兹对偶程序中进入所谓的数论高级（非阿贝尔）互易律的公式的方式，这是连接数论、分析和理论物理的桥梁。目前的项目与这两个主题相关：它的目标之一是使用无限维空间的几何来回答朗兰兹对偶引发的问题；另一种方法是利用更经典的有限维几何对象及其量化来研究表征。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分辨率的表示理论。另一个目标是在p进群的调和分析中进一步发展l进束法。研究者还将继续探索表示理论中的d模方法，重点关注Hodge d模的应用以及所导出的局部化定理在正特征中的推广。

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.