Arc Spaces in the Langlands Program and Geometric Representation Theory

这个研究项目涉及数论和物理之间的接口问题，在一个称为几何表示理论的主题。 该研究使用代数和数论的技术来研究朗兰兹计划中的结构，该计划正在建立连接数论，分析和理论物理的桥梁。 该项目涉及弧空间，数学结构最初旨在研究空间的奇点，最近发现有富有成效的应用程序在朗兰兹计划。最近的研究工作提供了一个新的方法来应用技术的无穷维几何的几何表示理论。这个项目的目的是采用这些结果在一个层理论的方法，对象表示理论的意义，是众所周知的是难以分析的功能水平。在一个相关的项目中，这些新工具将被用来构建一个p-adic函数层字典。在另一个方向上，研究项目探索了一种几何方法，以拉弗格的“功能性的核心”。“这样的构造将应用于经典和几何朗兰兹程序。

This research project concerns questions at the interface between number theory and physics, in a topic called geometric representation theory. The research uses techniques from algebra and number theory to investigate conjectures in the Langlands program, which is building bridges connecting number theory, analysis, and theoretical physics. The project involves arc spaces, mathematical structures originally designed to study spaces with singularities and recently discovered to have fruitful applications in the Langlands program. Recent work of the investigator provides a new way to apply techniques of infinite dimensional geometry in geometric representation theory. This project aims to employ these results in a sheaf-theoretic approach to objects with representation theoretic significance that are known to be difficult to analyze on the level of functions. In a related project, these new tools will be used to construct a p-adic function-sheaf dictionary. In another direction, the research project explores a geometric approach to Lafforgue's "kernel of functoriality." Such a construction will have applications both to the classical and the geometric Langlands programs.