Arc Spaces in the Langlands Program and Geometric Representation Theory

这个研究项目涉及数论和物理学之间的接口问题，在一个名为几何表示理论的主题中。这项研究使用了代数和数论的技术来调查朗兰兹程序中的猜想，这是连接数论、分析和理论物理的桥梁。该项目涉及弧空间，最初设计用于研究奇异空间的数学结构，最近发现在朗兰兹计划中有富有成效的应用。研究者最近的工作为无限维几何技术在几何表示理论中的应用提供了一条新的途径。该项目旨在将这些结果应用于具有表征理论意义的对象，这些对象在功能层面上难以分析。在一个相关的项目中，这些新工具将用于构造p进函数表字典。在另一个方向上，该研究项目探索了一种几何方法来实现Lafforgue的“功能性内核”。这种构造可以应用于经典朗兰兹规划和几何朗兰兹规划。

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.