Arc Spaces in the Langlands Program and Geometric Representation Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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.