Invariant Theory, Moduli Space, and Automorphic Representations

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The Langlands program envisions a deep relation between basic questions about integers, like how the number of solutions to equations modulo p varies as the prime number p varies, and the structure of certain infinite dimensional representations of groups of matrices. This project will investigate how representations of different groups of matrices are related to one another, known as Langlands's functoriality, from a geometric perspective. In the course of the project, the principal investigator will train graduate and undergraduate students in cutting edge areas of mathematics including algebraic geometry, representation theory, and number theory. The Arthur-Selberg trace formula and its relative variants are among the main tools at our disposal to address Langlands's functoriality conjecture. In this project, the principal investigator, Dr. Ngo Bao Chau, will investigate the structure of certain moduli spaces appearing naturally in the study of the geometric side of the (relative) trace formula. These new moduli spaces can be seen as a generalization of the Hitchin fibration, which was instrumental in the proof of the fundamental lemma by Dr. Ngo. Their investigation will require a new understanding of invariant theory which is a classical topic in algebraic geometry. Dr. Ngo expects these results in invariant theory will shed light not only on the trace formula but also on related problems, including the determination of the kernel of the nonabelian Fourier transform responsible for the functional equation of automorphic L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.