Equidistribution, Period Integrals of Automorphic Forms, and Subconvexity

This project will investigate some open questions in arithmetic quantum chaos, a multidisciplinary field at the interface of number theory and theoretical physics. The mathematical study of quantum chaos involves the interplay between quantum mechanics and classical mechanics. The goal is to understand how chaotic behavior of the latter, such as ergodic billiard flow, is reflected by chaotic behavior of the former. One of the main tasks of this project is to investigate these relations in settings where there are additional ties to number theory: the study of prime numbers and their distribution. The PI will also supervise undergraduate and high school students as part of the UVA number theory REU, as well as continue to train PhD students. The PI will investigate problems in the analytic theory of automorphic forms that involve applications of subconvex bounds for L-functions. One major problem to be resolved is restricted arithmetic quantum unique ergodicity, which involves the study of the mass equidistribution of automorphic forms on curves on the modular surface. The PI will additionally prove new subconvex bounds for Rankin-Selberg L-functions. The methods to prove these results will rely on new instances of spectral reciprocity, which are identities for moments of L-functions, and will be proven using period formulae relating integrals of automorphic forms and their 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.