Analytic theory of automorphic forms

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The prime numbers have fascinated people for millennia because of their orderly definition but unpredictable behavior. It is only recently, in the past few decades, that primes have proved useful in cryptography, where it is often important to generate large primes. For instance, one can program a computer to find the first prime with 100 digits. Knowing how long this takes in general amounts to studying the distribution of the prime numbers. This distribution is intimately connected to properties of the Riemann zeta function, the most basic example of a so-called L-function. The PI will study many of the statistical properties of the Riemann zeta function and other more general L-functions. This project will study analytic properties of L-functions and automorphic forms. For instance, the PI plans to study the quantum unique ergodicity conjecture in some new contexts such as in higher rank and along an arbitrary curve. The PI has had prior success for the Eisenstein series restricted to a particular geodesic in the upper half plane. These problems fit into the general setting of how eigenfunctions of the Laplacian behave as the eigenvalue grows large, a topic of great interest in geometry, mathematical physics, etc. However, in the arithmetical setting these problems are related to quantitative analytic properties of L-functions.