Families of L-functions and automorphic forms

The PI plans to investigate a variety of problems in the analytic theory of automorphic forms, especially problems concerning higher rank groups such as the general linear group of degree 3 and higher. In particular, the PI will study some problems concerning the analytic study of automorphic forms restricted to small subsets, especially period integrals. Other components of the research include the study of mean values of L-functions including the Riemann zeta function. One of the goals of analytic number theory is to understand the statistical properties of algebraic objects. For instance, one can ask what is the probability that a randomly chosen large number is a prime. It is very fortunate for modern cryptography that it is easy to find large primes. Cryptography is a crucial tool in banking, commerce, and of course national security. Some of our best knowledge on the prime numbers comes from properties of the Riemann zeta function, the simplest L-function. Many other fascinating algebraic and geometric objects are encoded in other types of L-functions. The PI plans to study many of the statistical properties of these L-functions.