Families of L-functions and automorphic forms

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The PI plans to investigate a variety of problems in theanalytic theory of automorphic forms, especially problemsconcerning higher rank groups such as the general lineargroup of degree 3 and higher. In particular, the PI willstudy some problems concerning the analytic study ofautomorphic forms restricted to small subsets, especiallyperiod 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 understandthe statistical properties of algebraic objects. For instance,one can ask what is the probability that a randomly chosenlarge number is a prime. It is very fortunate for moderncryptography that it is easy to find large primes. Cryptographyis a crucial tool in banking, commerce, and of course national security. Some of our best knowledge on the prime numberscomes from properties of the Riemann zeta function, thesimplest L-function. Many other fascinating algebraicand geometric objects are encoded in other types of L-functions.The PI plans to study many of the statistical properties ofthese L-functions.