Collaborative Research: Automorphic Forms, Representations and L-functions

This collaborative proposal is concerned with developing a new higher rank version of a fundamental identity known classically as the Kuznetsov trace formula, which relates the spectrum of a certain differential operator to the geometry of the space on which the operator acts. The aim is to establish either asymptotics or strong bounds for all the different terms appearing in the formula. A first application is to obtain the symmetry types of certain thin families of L-functions in various higher rank situations. A broad range of further applications are expected. Additionally, the following research problems will be investigated: a search for a new class of Multiple Dirichlet Series will be executed; supercuspidal representations in higher rank will be studied; the Affine Linear Sieve will be combined with bilinear forms methods to exhibit thin orbits containing an infinitude of primes; and finally, effective infinite-volume counting problems will be attacked.

The theory of automorphic forms, representations, and L-functions is a central theme in modern number theory, and has provided links between such diverse areas of mathematics as algebraic geometry, representation theory, probability, combinatorics, and mathematical physics. Thus progress in the understanding of the aforementioned objects often has a significant impact in other fields. For example, cryptographic algorithms which secure wireless communication for the internet and cellular phones often rely heavily on deep properties of prime numbers. The proposal also includes a significant educational and dissemination component in the mentoring of undergraduate, graduate students, and postdocs working in these evolving parts of mathematics, with the hope of bringing traditionally under-represented goups into the field.