Applications of Automorphic Forms in Number Theory and Combinatorics

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The proposal is to support an international research conference for 60 or more participants entitled "Applications of Automorphic Forms in Number Theory and Combinatorics", to be held April 12-15, 2014, at Louisiana State University in Baton Rouge, Louisiana. This conference aims to bring together top experts along with junior researchers and graduate students in the active fields of number theory and automorphic forms. Distinguished speakers include Abel prize winners Jean-Pierre Serre and John Tate. From the synergy of a gathering of researchers with related interests but complementary knowledge and skills, advances are expected in research programs for the participants, dissemination of important current work by experts to a broad audience, and professional development for junior researchers and graduate students. Number theory has long fascinated and challenged curious minds; in recent years, it is becoming an indispensable tool for many practical applications including communication, coding theory, and cryptography. In the 1960's and 70's Langlands revolutionized number theory with broad conjectures linking algebraic number theory to automorphic forms, which, if proved, would unify large areas of mathematics. The proof of the Shimura-Taniyama-Weil conjecture and Serre's conjecture for degree-2 Galois representations over finite fields provided dramatic examples. Applications of these results have been seen in many areas of number theory including the theory of noncongruence modular forms. The coefficients of automorphic forms, central to this theory, often appear when counting interesting mathematical objects, including integer partitions. Among the applications of modular forms, one of the most unexpected is in the construction of expanders, which are also known as Ramanujan graphs. In recent years, there are applications of Ramanujan graphs to coding theory and cryptography. This conference will enable direct communication between both seasoned experts and junior researchers about current progress on these topics. This will not only encourage new collaborations and research projects, but we also expect distinguished experts to use their insights to guide the research trajectories of many junior participants. This conference will impact society in a few distinct ways. First, we expect significant professional development, especially for graduate students and junior researchers, through contact with distinguished plenary speakers, and participation in special sessions. In addition, we expect broad dissemination of the results of this conference both through our website, where we will post slides of talks and posters, and through a potential proceedings volume for the meeting. Finally, we expect this conference to draw publicity to the number theory currently being done in the south, and work being done by female number theorists. Conference website: https://www.math.lsu.edu/nt2014/