The proposal concerns two problems with applications to automorphic forms. The first problem is in representation theory: it is a conjecture in local harmonic analysis that is motivated by taking Arthur's conjectures together with results of Burger, Li and Sarnak. This problem is of interest as a question in representation theory; it also offers a testing ground for Arthur's conjectures and affords the possibility of a better understanding of the automorphic spectrum. The second problem is to study analytic number theory in the context of automorphic forms on groups of higher rank. The dream goal of this is a better understanding of higher moments of L-functions, but there are a number of easier and concrete problems, such as the development of large-sieve inequalities, whose solution would also have immediate consequences for analytic number theory.
The project concerns two questions in the field of ``automorphic forms.'' This is a relatively new field of mathematics, guided by the Langlands program -- it seeks to establish connections between certain (apparently) far-separated areas of mathematics. These connections have allowed work in automorphic forms to have a significant impact in other fields. Many cryptographic algorithms -- necessary for secure communication over the Internet -- are based on very subtle properties of prime numbers, and underlying many of these algorithms are difficult results from analytic number theory and automorphic forms. Another application of automorphic forms has been the construction of ``Ramanujan graphs'' -- these are graphs with remarkable connectivity, and have had application to communication networks and to theoretical computer science. The questions under consideration will deepen our understanding of automorphic forms. In addition to the type of application just discussed, these questions lie at the intersection of different fields of mathematics, and will encourage collaboration between experts in these different fields.
