Arthur's Conjecture, Spectural Theory, and Analytic Number Theory in Higher Rank

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DMS-0245606Vogan, David A.AbstractTitle: Arthur's Conjecture, Spectural Theory, and Analytic Number Theory in Higher RankAbstract. The proposal concerns two problems with applications toautomorphic forms. The first problem is in representation theory:it is a conjecture in local harmonic analysis that ismotivated by taking Arthur's conjectures together with resultsof Burger, Li and Sarnak. This problem is of interest as aquestion in representation theory; it also offers a testingground for Arthur's conjectures and affords the possibilityof a better understanding of the automorphic spectrum. Thesecond problem is to study analytic number theory in the contextof automorphic forms on groups of higher rank. The dream goal ofthis is a better understanding of higher moments of L-functions,but there are a number of easier and concrete problems, such asthe development of large-sieve inequalities, whose solution wouldalso 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 establishconnections between certain (apparently) far-separatedareas of mathematics. These connections have allowed work inautomorphic forms to have a significant impact in other fields.Many cryptographic algorithms -- necessary for secure communicationover the Internet -- are based onvery subtle properties of prime numbers, and underlyingmany of these algorithms are difficult results from analytic numbertheory and automorphic forms. Another applicationof automorphic forms has been the construction of ``Ramanujan graphs''-- these are graphs with remarkable connectivity, and have hadapplication to communication networks and to theoretical computerscience. The questions under consideration will deepenour understanding of automorphic forms. In additionto the type of application just discussed,these questions lie at the intersection of different fields ofmathematics, and will encourage collaboration between expertsin these different fields.