FRG: Collaborative Research: Combinatorial representation theory, multiple Dirichlet series and moments of L-functions.

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Great progress has been made in recent years in the theory of multiple Dirichlet series. A variety of previously studied examples have been organized into a coherent framework. The emergent structures serve to both suggest natural generalizations---often with applications to analytic number theory---and point towards unexpected connections with such diverse areas of mathematics as the spectral theory of automorphic forms, arithmetic of function fields, the geometry of affine root systems and combinatorial representation theory. Many applications in analytic number theory have been found and many more are expected. These include moment estimates and convexity breaking for L-functions over an arbitrary number field, nonvanishing results for L-functions over number fields and function fields and results on the nature of the mysterious Whittaker coefficients of metaplectic Eisenstein series and higher order theta functions. Moreover, during the past several years the combined efforts of the investigators have demonstrated that Weyl group multiple Dirichlet series have a beautiful structure that was previously unknown, and by elucidating this structure, new connections with other areas of mathematics are rapidly emerging. The grant will fund continued investigation of these rapidly developing areas. In addition, two workshops are planned for the dissemination of these results and new techniques to research mathematicians and graduate students.Number theory began thousands of years ago and was initially inspired by questions about prime numbers. Dirichlet series are infinite series, such as the Riemann zeta function, and are a primary tool in the study of prime numbers. More recently they have come to fore by providing interconnections between many diverse areas of pure mathematics and physics. Multiple Dirichlet series are simply Dirichlet series in several variables -- they have the merit that the number theoretic quantities they measure can themselves be Dirichlet series, in particular L-functions, which are fundamental objects that can be associated with many classes of number-theoretic data, such as elliptic curves, representations of Galois groups, or modular forms.