The research of the PI (Miller) focuses on the analytic theory of automorphic forms. The proposed research concentrates on automorphic distributions, a technique he has been investigating with Wilfried Schmid (Harvard University). They hope to use these distributions to develop summation formulas, similar to Poisson's and Voronoi's, whereby arbitrary sums weighted by modular form coefficients and additive twists can be dualized. The PI plans to investigate various applications of the distributions and these summation formulas. Many concern developing new constructions for automorphic L-functions, in particular approaches for which archimedean and ramified computations are possible. Another is to problems in analytic number theory, such as the subconvexity problem for cusp forms on GL(n). The PI also plans to refine a numerical algorithm which identifies modular forms, by reversing summation formulas.
The study of automorphic forms slices across many important areas of modern mathematical research, including number theory, representation theory, geometry, analysis, and mathematical physics. Through L-functions, Langlands has conjectured many deep and interesting structural relationships between automorphic forms which have implications in the above areas. As an example, the work of Wiles et al demonstrates the link between certain automorphic forms and the ancient problem of solving equations between squares and cubes. The proposed research aims to apply and develop new tools for automorphic forms and L-functions from analysis, which is the branch of mathematics expanding calculus, and representation theory, the concrete study of symmetry. Current applications of automorphic forms and L-functions are manifest in constructing the sophisticated codes which enable high-speed and secure transactions over the internet.
