School of Mathematics

I present some new dispersive estimates for Schroedinger's equation with a time-dependent potential, together with applications.

Abstract: Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)

Application of Plancherel's theorem to integral kernels approximating compact period functionals yields estimates on (global) automorphic Levi-Sobolev norms of the functionals. The utility of this viewpoint can be illustrated in reconsideration of several examples: Lax-Phillips' pseudo-Laplacians discretizing (part of) the continuous spectrum, Colin de Verdiere's meromorphic continuation of Eisenstein series, Hejhal's discussion of Haas' numerical analysis of the spectrum of the automorphic Laplacians, and construction of other self-adjoint operators on spaces of automorphic forms.

We develop a simple geometric variant of the Kabatiansky-Levenshtein approach to proving sphere packing density bounds. This variant gives a small improvement to the best bounds known in Euclidean space (from 1978) and an exponential improvement in hyperbolic space. Furthermore, we show how to achieve the same results via the Cohn-Elkies linear programming bounds, and we formulate a few problems in harmonic analysis that could lead to even better bounds. This is joint work with Yufei Zhao.