Title: Automorphic methods in the geometry of numbers

Speaker: Seungki Kim(University of Cincinnati)

Abstract: The geometry of numbers concerns itself with metric properties of lattices in a Euclidean space: the length of the shortest nonzero vector of a lattice L, the covering radius of L, upper and lower bounds on these quantities, and so on. Lattices are naturally identified with points of SL(n,Z)\SL(n,R), and lattice-point counting functions are pseudo-Eisenstein series, so the methods of automorphic forms naturally come into play. I'll give a brief introduction to the field, and then discuss my recent work with Seokho Jin (Chung-ang U.) on truncated L^2-norm formula on the number of sublattices of fixed rank and bounded height. Curiously, the L^2-norm diverges in general, so the use of truncation and the Maass-Selberg relations is somewhat forced. As an application, we present a progress on an analogue of the Gauss circle problem for rational points on Grassmannians.

Website: https://buciumas.github.io/postech-representation-theory/index