University of Georgia
Mathematics Department Colloquium
Professor Lizhen Ji
University of Michigan-Ann Arbor
October 5, 2000
Spectral theory and geometry of locally symmetric spaces
Abstract: Locally symmetric spaces form an important
class of Riemannian manifolds and arise from various subjects such as algebraic
geometry and number theory as moduli spaces. They also play an important
role in representation theory. Many natural locally symmetric spaces are
quotients of symmetric spaces by arithmetic subgroups, and are noncompact.
For example, the moduli space of elliptic curves is a noncompact Riemann
surface, the modular curve. For such spaces, an interesting problem is
to understand their spectral theory and relations to the geometry. One
such relation is asked in Kac's famous paper "Can one hear the shape of
a drum?" The spectrum consists of a discrete part and a continuous part,
which is given by the Eisenstein series and scattering matrices. We will
describe a solution of the trace class conjecture in the theory of Selberg
trace formula and Weyl upper bound
on the discrete spectrum. We will also discuss relations between
the scattering matrices and the sojourn times of geodesics going to infinity.