Spectral Asymptotics on Compact Manifolds and Related Problems in Analytic Number Theory

Abstract: We review and discuss the remainder term of Weyl's Law for the spectral counting function on closed compact Riemannian manifolds. Since the celebrated theorem of Hormander (1968) there has been much effort towards improving the estimate of the remainder term in Weyl's law for specific classes of manifolds. Interestingly, in some cases these problems and the methods used to attack them are related to classic lattice counting problems and open conjectures in analytic number theory including the Gauss Circle Problem, the Dirichlet Divisor Problem, and the calculation of the mean-square average of the Riemann zeta function on the critical line. I will discuss my own results for Weyl's law on Heisenberg Manifolds (the first, natural, noncommutative generalization of tori) and new results I have obtained in the Dirichlet Divisor Problem.

The speaker is a UFA candidate in the Department of Mathematics and will also give the DG-MP-PDE Seminar, Tuesday, October 2nd at 2:00 p.m. in WMAX 110 (PIMS Facility).

Refreshments will be served at 3:45 p.m. (Math Lounge, MATX 1115).