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## Spectral asymptotics for the Laplacian and lattice point counting

### John Toth

McGill University

###
Thursday, January 24, 2002

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** Let *(M,g)* be a compact Riemann manifold and
*-\Delta* its Laplace-Beltrami operator. The spectral counting
function, *N(\lambda)*, counts (with multiplicities) the number of
Laplace eigenvalues less than *\lambda*. The Weyl formula states that
to leading order:

N(\lambda) \sim_{\lambda \rightarrow \infty} c
\lambda^{n/2}.

We will review some known results concerning the
error term *R(\lambda) := N(\lambda) - c \lambda^{n/2}* and then
discuss recent work (joint with Y. Petridis) on bounds for
*R(\lambda)* in the case of a flat torus or a Heisenberg manifold. In
these cases, asymptotics for *N(\lambda)* can be reduced to certain
lattice point counting problems.

This talk will be accessible to undergraduates.