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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.
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