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Friday, April 22, 2016

**Abstract:** For a compact oriented n-dimensional manifold $M$, the Laplace-Beltrami operator $\Delta$ acting on $L^2(M)$ has discrete spectrum, with eigenvalues limiting to infinity. One may hope for a more refined growth estimate, by asking how many eigenvalues of $\Delta$ lie below a fixed value of $\lambda$. In 1911, Hermann Weyl proved an asymptotic statement of the form $$ \#(\lambda) \sim \frac{\text{Vol}(M)}{(4\pi)^{n/2}\Gamma((n/2)+1)} \lambda^{n/2} $$ as $\lambda\rightarrow \infty$, where $\#(\lambda)=\max\{j:\lambda_j\leq \lambda \}$ is the counting function. That eigenvalue growth is determined by the volume of the manifold was one of the early results indicating that spectral invariants are closely tied to geometry. Here we shall present a proof of this statement based on the short time behavior of the heat kernel $e^{-t\Delta}$ of $M$, and a Tauberian theorem of Karamata.