NSF-CBMS Regional Conference in the Mathematical Sciences:

"Advances in Inverse Spectral Geometry"

Outline of Lectures

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Conference Schedule  

Bob Brooks: Mutually isospectral Riemann surfaces.

In their foundational paper, Gordon and Wilson constructed isospectral deformations on 2-step nilpotent manifolds. At approximately the same time, Sunada gave a finite-groups approach to constructing isospectral manifolds. These two results were combined by DeTurck and Gordon to give a Sunada-type approach to isospectral manifolds. We will discuss another way to combine these results, to give a construction of isospectral surfaces which may be thought of as discretizing the nilpotent deformation approach. The result is the construction of large numbers of mutually isospectral Riemann surfaces. (This is joint work with R. Gornet & B. Gustafson.)

Peter Perry: Isospectral sets of three-manifolds.

Let (M,g) be a closed Riemannian three-manifold. One measure of the geometric content of the spectrum is the isospectral set, i.e., the set of all closed Riemannian manifolds (M',g') with the same Laplace spectrum as (M,g). We review recent work of Brooks, Perry, and Petersen showing that for a spectrally determined neighborhood of certain metrics g, the isospectral set is compact in a sense we make precise.

David Webb: One can't hear orientability of bordered surfaces.

(Joint work with Pierre Bérard.)

We exhibit a pair of flat bordered surfaces which are isospectral for the Neumann boundary conditions, one of which is orientable while the other is nonorientable. The surfaces are constructed using the orbifold version of Sunada's theorem, and Neumann isospectrality is verified explicitly by transplantation of eigenfunctions. By using a fundamental tile which is as symmetrical as possible, we show that our construction yields a pair of Neumann isospectral bordered surfaces which are not Dirichlet isospectral.

Scott Wolpert: High-energy limits of eigenfunctions for hyperbolic surfaces.

For a manifold a sequence of eigenfunctions gives rise to a geodesic flow invariant measure on the unit tangent bundle. A simple construction of such measures for hyperbolic surfaces will be presented. It is an open problem to characterize all possible limiting measures. For PSL(2;Z) a result of Luo-Sarnak about such limits will be recast to give an asymptotic formula combining the divisor function and the Riemann zeta function.

Steve Zelditch: Wave trace invariants.

The trace of the wave group $e^{it \sqrt{\Delta}}$ on a compact Riemannian manifold (M,g) with simple length spectrum gives rise to a sequence of spectral invariants $a_{\gamma k}$ associated to each closed geodesic $\gamma$. We will characterize the wave invariants in the same sense that the heat invariants are characterized as integrals of homogeneous curvature polynomials, explain how they can be calculated and discuss some possible applications.

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