Mathematical Sciences: Limiting Behavior of the Spectrum of the Laplacian in a Region with Many Small Randomly Placed Obstacles.

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The principal investigator will work on questions at the interface of differential geometry, probability, and mathematical physics. The starting point for this research is the work of Varadhan and Donsker, in the setting of Euclidean space. Start with a Poisson distribution of spheres of fixed radius, and consider a large sphere of radius N. Let D be the Laplace operator. With Dirichlet boundary conditions on both the large sphere and the smaller ones, there is a probability distribution of eigenvalues (appropriately normalized) of -D in the large sphere. As N increases, the random measures converge to a deterministic measure. One question then involves the limiting behavior of this measure on the interval (0,v) as v converges to 0. The investigator has extended known results in Euclidean space to hyperbolic spaces. Proofs in both types of spaces involve probability arguments and heuristics. The Investigator Intends to extend these results into other less restrictive spaces, where for example, the invariance properties of the underlying process would no longer be available.