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

首席研究员将在微分几何、概率和数学物理的界面上研究问题。本研究的出发点是Varadhan和Donsker在欧几里得空间背景下的工作。从半径固定的球的泊松分布开始，考虑一个半径为n的大球，设D为拉普拉斯算子。在大球面和小球面上都有Dirichlet边界条件，在大球面上有d的特征值（适当归一化）的概率分布。随着N的增加，随机测度收敛于确定性测度。一个问题涉及到当v收敛于0时，这个测度在区间（0,v）上的极限行为。研究者将欧几里得空间的已知结果推广到双曲空间。这两种空间的证明都涉及概率论证和启发式。研究者打算将这些结果扩展到其他限制较少的领域，例如，底层工艺的不变性将不再可用。

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.