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 这项研究的出发点是 瓦拉德汉和唐斯克的工作，在欧几里得的设置 空间 从一个泊松分布的球面开始， 半径，并考虑一个半径为N的大球体。 让D成为 拉普拉斯算子。 Dirichlet边界条件， 大的球体和小的球体，有可能 -D的特征值（适当归一化）在 大的球体。 随着N的增加，随机测度收敛 一个确定性的措施。 一个问题是， 该测度在区间（0，v）上的极限行为为v 收敛到0。 研究者将已知结果扩展到 欧氏空间到双曲空间。 两种类型的证明 空间涉及概率论和逻辑学。 的 研究者打算将这些结果扩展到其他更少的研究中。 限制空间，例如，不变性 基本进程的信息将不再可用。

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.