Analysis and Topology of Manifolds

拟议的研究涉及拉普拉斯算子在微分形式(包括函数的情况)上的分析与其与底层空间的几何和拓扑的离散类似之间相互作用的各个方面。特别是，主要研究人员将致力于以下具体主题：L^2 Betti数及其相关的谱不变量；退化双曲流形的谱理论；L^2有限体积双曲流形的挠率；不感觉边界和非局部边界条件的原理；适度正则性度量的谱；紧致黎曼流形上函数的拉普拉斯最小特征值的上界。作为拟议研究主题的微分和差分运算符出现在组合数学、通信网络、微分几何、数学物理、数值分析、概率和拓扑学等不同的数学领域。来自这些领域之一的直觉和想法可以(而且确实)为其他看似无关的领域的问题提供强大的、意想不到的洞察力。正因为如此，拉普拉斯算子是一个基本的研究对象。拟议项目的一个方面是将来自连续环境中的分析的想法应用于最终导致拓扑结果的组合问题。

The proposed research deals with various aspects of interactions between the analysis of the Laplace operator on differential forms (including the case of functions) and its discrete analogs with the geometry and topology of underlying spaces. In particular, the principal investigator will work on the following specific topics: L^2 Betti numbers and related spectral invariants; spectral theory of degenerating hyperbolic manifolds; L^2 torsion of finite volume hyperbolic manifolds; the principle of not feeling the boundary and nonlocal boundary conditions; spectrum for metrics of modest regularity; upper bounds for the smallest eigenvalue of the Laplacian on functions on a compact Riemannian manifold. The differential and difference operators that are the subject of the proposed research appear in such diverse areas of mathematics as combinatorics, communication networks, differential geometry, mathematical physics, numerical analysis, probability, and topology. Intuitions and ideas coming from one of these fields can (and do) provide powerful and unexpected insight for problems in other seemingly unrelated fields. Because of this, the Laplace operators are a fundamental object of study. One aspect of the proposed project is the application of ideas from analysis in a continuous setting to a combinatorial problem which ultimately leads to a topological result.