Analysis and Topology of Manifolds

提出的研究涉及微分形式（包括函数的情况）上的拉普拉斯算子的分析及其离散类似物与底层空间的几何和拓扑之间相互作用的各个方面。特别是，首席研究员将研究以下具体主题：L^2贝蒂数和相关的光谱不变量；退化双曲流形的谱理论有限体积双曲流形的L^2扭转无边界感原则和非局部边界条件；适度正则度规的谱紧黎曼流形上函数的拉普拉斯最小特征值的上界。作为本研究主题的微分算子和差分算子出现在数学的不同领域，如组合学、通信网络、微分几何、数学物理、数值分析、概率论和拓扑学。来自这些领域的直觉和想法可以（并且确实）为其他看似不相关的领域的问题提供强大而意想不到的见解。因此，拉普拉斯算子是一个基本的研究对象。该计划的一个方面是将连续分析的思想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.