Analysis and Topology of Manifolds

流形分析与拓扑

• 批准号: 9704743
• 负责人: Jozef Dodziuk
• 金额: $ 6.12万
• 依托单位: CUNY Queens College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704743&HistoricalAwards=false
• 关键词: Analysis; Topology; Manifolds

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.

拟议的研究涉及的拉普拉斯算子的微分形式（包括函数的情况下）和它的离散类似物与几何和拓扑结构的基础空间的分析之间的相互作用的各个方面。特别是，首席研究员将致力于以下特定主题：L^2贝蒂数和相关的谱不变量;退化双曲流形的谱理论;有限体积双曲流形的L^2挠;感觉不到边界和非局部边界条件的原则;适度正则性度量的谱;紧黎曼流形上函数的拉普拉斯算子的最小特征值的上界。 微分和差分算子的主题提出的研究出现在这样的不同领域的数学组合学，通信网络，微分几何，数学物理，数值分析，概率和拓扑。来自这些领域之一的直觉和想法可以（并且确实）为其他看似无关的领域的问题提供强大而意想不到的见解。 正因为如此，拉普拉斯算子是一个基本的研究对象。所提出的项目的一个方面是应用的想法，从分析在一个连续的设置组合的问题，最终导致一个拓扑结果。