Problems in Geometric and Quantitative Topology

几何和定量拓扑问题

• 批准号: 1105657
• 负责人: Shmuel Weinberger
• 金额: $ 27.47万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105657&HistoricalAwards=false
• 关键词: Problems; Geometric; Quantitative; Topology

This proposal deals with topological problems in a way that keeps track of sizes of solutions to the problems. These include measurements of displacement (bounded topology), of Lipschitz constants and distortions, and related metric invariants. Applications of such ideas arise within pure topology, stratified spaces, homology manifolds, Riemannian geometry, geometric group theory, and theoretical computer science. The proposal involves a mix of problems from these areas, some related to singularities and group actions which involve refining and applying previously developed tools, while others are motivated by rigidity theory, and conjectures of Gromov that seem to require new methods.Topology is ordinarily thought of as an area of mathematics that is qualitative in nature. Progress on a number of current central problems requires more precise quantitative control of the solutions of related problems that were solved in earlier generations by powerful but inexplicit algebraic methods. Such control brings topology in closer interaction with applications in other fields of mathematics as well as computer science and engineering. This proposal suggests an attack on some of these problems.

这个建议处理拓扑问题的方式，保持跟踪的大小的解决方案的问题。 这些包括位移（有界拓扑），Lipschitz常数和失真，以及相关的度量不变量的测量。这些思想的应用出现在纯拓扑、分层空间、同调流形、黎曼几何、几何群论和理论计算机科学中。 该建议涉及的问题从这些领域的混合，一些涉及到奇点和群作用，其中涉及到完善和应用以前开发的工具，而其他人的动机是刚性理论，和programmures的格罗莫夫，似乎需要新的方法。拓扑通常被认为是一个领域的数学是定性的性质。 一些当前的中心问题的进展需要更精确的定量控制的解决方案的相关问题，解决了在前几代强大的，但不明确的代数方法。 这种控制使拓扑学与其他数学领域以及计算机科学和工程领域的应用有了更密切的相互作用。 这一提议建议对其中一些问题进行攻击。