Stability and Instability in Topology

拓扑的稳定性和不稳定性

• 批准号: 1406209
• 负责人: Benson Farb
• 金额: $ 57.6万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406209&HistoricalAwards=false
• 关键词: Stability; Instability; Topology

AbstractAward: DMS 1406209, Principal Investigator: Benson FarbThe principal investigator proposes a program of research, education and outreach that is meant to have a maximal impact at all levels. The general research project of the principal investigator involves the topological structure of so-called "locally symmetric spaces." These spaces appear throughout mathematics and physics, and encode deep, complicated and applicable information. The principal investigator and his collaborators have found a new pattern that occurs in these spaces, and they have built a conjectural picture of other patterns that should exist. The principal will continue to engage with students at all levels, from PhD to undergraduate to K-12. This outreach includes in particular the training of a number of students from under-represented groups. It also includes public outreach, communicating of the power of mathematics and science, and how it impacts our daily lives, via public lectures.In more technical terms, the principal investgiator proposes work with A. Putman and T. Church to expose new phenomena and introduce new methods to the study of the cohomology of arithmetic groups. These methods are meant to show, in particular, that the top-dimensional cohomology of certain arithmetic groups over rings of integers in a number field vanishes rationally when the number field has class number one, and otherwise has exponentially growing (in dimension) cohomology, with growth rate depending explicitly on class number. This work is part of a broader conjectural picture of low-codimension cohomology of arithmetic groups, made by the PI with Church and Putman.

摘要奖项：DMS 1406209，首席研究员：Benson Farb 首席研究员提出了一项研究、教育和推广计划，旨在在各个层面产生最大的影响。首席研究员的一般研究项目涉及所谓“局部对称空间”的拓扑结构。这些空间贯穿数学和物理学，编码深刻、复杂且适用的信息。 首席研究员和他的合作者发现了这些空间中出现的新模式，并且他们对应该存在的其他模式建立了推测图景。 校长将继续与从博士到本科再到 K-12 各个级别的学生进行接触。这种外展活动特别包括对来自代表性不足群体的一些学生进行培训。 它还包括通过公开讲座进行公共宣传，传播数学和科学的力量，以及它如何影响我们的日常生活。用更专业的术语来说，首席研究员建议与 A. Putman 和 T. Church 合作，揭示新现象并引入新方法来研究算术群的上同调。 这些方法特别旨在表明，当数域具有第一类时，数域中整数环上的某些算术群的顶维上同调会合理消失，否则具有指数增长（在维数上）上同调，且增长率明确取决于类数。这项工作是 PI 与 Church 和 Putman 共同提出的算术群低余维上同调更广泛猜想图的一部分。