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.

Abstractaward：DMS 1406209，首席研究员：Benson Farbththe首席研究员提出了一项研究，教育和外展计划，该计划旨在在各级影响最大。主要研究者的一般研究项目涉及所谓的“本地对称空间”的拓扑结构。这些空间出现在整个数学和物理学中，并编码深层，复杂且适用的信息。 首席研究员及其合作者发现了在这些空间中发生的新模式，他们建立了应该存在的其他模式的猜想。 校长将继续与从博士到本科再到K-12的各级学生互动。该外展活动尤其包括培训来自代表性不足的小组的许多学生。 它还包括公开宣传，交流数学和科学的力量，以及它如何通过公开讲座影响我们的日常生活。在更多的技术术语中，主要投资者建议与A. Putman和T. Church合作，以暴露新现象，并为研究杂志群体的同学研究引入新方法。 这些方法尤其是要表明，当数字字段具有第一类时，某些算术组在整数中的顶级算术共同体学在一个数字字段中逐渐消失，并且否则（在维度上）的共同体学（在维度上）的共同体学（在维度上）的同胞增长，而增长速度依赖于班级数字。这项工作是PI与教堂和Putman制造的算术群体低补充算术群体的更广泛猜想的一部分。