Stability and Instability in Topology

奖项：DMS 1406209，首席研究员：Benson farb首席研究员提出一项研究、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的“局部对称空间”的拓扑结构。这些空间出现在整个数学和物理中，并编码深刻、复杂和适用的信息。首席研究员和他的合作者发现了在这些空间中出现的一种新模式，他们已经建立了一个应该存在的其他模式的推测图。校长将继续与各个层次的学生接触，从博士到本科生再到K-12。这种外联活动特别包括培训一些来自代表性不足群体的学生。它还包括通过公开讲座向公众宣传数学和科学的力量，以及它如何影响我们的日常生活。在更专业的术语中，首席研究员建议与A. Putman和T. Church合作，以揭示新的现象并引入新的方法来研究算术群的上同调。这些方法的目的是特别地表明，当数域具有类1时，数域上整数环上的某些算术群的上维上同调理性地消失，否则具有指数增长（维数）上同调，其增长率显式地依赖于类数。这项工作是PI与Church和Putman对算术群的低余维上同调的更广泛猜想的一部分。

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.