Stability and Instability in Topology

摘要奖：DMS 1406209，首席研究员：Benson Farb首席研究员提出了一项旨在在所有层面产生最大影响的研究、教育和推广计划。首席研究员的总体研究项目涉及到所谓的“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，并编码了深刻、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，他们建立了一幅应该存在的其他模式的猜想图景。校长将继续与所有级别的学生接触，从博士到本科生再到K-12。这一外联活动特别包括培训来自任职人数不足群体的一些学生。它还包括公开宣传，通过公开讲座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，主要投资人建议与A·普特曼和T·丘奇合作，揭示新现象，并引入新方法来研究算术群的上同调。特别地，这些方法旨在证明当数域上的整数环上的某些算术群的顶维上同调为有理零时，当数域的类数为1时，数域上的上同调为指数增长的(在维上)上同调，并且增长率明显地依赖于类数。这项工作是由PI与丘奇和普特曼共同提出的关于算术群的低上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.