Stability and Instability in Topology

AbstractAward：DMS 1406209，首席研究员：本森Farb首席研究员提出了一个研究，教育和推广计划，这意味着在各级产生最大的影响。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。“这些空间出现在数学和物理学中，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这些空间中出现的一种新模式，他们已经建立了其他应该存在的模式的结构图。 校长将继续与各级学生接触，从博士到本科再到K-12。这一外联活动特别包括培训一些来自代表性不足群体的学生。 它还包括公共宣传，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。Putman和T. 教会揭露新的现象和介绍新的方法来研究算术群的上同调。 这些方法的目的是显示，特别是，某些算术群的顶维上同调的整数环在数域理性消失时，数域有类数1，否则有指数增长（在维度）上同调，增长率显着依赖于类数。这项工作是由PI与Church和Putman一起制作的算术群的低余维上同调的更广泛的数学图像的一部分。

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.