Stability and Instability in Topology

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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.