TAPIRS

Abstract. Perhaps one of the most well-known theorems in graph theory is the celebrated Graph Minor Theorem of Robertson and Seymour. This theorem states that in any infinite collection of finite graphs, there must be a pair of graphs for which one is obtained from the other by a sequence of edge contractions and deletions. In this talk, I will present work of Nick Proudfoot, Dane Miyata, and myself which proves a categorified version of the graph minor theorem. As an application, we show how configuration spaces of graphs must display some strongly uniform properties. We then show how this result can be seen as a vast generalization of a variety of classical theorems in graph configuration spaces. This talk will assume minimal background knowledge, and will display few technical details.

Abstract. In this talk, I will survey known results about the (co)homology of SL_n(Z) and its finite index subgroups. I will focus on homological stability and representation stability phenomena. In addition to stability results where the homological degree remains fixed, I will talk about new (partially conjectural) stability patterns near the virtual cohomological dimension. These patterns involve comparing cohomology groups in different cohomological degrees. This talk will include joint work with Church, Nagpal, Patzt, Putman, Reinhold, and Wilson.

Abstract. The reason why the homology of the configuration space of points in the plane forms a finitely generated FI--module (and thus exhibits representation stability) is that even as the number of points goes to infinity, the jth homology is generated by cycles in which at most 2j of the points move. What about the configuration space of disks of width 1 in an infinite strip of width w? This disks in a strip space behaves more like the no-k-equal configuration space of the line, where k-1 but not k points may be collocated; we show that the homology of this no-k-equal space forms a finitely generated FI_d--module (or exhibits generalized representation stability) as defined by Sam--Snowden and Ramos. The method is to compute homology combinatorially using discrete Morse theory. Unlike other examples of homology with generalized representation stability, here the asymptotic behavior depends on the degree of homology.

Abstract. The aim of this talk is to go over the current state of research for the representation stability for the homology of some arithmetic groups and to discuss its connections to number theory and quantum codes. In particular, we will outline an approach due to Collinet which uses the strong approximation theorem and Hasse principle to show homological stability for some unitary groups.

Abstract. We define a new combinatorial category analogous to FI for 0-Hecke modules, indexing sequences of representations of H_n(0) as n varies. We then provide examples of finitely generated modules for this category and use these to discuss properties such modules possess; including a new form of representation stability and eventually polynomial growth.