Schedule & Abstracts

Abstract: Given a space X we can consider the configuration space of n distinct points from X. When X is a Euclidean space, the singular cohomology of these configuration spaces has rich structure. If we instead consider configurations of points in G-representations where G is a finite group, then the configuration space inherits an action of the group G. In this talk we’ll review some of the classical computations by Arnold and Cohen to compute the singular cohomology, and then discuss new techniques used to compute the Bredon G-equivariant cohomology computations. This is joint work with Dan Dugger.

Abstract: I will survey the model theory of locally approximating groups of homeomorphisms of compact manifolds, which are groups of homeomorphisms which are "sufficiently dense" in the full group of homeomorphisms, with the compact-open topology. These groups always interpret first order arithmetic; using arithmetic, one can prove that all finitely generated subgroups of locally approximating groups are definable, with parameters. Under some further conditions, one can prove that these groups are prime models of their theories. I will also discuss action rigidity for these groups: if an arbitrary group G is elementarily equivalent to a locally approximating group of homeomorphisms of a compact manifold M, then for any locally approximating group action of G on a manifold N, we must have that M and N are homotopy equivalent to each other. In low dimensions, we may in fact conclude that M and N are homeomorphic to each other. This represents joint work with J. De la Nuez Gonzalez.

Abstract: It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups.  This is joint work with Jason Behrstock and Jacob Russell.

Abstract: An exotic n-sphere is a smooth n-dimensional manifold which is homeomorphic, but not diffeomorphic, to the n-sphere with its standard smooth structure. In the 1960s, Kervaire and Milnor tied the classification of exotic spheres to the stable homotopy groups of spheres, and in the 1980s, Schultz and Stolz explained how certain relations in the stable homotopy groups of spheres detect smooth $\mathbb{Z}/p$-actions on exotic spheres. In this talk, I will advertise some open problems related to exotic spheres and summarize recent work with Behrens, Bhattacharya, Bobkova, Botvinnik, and Mahowald approaching these problems using new techniques from stable homotopy theory.