I am currently a visiting scholar at Universität Hamburg. My research interests lie in algebraic topology and homotopy theory, in particular string topology of classifying spaces and its applications to group homology and cohomology.
I received my PhD from Stanford University in 2010. My advisor there was Professor Ralph Cohen.
This paper establishes an unexpected connection between finite groups of Lie type and string topology of classifying spaces of compact Lie groups: the cohomology of a finite group of Lie type is a module over the cohomology of the free loop space of the classifying space of the corresponding compact Lie group when the latter cohomology groups are equipped with a string topological multiplication. This module structure gives, among other things, a new and structured way to approach the Tezuka conjecture asserting that under certain conditions, the two cohomologies are isomorphic.
The cohomology of the degree-n general linear group over a finite field of characteristic p, with coefficients also in characteristic p, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in n. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in n. We also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood. The paper was inspired by my previous computations of higher string topology operations.
Examples of non-trivial higher string topology operations have been rare in the literature. This paper ameliorates the situation by providing explicit computations of a wealth of such operations. It also begins the work of applying the string topological methods enabled by my work with Hepworth to the study of automorphism groups of free groups: As an application of the computations, one obtains a wealth of interesting homology classes in the twisted homology groups of automorphism groups of free groups, the ordinary homology groups of holomorphs of free groups, and the ordinary homology groups of affine groups over the integers and the field of two elements. The elements constructed live in the unstable range where the homology of these groups remains poorly understood.
The main result of the paper is the extension of string topology of classifying spaces into a new kind of field theory which has operations parametrized by the homology groups of automorphism groups of free groups with boundaries in addition to operations parametrized by the homology groups of mapping class groups of surfaces. The paper was the subject of a series of three invited lectures at Loop spaces in geometry and topology, a large international conference held at Centre de Mathématiques Henri Lebesgue, Nantes, France, September 1–5, 2014.
A fundamental result in equivariant K-theory, the classical Atiyah–Segal completion theorem relates the G-equivariant K-theory of a finite G-CW complex X to the non-equivariant K-theory of the Borel construction of X. Here G is a compact Lie group. In this paper, which formed an important part of my PhD research, I prove that the Atiyah–Segal completion theorem also holds in twisted K-theory, a form of K-theory which in recent years has been the focus of intense study because of its connections to string theory.