I am a postdoc at the KTH Royal Institute of Technology. Before coming to KTH, I used to be a postdoc at the University of Copenhagen Department of Mathematical Sciences, where I was a member of the topology group and part of the Centre for Symmetry and Deformation. I received my PhD from Stanford University in 2010. My advisor there was Professor Ralph Cohen.
My research interests lie in algebraic topology and homotopy theory, in particular string topology, parametrized homotopy theory, and field theories. Currently I am especially interested in string topology of classifying spaces.
Slides for the talk I gave at the Goodwillie birthday conference in Dubrovnik on my work on string topology of classifying spaces. Slides for an earlier short talk I gave on the same subject at the Fourth Arolla Conference on Algebraic Topology.
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.
This paper forms the foundation for my current research. The main result of the paper is the extension of string topology of classifying spaces into an entirely 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.