See also the Brave New Rings seminar page.
We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.
We show that there are uncountably many A_infty ku-ring spectrum structures on ku/p.
We construct a tensor model structure on the category of chain complexes of R-modules for a sheaf of rings R in a Grothendieck topos. If the topos has enough points, then the homotopy category is equivalent to the derived category. We study some t-structures on the derived category which can be lifted to t-model structures on the category of chain complexes of R-modules.
On joint work with Ulrich Bunke and Thomas Schick.
I'm going to report on my Diplomarbeit where I investigated the Lusternik-Schnirelmann-category of the ordered and unordered configuration spaces of k points in a real vector space of dimension n. This is closely related to the sectional category of the associated fibrations which attracted some attention in case n=2. In this case calculations have been done by Vassiliev and recently by DeConcini et al. and Arone. In the general situation we obtain bounds for all pairs (n,k) and precise results for example if k is a power of 2. The precise determination for arbitrary k is still open, even for n=2. I plan to include an elementary introduction to both Lusternik-Schnirelman-Theory and configuration spaces.
We study the behavior of Nil-subgroups of K-groups under localization. As a consequence of our results we obtain that the K-theoretic relative assembly map from the family of finite groups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map that appears in the Farrell-Jones conjecture in terms of group homology and the K-groups of finite cyclic subgroups.
This is joint work with Dan Isaksen. We construct a motivic J-homomorphism from the bigraded homotopy groups of the infinite general linear group to the bigraded motivic stable homotopy groups of spheres. It evaluates to the usual complex (real) J-homomorphism after taking complex (real) points. In my talk, I will illustrate the construction, discuss its relation to algebraic K-theory and give some examples using the motivic Hopf maps.