I will discuss joint work with Randy McCarthy, in which we explicitly construct the Taylor tower (at *) of the functor sending a simplicial set X. to the reduced (over R) algebraic K-theory of the ring R extended by M[X.], where M is an R bimodule. The construction is in terms of `topological Witt vector with coefficients' invariants, generalizing topological Hochschild homology, which offer a good approach to calculating the reduced algebraic K-theory of trivial square zero extensions of rings by bimodules.
I will discuss joint work with Mark Behrens on the construction of various spectra of topological automorphic forms, with particular attention to specific instances arising from quaternion algebras and higher-dimensional abelian varieties with complex multiplication.
This is a joint project with Mark Feshbach. The goal of the project is to find a suitable n-category framework for describing cobordisms with corners and topological quantum field theory as a functor from that higher category of cobordisms with corners to a higher category of vector spaces. The problem is quite old: almost every higher category theorist has it in the back of her head, but nobody seems to have gotten through to a satisfactory solution. In the talk, I will describe the set up we have found.
We present an application of on-going work with Hopkins and Ravenel on the structure of the Lubin-Tate ring $E_{n*}$ as a $G$-module, for $G$ a finite subgroup of the Morava stabilizer group. Restricting attention to $p=2$, we get an easy method, generic in $n$, that allows us to compute the homotopy groups of the homotopy fixed points, and these in turn are closely related to the homotopy groups of the Real Johnson-Wilson theories ER(n).
Real K-Theory is 8-periodic. This periodicity can be seen algebraically from the periodicity of Clifford algebras: Clifford algebras form a 2-category, and in that 2-category, the generator Cl(1) has order 8. The analogous algebraic objects for elliptic cohomology might be called "higher Clifford algebras" and ought to form a 3-category. We introduce a candidate such 3-category whose objects are invertible conformal nets. We show that the generating net, the net of free fermions, will have order at least 24. This is joint work with Arthur Bartels and Andre Henriques.
Classically, vector bundles on the conjugation quotient G//G have a universal property with respect to Rep(G): they form its Drinfeld center. Recent joint work with David Ben-Zvi and David Nadler generalizes this result, extending work of Hinich to derived algebraic geometry. We describe a homotopy-theoretic analogue of the Drinfeld center of a monoidal stable infinity category as a Hochschild cohomology category. For the category of sheaves on X, we prove that its center is equivalent to sheaves on the derived loops LX. The structure of this category of sheaves defines an extended 2-dimensional topological quantum field theory.
While computing algebraic K-theory groups is often very difficult, understanding fixed point spectra of topological Hochschild homology can aid in such computations.Taking homotopy groups of these spectra, we arrive at TR-groups, an integer-graded theory with a rigid algebraic structure. For many algebraic K-theory computations, however, it is beneficial to further exploit the S^1-equivariant structure of topological Hochschild homology. The topological Hochschild S^1-spectrum has naturally associated equivariant homotopy groups graded by the real representation ring of S^1, which provide an RO(S^1)-graded TR-theory. In this talk we will discuss how these RO(S^1)-graded TR-groups arise naturally in algebraic K-theory computations and describe explicit computations of RO(S^1)-graded TR-groups.
It is well known that both very special gamma-spaces and grouplike $E_\infty$ spaces both model connective spectra. We follow May and Thomason's work to show that gamma-spaces and $E_\infty$ spaces are Quillen equivalent with appropriate model categories. Further we show that with suitable model category structures equivariant gamma-spaces (Shimakawa) are Quillen equivalent to equivariant $E_\infty$ spaces. We then define the units of equivariant ring spectra in terms of equivariant gamma-spaces. Thus showing that the space of units of equivariant ring spectra is an equivariant infinite loop space.
University of Chicago Algebraic Topology Seminar
University of Chicago Algebraic Topology Seminar