Given any model category, or more generally, any homotopy theory, one can obtain from it a simplicial category which encodes all the homotopy theoretic information of the original homotopy theory. Having a model category structure on the category of all (small) simplicial categories is then a first step in studying the "homotopy theory of homotopy theories." While this model category structure does exist with appropriate weak equivalences, it would be helpful to find Quillen equivalent model category structures in which calculations are easier. I will discuss the simplicial category model category structure as well as two proposed model structures which should be Quillen equivalent to it: the complete Segal space model category structure on simplicial spaces and the Segal category model category structure on Segal precategories. From there, I will explain current work in progress towards finding Quillen equivalences between these model categories.
Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X - or the same kind of thing with more than two spaces. Thus, the self-similarity of X is described by a system of simultaneous equations. I will formalize this idea and explain the notion of a `universal solution' of such a system. The theory has some surprising consequences.
I will discuss some recent progress on understanding the relationship of the K(2)-local EO_2 resolution of the sphere to a resolution of Goerss, Henn, Mahowald, and Rezk. The (second) Morava stabilizer group is the p-adic Lie group of automorphisms of the formal completion of an elliptic curve. There is a submonoid of endomorphisms of the curve itself which give rise to a dense submonoid of the Morava stabilizer group. Relating the the dense submonoid to the whole group gives rise to some interesting homological algebra which relates higher derived functors of a completion functor to the Poincare duality satisfied by the group.
We will examine $n$-categories in which every $k$-cell is equipped with a specified dual, for all $0 < k < n$, and $n$-categories in which, in addition, every $n$-cell has a specified ``formal dual''. We show that an $\omega$-category with all duals is an $\omega$-groupoid. To avoid questions about the definition of $\omega$-category, we show that this result can be obtained using a bare minimum of $\omega$-categorical structure. The result then holds for any definition of $\omega$-category having this bare minimum. I will briefly introduce the hypotheses relating n-categories and TQFTs. We study the example of cobordisms of all dimensions; although the full structure of an $\omega$-category of cobordisms has not yet been found, we exhibit enough structure to show that this example fits into the above framework.
The P.I. will discuss how fat graphs and combinatorics can be used to study the mapping class group of a surface. Construction of a combinatorial CW structure. Combinatorial cochain complex and combinatorial cycles. Stabilisation as a combinatorial 2-category.
This lecture is an expository stroll through part of the world of braid groups. Topics are interactions between low dimensional topology, homotopy theory, Lie algebras, cohomology of groups, as well as representations of braid groups.
A bundle of spectra determines a twisted generalized cohomology theory. These theories, mainstays of parametrized stable homotopy theory, have received much attention of late because of their natural appearance in string theory and because of their connections with elliptic cohomology and with the representation theory of loop groups. These theories are often computable; in particular, many familiar spectral sequences (Rothenberg-Steenrod, Bousfield-Kan/Mayer-Vietoris, Atiyah-Hirzebruch, et al) have tractable parametrized analogs. I will discuss primarily the application of the twisted Rothenberg-Steenrod spectral sequence to the computation of the twisted K-homology of Lie groups. Time permitting I will also describe twisted Spin^c bordism and its conjectural relation to geometric mirror symmetry.
I shall describe how the simplicial bar construction on an operad in based spaces or spectra has a natural cooperad structure. We shall apply this to Goodwillie calculus and show that the Goodwillie derivatives of the identity functor on based spaces form an operad. I will construct modules over this operad and see how this all relates to Koszul duality.
It is still an open conjecture whether the elliptic cohomology theory of 2-dimensional supersymmetric Conformal Field Theories (CFT) constructed by St.Stolz and P.Teichner is in fact a geometric model for the theory of topological modular forms, TMF. One strategy to derive more evidence would be to determine a connective version of CFT and study its relation to tmf. A first step in this direction is done once one can determine the connective version of the analogous lowerdimensional theory of (1-dim. susy) Euclidean Field Theories, EFT, which is already proven to be equivalent to K-theory. In this talk I will present a solution for the 1-dimensional case which seems suitable for generalization.
University of Chicago Algebraic Topology Seminar
University of Chicago Algebraic Topology Seminar