University of Chicago
Department of Mathematics

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.

The category of Mackey functors for a finite group $G$ is a symmetric monoidal closed abelian category. The unit for the tensor product on this category is the Burnside Mackey functor $B$. Purely formal results about symmetric monoidal closed abelian categories allow us to localize the category of Mackey functors at any Mackey functor prime ideal $P$ of $B$. These localizations ought to be valuable tools for studying the contribution of each subgroup $H$ of $G$ to any given Mackey functor, such as the representation ring. Unfortunately, the formal approach to localization is so abstract that it provides very little insight into the behavior of these localizations of the category of Mackey functors. In this talk, we provide a much more concrete alternative description of these localizations. This approach is far more tractible and more easily related to geometry.

The area of string topology began with a construction by Chas and Sullivan of previously undiscovered algebraic structure on the homology H_*(LM) of the free loop space of an oriented manifold M. Among other results, Chas and Sullivan showed that H_*(LM), suitably regraded, carries the structure of a graded-commutative algebra. The product pairing was subsequently extended by Cohen and Godin into a form of topological quantum field theory (TQFT). Open-closed string topology, first sketched by Sullivan, arises when considering spaces of paths in M with endpoints constrained to lie on given submanifolds (the so-called D-branes). In this talk, I describe a way to extend the TQFT structure of string topology into an analogue of TQFT which incorporates open strings. The method of construction is homotopy theoretic, and it makes use of constrained mapping spaces from fat B-graphs (which I define) into the ground manifold M. I also discuss how certain spaces of fat B-graphs model the classifying space of constrained diffeomorphism groups of an open-closed cobordism.

Inspired by work of Grojnowski and Greenlees on Rational Circle- Equivariant Elliptic Cohomology, we give a new construction of an equivariant theory valid for any compact Lie group which agrees with Greenlees' when C is an elliptic curve over the rationals and Grojnowski's in the case of a complex- analytic curve (both of which are only defined in case G=S^1). We then go on to construct a natural Thom isomorphism in elliptic (co)homology for complex vector bundles with certain vanishing conditions on their Borel-equivariant Chern classes. Applications include a short and elegant proof of the Witten rigidity theorem.