Cascade Topology Seminar: Abstracts

Matthew Ando (Illinois): "Orientation theory with applications"

I will begin with a review of the Umkehr map, which is surprisingly powerful in connecting generalized cohomology theories to geometry and analysis. It corresponds to integration in ordinary cohomology and the index of elliptic operators in K-theory. In elliptic cohomology it is related to the one-loop amplitude in string theory. The construction of the Umkehr map depends on a Thom isomorphism, or orientation. After a review of this story, I will discuss the question of the existence of Thom isomorphisms or orientations in generalized cohomology theories from a topological and algebro-geometric point of view.

Chris Douglas (Stanford): "Topological Field Theory and Twisted K-Theory"

I will begin by describing the basic elements of quantum field theory, namely the space of fields, the action, the quantum Hilbert space, and the partition function. Topological field theories abstract this structure by focusing attention only on the Hilbert space and the partition function. However, carefully considering the underlying spaces of fields can ease the construction of certain topological field theories. Freed, Hopkins, and Teleman pioneered this approach and used it to build a topological field theory whose state space is twisted equivariant K-theory. I will describe how the techniques of parametrized homotopy theory can help us compute certain aspects of the structure of this field theory.

Gabriel Indurskis (UBC): "Finding p-reps and Culler-Shalen seminorms of 3-manifold groups"

A well-known method due to Robert Riley characterizes the p-reps of the fundamental group of the exterior of a 2-bridge knot by the roots of a one-variable polynomial. (A p-rep of such a group is a representation with values in SL(2,C) which is parabolic on the peripheral subgroup.) We describe how to generalize this method via a "detour" through the eigenvalue variety to find the p-reps for the infinite family of manifolds obtained by Dehn filling of one boundary component of the Whitehead link exterior. As an application, we completely determine the Culler-Shalen seminorm for these manifolds.

Jacob Lurie (Harvard): "Cohomology Theories, Formal Groups, and Elliptic Spectra"

In this talk, we will give an exposition of the relationship between generalized cohomology theories and formal groups. The underlying theme is that algebraic geometry can provide as useful framework for organizing and interpreting the results of calculations in algebraic topology. We will close by discussing a subject where this point of view is indispensable: the theory of elliptic cohomology.

Dale Rolfsen (UBC): "Obstruction to nice foliations on 3-manifolds"

Victor Turchin (Oregon): "Knots and Poisson algebras operad"

The homology of the space of long knots can be described in terms of the Hochschild homology of the Poisson algebras operad. For example, the bialgebra of chord diagrams (responsible for the finite type knot invariants) is a subspace of the Hochschild homology in question. This description appears naturally both in the Vassiliev (theory of discriminants) and Goodwillie-Weiss-Sinha (calculus of functors) approaches. The talk will start by a definition of an operad.