Past Topology Seminar

In this lecture, I will give a more precise statement of the Baez-Dolan cobordism hypothesis, which gives a description of framed bordism (higher) categories by a universal mapping property. I'll also describe some generalizations of the cobordism hypothesis, which take into account the structure of diffeomorphism groups of manifolds and which apply to manifolds which are not necessarily framed.

In this lecture, I'll give an overview of some ideas from higher category theory which are needed to make sense of the Baez-Dolan cobordism hypothesis. If time permits, I'll present Rezk's theory of complete Segal spaces (a model for the theory of higher categories in which most morphisms are assumed to be invertible) and explain how bordism categories can be realized in this framework.

In this lecture, I will review Atiyah's definition of a topological quantum field theory. I'll then sketch the definition of a more elaborate structure, called an "extended topological quantum field theory", and describe a conjecture of Baez and Dolan which gives a classification of these extended theories.

I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations.

This is joint work with Rob Meyerhoff.

The notion of a sutured 3-manifold was introduced by Gabai. It is a powerful tool in 3-dimensional topology. A few years ago, Andras Juhasz defined an invariant of sutured manifolds called sutured Floer homology.

I'll discuss a simpler invariant obtained by taking the Euler characteristic of this theory. This invariant turns out to have many properties in common with the Alexander polynomial. Joint work with Stefan Friedl and Andras Juhasz.

There is a well-known relationship between the theory of formal group schemes and stable homotopy theory, with Ravenel's chromatic filtration and the nilpotence theorem of Hopkins, Devinatz and Smith playing a central role. It is also familiar that one can sometimes get a more geometric understanding of homotopical phenomena by examining how they interact with group actions. In this talk we will explore this interaction from the chromatic point of view.

Baumslag-Solitar groups are very simple groups which are not Hopfian (they are isomorphic to proper quotients). I will discuss these groups, as well as their obvious generalizations, with emphasis on their automorphisms and their generating sets

We study non-commutative analogues of Serre-ï¬~Abrations in topology. We shall present several examples of such ï¬~Abrations and give applications for the computation of the K-theory of certain C*-algebras. (Joint work with Ryszard Nest and Herve Oyono-Oyono.)

$PD$-complexes model the homotopy theory of manifolds.

In dimension 3, the unique factorization theorem holds to the extent that a $PD_3$-complex is a connected sum if and only if its fundamental group is a free product, and the indecomposables are aspherical or have virtually free fundamental group [Tura'ev,Crisp]. However in contrast to the 3-manifold case the group of an indecomposable may have infinitely many ends (i.e., not be virtually cyclic). We shall sketch the construction of one such example, and outline some recent work using only group theory that imposes strong restrictions on any other such examples.

It is a classical result that the Alexander polynomial of a fibered knot has to be monic. But in general the converse does not hold, i.e. the Alexander polynomial does not detect fibered knots. We will show that the collection of all twisted Alexander polynomials (which are a natural generalization of the ordinary Alexander polynomial) detect fibered 3-manifolds.

As a corollary it follows that given a 3-manifold N the product S1 x N is symplectic if and only if N is fibered.