The cohomology rings of homogeneous spaces tend to have positive structure constants with respect to natural bases. In ordinary cohomology theory, this fact can be proved by using the (transitive) group action and a generic transversality argument. (joint work with with Anderson and Miller)

(Joint with Jim Belk) V has subgroups that are so close to being a BN-pair that the classical proof for simplicity of linear groups with irreducible Coxeter system goes through almost without change. It turns out that the subgroup F plays the role of the solvable Borel subgroup.

There are a number of distinct definitions of the notion of a weak n-category; each definition has its own advantages and disadvantages. One desirable property is to have for two n-categories C and D a suitable notion of the "n-category of functors" from C to D. I'll describe a model with this property, and in doing so try to shed some light on what makes the notion of n-category so difficult to pin down.

I give geometric invariants of cobordisms of fold maps, and show relations between the signature of the source manifold and the singular set, when the dimension of the target or the source manifold, or the codimension is small. Also show relations between the singular fibers of a fold map and the source manifold, for example, if the singular fibers are not so complicated in a sense, then the source manifold is null-cobordant.

Kuratowski's 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets that can be generated from a fixed set by taking closures and complements. In this talk I will consider the analogous questions for any possible subset of the operations {closure, complement, interior, union, intersection}, and any number of initially given sets. The topic is somewhat recreational and requires only undergraduate point-set topology.

The study of contact structures on 3-manifolds changed radically by Giroux's rather surprising correspondence with open book decompositions. Abstractly, an open book can be thought of as a pair $(\Sigma, \phi)$ where $\Sigma$ is a compact bordered surface and $\phi$ is an automorphism of $\Sigma$. Work by Goodman, Honda-Kazez-Mati\'{c}, and Giroux and Baldwin showed that one could extract rather strong geometric information directly from the monodromy $\phi$, and that this data was encoded by certain monoids in the mapping class group of $\Sigma$. We cement this analogy and show that most properties of interest to a contact geometer have a corresponding monoid in the mapping class group.