University Illinois Chicago

Abstract: This is joint work with Peter Shalen.

A Margulis number for a hyperbolic 3-manifold is a number $\lambda$ such that the $\lambda/2$-thin part of the manifold is homeomorphic to a union of cusp neighborhoods and tubes about short geodesics. There is a universal constant which serves as a Margulis number for all hyperbolic manifolds. For each hyperbolic manifold, the supremum of its Margulis numbers is a topological invariant.

Our result provides an explicit Margulis number for any hyperbolic Haken manifold. The proof involves a variety of ideas for estimating Margulis numbers, including an application of the Patterson-Sullivan construction in a new setting.

UC Santa Barbara

In 1971, Roger Penrose pointed out a connection between the representation theory of SO(3) and the four color conjecture. Graphs are used to represent morphisms between representations of SO(3), and the evaluation of a closed graph in the plane then turns out to count the number of way to four-color the regions of the graph. There are similar stories for other Lie groups, using a slightly more complicated weighted count of certain types of colorings. I will describe how to compute the details in the rank two case, using a new algebra that is in some sense a more fundamental version of the Temperley-Lieb algebra.

Stanford University

Abstract: Symplectic field theory is a powerful and beautiful approach to studying contact and symplectic manifolds, and this framework provides invariants of Legendrian knots in contact manifolds. In special cases, these constructions can be modeled combinatorially, leading to independent, rigorous, and computable invariants. In this talk, I'll describe a combinatorial construction of Legendrian contact homology for knots in Seifert fibered spaces, as well as the motivating geometry behind the scenes.

UC Riverside

Abstract: Given a smooth 4-manifold M, there is an estimate on the minimal genus among representatives of a class of H_2(M) in terms of an adjunction inequality involving Seiberg-Witten basic classes. In spite of the importance of such inequality in various problems (e.g. the solution of Thom Conjecture) it is known that in general such inequality is not sharp. In particular, in 1998, Peter Kronheimer proved that such inequality can be sharpened for 4-manifolds of the form S^1 X N^3 using the Thurston norm of N. It is not clear how to extend Kronheimer's approach to other classes of manifolds. Here we discuss how, using an approach that is quite different from Kronheimer's, we can recast and extend such result to 4-manifolds that are circle bundles over a 3-manifold whose fundamental group satisfies certain group-theoretic properties. More specifically, this group must be virtually RFRS; for example in the case of Haken hyperbolic manifolds (with b_1 >1) this is a consequence of Dani Wise's program. The talk is based on joint work with Stefan Friedl.