|Speaker||Time||Title and Abstract|
| Marc Culler
University Illinois Chicago
|11:00-12:00|| Title: Margulis numbers for Hyperbolic Haken manifolds.
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.
|lunch||12:00-1:30||various eateries are within walking distance|
| Stephen Bigelow
UC Santa Barbara
|1:30-2:30|| Title: Algebras that count graph colorings
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.
|2:45-3:45|| Title: Legendrian contact homology for Seifert fibered spaces
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.
|4:15-5:15|| Title: Refined adjunction inequalities for 4-manifolds with a circle action
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.
|dinner||5:30||Beer, wine, and food!|
Click here for a nice map of the Claremont Colleges. On the map, Milikan is numbered 8 (but so are other buildings!) Notice on the map, the indications to reach the 210 freeway, or the 10 freeway. To reach Pomona College from the 10 freeway, exit at Indian Hill, go north, turn right (east) on 6th street and then turn left (north) on College. To reach Pomona College from the 210 freeway, if traveling East, exit at Towne Avenue, turn right (south) on Towne, turn left (east) on Foothill Blvd, and turn right (south) onto College Avenue. If traveling West on the 210 freeway, exit at Baseline/Padua, turn right (west) onto Baseline, turn left (south) onto Padua at the first light, turn right (west) onto Foothill Blvd at the third light, turn left (south) onto College Avenue.