Group Theory, Number Theory, and Topology Day, UW-Madison, 24 Jan 2013

Yes, all three subjects in one day!  UW-Madison is happy to host a day of lectures by researchers at the interface between group theory, number theory, and topology.

9:30:  Coffee and bagels in the 9th floor lounge of Van Vleck Hall.

10:45 - 11:45:  Nathan Dunfield (University of Illinois at Urbana-Champaign)
Room:  Van Vleck B115
Title:  Integer homology 3-spheres with large injectivity radius
Abstract:  Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.

1:20-2:20:  Alan Reid (University of Texas at Austin)
Title: Genus of groups and 4 questions of Baumslag
Room: Van Vleck B231
Abstract: Let G be a f.g. residually finite group.  The genus of a group
G consists of the set of those f.g. residually finite groups that have the
same collection of finite quotients as G.  This talk will describe recent
work on the genus of free groups as well as questions of Baumslag concerning
the genus of parafree groups, and more general residually torsion-free
nilpotent groups.

2:30 -3:30:  Tamar Ziegler (Technion)
Title:An inverse theorem for the Gowers norms
Room: Van Vleck B129
Abstract: Gowers norms play an important role in solving linear equations in subsets of integers - they capture random behavior with respect to the number of solutions. It was conjectured that the obstruction to this type of random behavior is associated in a natural way to flows on nilmanifolds. In recent work with Green and Tao we settle this conjecture. This was the last piece missing in the Green-Tao program for counting the asymptotic number of solutions to rather general systems of linear equations in primes.

What's more, on January 25 there will be three more talks on broadly similar themes: a geometry seminar by Anne Thomas (Sydney) on the geometry of right-angled Artin groups, an algbebraic geometry seminar by Anatoly Libgober (UIC) involving fundamental groups of algebraic varieties, and a colloquium by Alexander Fish (Sydney) about dynamics and additive combinatorics. 

Some funding may be available for lodging in Madison for visitors, especially graduate students; please contact Jordan Ellenberg or Nigel Boston if you're interested in coming!  The funding source is the NSF-RTG grant "Number Theory and Algebraic Geometry at the University of Wisconsin."