Geometry/Topology Seminar

Fall 2008

Thursdays (and sometimes Tuesdays) 2-3pm, in Eckhart 308

- Thursday September 18 at 2pm in Ry358
- Mikhail Belolipetsky, Durham University
- Class field towers, Pisot numbers, and counting manifolds
- Abstract: I am going to talk about some recent results of a joint work with A. Lubotzky.
- Thursday October 2 at 2pm in E308
- Jan Essert, Universitaet Meunster
- Thursday October 9 at 2pm in E308
- Karsten Grove, University of Notre Dame
- An exotic T₁S⁴ with positive curvature?
- Abstract: I will explain how the study of positively curved manifolds with large isometry group has lead to the construction of a new example. The example is a cohomogeneity one manifold with the same homeomorphism type as the unit tangent bundle of the 4-sphere. This is joint work with Luigi Verdiani and Wolfgang Ziller.
- Thursday October 16 at 2pm in E308
- Ralf Spatzier, University of Michigan
- Higher Rank Abelian Anosov Actions on Tori
- Abstract: I will discuss recent work on the classification of Anosov actions by higher rank abelian groups on tori, and in particular recent joint work with Boris Kalinin and David Fisher.
- Thursday October 23 at 2pm in E308
- Mohammed Abouzaid, MIT
- Topological models for Lagrangian Floer Homology
- Abstract: A general philosophy in the categorical approach to symplectic topology says that invariants of a cotangent bundle should be expressible in terms of the base. I will describe a geometric construction which expresses one such invariant involving Lagrangians which are well-behaved at infinity as a concrete collection of modules over the chains of the based loop space, thought of as an algebra with respect to concatenation. Time permitting, I will explain a conjectural generalization to plumbings which turns out to be a theorem for punctured surfaces.
- Thursday October 30 at 2pm in E308
- Benson Farb, University of Chicago
- The Grothendieck-Katz p-curvature Conjecture (or, my brush with arithmetic geometry)
- Abstract: In this talk I'll explain the Grothendieck-Katz p-curvature conjecture. I'll then describe some recent progress, joint work with Mark Kisin, where we prove the conjecture for many locally symmetric varieties. The new ingredient is the use of Margulis's deep rigidity theorems, which are combined with previously known results of Katz and Andre.
- Thursday November 6 at 2pm in E308
- Pierre Py, University of Chicago
- Quasi-morphisms continuous for the C⁰-topology on groups of area-preserving diffeomorphisms of surfaces
- Abstract: I will recall the notion of a quasi-morphism on a group and describe a few examples of quasi-morphisms defined on groups of Hamiltonian diffeomorphisms of surfaces. Then we will try to answer the following question: which quasi-morphisms on these groups are continuous for the C⁰-topology? This question is related to the problem of the simplicity of the group of compactly supported area-preserving homeomorphisms of the disc. This is based on a joint work with M. Entov and L. Polterovich.
- Thursday November 13 at 2pm in E308
- Christopher Leininger, University of Illinois at Urbana-Champaign
- Pure braid relations
- Special Day and Room
- Wednesday November 19 at 2pm in Ry358
- Yitwah Cheung, San Francisco State University
- Special divergent trajectories for a homogeneous flow
- Abstract: Let L be a unimodular lattice in R^d+1 and consider the evolution L_t of this lattice under the action of diag(e^t,...,e^t,e^-dt). The kth successive minimum of a lattice L, denoted \lambda_k(L), is the smallest possible length for the longest vector in a linearly independent subset of L of order k. When k=1, this is simply the shortest nonzero vector in L. Clearly, \lambda₁(L) \leq ... \leq \lambda_d+1(L). It's also not hard to see that the product of the successive minima is universally bounded above and below by some positive constants. Schmidt conjectured that for any k=1,...,d-1 there exist L such that \lambda_k(L_t) tends to zero and \lambda_k+2(L_t) tends to infinity as t tends to infinity (with \lambda_k+1(L_t) oscillating between the two). We construct examples satisfying the Schmidt conjecture that have the “simplest combinatorics” in some suitable sense that will be made precise in the talk. This is joint work with Barak Weiss. We note that earlier examples satisfying Schmidt's conjecture have been constructed by Moshchevitin.
- Thursday November 20 at 2pm in E308
- Martin Bridgeman, Boston College
- Positive-definiteness of the Weil-Petersson extension on quasifuchsian space
- Abstract: We consider a natural two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmüller space. We describe completely the positive definite locus of G, showing that it is a positive definite metric off the fuchsian diagonal of quasifuchsian space. We show that G is equal to the pullback of the pressure metric from dynamics. We use the properties of G to prove that critical points of the Hausdorff dimension function on quasifuchsian space must have Hessian which is positive definite on at least a half-dimensional subspace.

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Ben McReynolds, dmcreyn at math.uchicago.edu
Nathan Broaddus, broaddus at math.uchicago.edu