Fall 2011

September 29, 2011

Schwarz-Christoffel Mappings and Hypergeometric Differential Equations — Alex Wright

Abstract: Schwarz-Christoffel mappings are mappings from the upper half plan to a polygon. These are abstractly guaranteed by the Riemann Mapping theorem. After discussing the fundamental theory, we will move on to hypergeometric differential equations (HGDE), which are the easiest ODEs you don’t learn in first year calculus. HGDE were studied by the likes of Euler, Gauss and Riemann, and they continue to be of widespread interest today, for example the study of lattices in $PU(n,1)$, in the theory of period mappings and VHS, in number theory, and in rational billiards. Our discussion of HGDE will focus on Schwarz triangle mappings: The ratio of two solutions to a HGDE is a Schwarz-Christoffel mapping from the upper half plane to a triangle, and the monodromy of the HGDE is a triangle group.

October 6, 2011

Commensurations of Nilpotent Groups — Daniel Studenmund

Abstract: The abstract commensurator of a group $G$ is the group of (equivalence classes of) isomorphisms between its finite index subgroups. Some groups have commensurator group much larger than their automorphism group, while other groups are completely rigid. For example, the commensurator group of $\mathbb{Z}^n$ is $\mathrm{GL}_n(\mathbb{Q})$, while every commensuration of a mapping class group is induced by conjugation. We will give a general technique to compute commensurators of finitely generated nilpotent groups, and apply this to higher dimensional discrete Heisenberg groups.

October 13, 2011

Computing filtered Hamiltonian Floer homology — Strom Borman

For many quantitative applications of Floer theories, one is required to compute the homology with respect to some filtration and in practice this can be difficult. In this talk I will outline a strategy for turning certain filtered Hamiltonian Floer homology computations into contact homology computations. The proof of this strategy requires a general compactness theorem, which includes `stretching the neck’ for Hamiltonian Floer trajectories, and generalizations of Bourgeois–Oancea’s work relating symplectic homology with contact homology. This is joint work in progress with Y. Eliashberg and L. Polterovich, and is part of a larger project with L. Diogo and S. Lisi.

October 20, 2011

Actions of higher rank lattices on low-dimensional manifolds — Wouter van Limbeek

A famous conjecture made by Zimmer states that lattices in ‘large’ semisimple Lie groups cannot act nontrivially on low-dimensional manifolds. Ghys proved Zimmer’s conjecture for $C^1$ actions on the circle. Using Margulis superrigidity and the normal subgroups theorem, Farb and Shalen obtained results on real analytic actions on surfaces. I will discuss these theorems, and sketch how Farb and Shalen’s approach may give results in higher dimensions.

October 27, 2011

Finiteness Conditions in Group Cohomology — Jenny Wilson

The aim of this talk is to address the questions: What is the cohomological dimension of a group? What is virtual cohomological dimension? What are the topological implications of these quantities? There will be many examples. Time permitting, I will also say what it means for a group to be of type $FP_n$, $FP$, or $FL$, and some consequences of these properties.

November 3, 2011

Livshitz Periodic Point Theorem (s) and Applications —- Ilya Gekhtman

Given a hyperbolic dynamical system $T:X \to X$ on a compact space,and a function $g:X \to \mathbb{R}$ does the cohomological equation $g=f\circ T-f$  have solution $f$? A. Livshitz proved in 1972 that for a large class of systems a sufficient condition is for all averages of g over periodic orbits of T to vanish. I will state Livshitz result more precisely, outline a proof, state some generalizations, and discuss applications to proving nonarithmeticity of length spectra of infinite volume hyperbolic manifolds and other stuff.

November 10, 2011

Almost every pair of interval exchange transformations are disjoint— Jon Chaika

We will outline the proof that almost every interval exchange is different. A key step in the proof is that any sequence of density one contains a rigidity sequence for almost every interval exchange. Relevant terminology will be defined.

November 17, 2011

What is the difference between negative curvature and hyperbolic?— Tam Nguyen Phan

I will give some answer to the above question with focus on complete, finite volume manifolds.

November 24, 2011

Thanksgiving.  No seminar.

December 1, 2011

Several faces of the Euler class — Tshishiku, Bena

I will define the Euler class of a circle bundle over a surface in three ways:

1. topology (obstruction theory)

2. algebra (group cohomology and quasi-morphisms)

3. geometry (de Rham cohomology)

and discuss why they are the same. Time permitting I will mention applications to the classification of 3-manifolds and the Milnor-Wood inequality.