# Fall 2010

Nov. 30 2010

Representation Stability — Jenny Wilson

Abstract: Over the past few months, Benson, Tom, and others have been developing
the theory of ‘representation stability’. I will give a brief review of the representation theory of the symmetric group, and talk about what ‘representation stability’ means in that context.  As an example, I’ll describe the braid group and the pure braid group, and outline a proof that the $i^{th}$ cohomology group of the pure braid group is representation stable

Nov. 23 2010

Some Quasimorphisms — Egor Shelukhin

Abstract:
I have prepared a short talk on the note by Dupont-Guichardet showing the coinicidence of two different definitions of what we now understand as differentials of certain quasimorphisms. I can also say stuff about a particular special case –  the Maslov quasimorphism on the universal cover of the linear symplectic group – following the
foundational paper of Barge-Ghys

Nov. 9 2010

Bounded Cohomology — Kathryn Mann

Abstract:  An introduction to bounded cohomology, assuming you the listener knows nothing of the subject.  I’ll define bounded cohomology for a topological space and a group, prove some elementary results, like bounded cohomology of any amenable group vanishes, and give some motivation and context.

Bounded cohomolgy crops up in the guise of rigidity (such as Gromov’s minimal volume), in the study of quasimorphisms of groups, and in understanding groups acting on the circle (or other manifolds).  The goal of the talk is to give you some appreciation for the subject.

Nov. 2 2010

Teichmuller curves –Alex Wright

Abstract: An introduction to Teichmuller curves, focusing on the three different perspectives we use to think about these things.

Oct. 26 2010

Understanding distances and geodesic structure through subsurface projections — Spencer Dowdall

Subsurface projections were introduced by Masur and Minksy in 2000 and have become essential tool for understanding the geometry of Teichmuller space and the curve complex. Roughly, subsurface projection is a way to take any simple closed curve on a surface and project it to a simple closed curve on a proper subsurface. I’ll define this procedure more carefully and discuss some cool results about geodesics and distances.

Oct. 19 2010

Mixing of Flows With Hyperbolic-ish properties — Ilya Gekhtman.

Abstract: Mixing is a dynamical property, stronger than ergodicity, which can be described as “asymptotic independence”. A flow $g_{t}$ is mixing with respect to a probability measure $\mu$ if for any measurable sets $A$ and $B$, $\mu(A\cap g_{t}B) \longrightarrow \mu(A)\mu(B)$ as $t \longrightarrow \infty$. Among a plethora of other applications in math and beyond,the construction of sufficiently nice mixing measures for geodesic flows allows one to compute multiplicative bounds for a variety of asymptotic quantities on manifolds, such as orbit growth, volume of balls of radius $R$, and number of closed geodesics of length at most $R$. I will talk about a neat trick due to Martine Babillot that can be used to prove mixing of geodesic flows with sufficient hyperbolicity, including product measures on (not necessarily compact or finite volume)  manifolds of pinched  negative curvature (in particular the Bowen Margulis measure), and Knieper’s measure of maximal entropy for compact rank-1 symmetric spaces. An adaptation of the argument can also be used to prove mixing for analogues

Bowen-Margulis measures for the Teichmuller and Weil-Petersson geodesic flows on quotients of Teichmuller space by convex cocompact subgroups of the mapping class group.

Oct. 12 2010

Formulations of the Ehrenpreis Conjecture — Daniel Studenmund

Abstract:  The Ehrenpreis Conjecture states that if $X$ and $Y$ are any two closed hyperbolic surfaces, then for all $\epsilon > 0$ there are finite covers $Z_1 \to X$ and $Z_2 \to Y$ so that $Z_1$ and $Z_2$ are $(1+\epsilon)$-quasi-isometric.  We’ll discuss (and hopefully prove) the conjecture for the case of the torus, using the fact that Teich($T^2$) = $H^2$.  Then we’ll introduce the Teichmuller space of the universal hyperbolic solenoid in order to provide a restatement of the conjecture for hyperbolic surfaces.  Be warned that this is not my area of expertise, so the focus will be on exploring fun concepts rather than proving anything technical.

Oct 5 2010

Circle bundles over surfaces and the Euler class — Kathryn Mann

Circle bundles over surfaces are completely classified by their Euler number (which I will define in my talk).  But the Euler number is so much more than just a bundle classification.  For instance:

1. The Milnor-Wood inequality: Flat bundles over a genus g surface are precisely those with Euler number at most $2g-2$

2. Ghys’ theorem: Flat bundles with the same bounded Euler class are all semiconjugate.

In this talk, I’ll tell you everything you wanted to know (maybe, no promises here) about circle bundles over surfaces and the Euler number, hopefully even including some recent work of Michelle Bucher and Nicholas Monod.  I’ll then mention some interesting differences that pop up when you replace circle bundles with torus bundles.