# Winter 2013

 January 17, 2013 Quantitative Shrinking Targets for IETs and Rotations — Jon Chaika In this talk we present some quantitative shrinking target results. Consider $T:[0,1] \to [0,1]$. One can ask how quickly under $T$ a typical point $x$ approaches a typical point $y$. In particular given $\{a_i\}_{i=1}^{\infty}$ is $T^ix \in B(y,a_i)$ infinitely often? A finer question of whether $T^ix \in B(y,a_i)$ as often as one would expect will be discussed. That is, does $\displaystyle \underset{N \to \infty}{\lim}\frac{\underset{n=1}{\overset{N}{\sum}} \chi_{B(y,a_n)(T^nx)}}{\underset{n=1}{\overset{N}{\sum}} 2a_n}=1$             for almost every $x$? These results also apply to a billiard in the typical direction of any rational polygons. This is joint work with David Constantine.
 January 24, 2013 Note: FFSS this week will start at 3pm in E308. Extension Problems for Groups of Diffeomorphisms — Katie Mann If $M$ is a manifold with boundary $\partial M$, restricting an (isotopically trivial) diffeomorphism of $M$ to its action of the boundary gives a surjective homomorphism from $\mathrm{Diff}_0(M)$ to $\mathrm{Diff}_0(\partial M)$. Does this map ever admit a section? The answer is usually no, and the reasons are usually interesting. In this talk, we describe obstructions to sections $\mathrm{Diff}_0(\partial M) \to \mathrm{Diff}_0(M)$ and in a number of cases obstructions to lifting finitely generated subgroups of $\mathrm{Diff}_0(\partial M)$ to groups of diffeomorphisms of $M$. Our methods range from the dynamical to the cohomological, and will be illustrated as much as possible with concrete examples. Notes in pdf: Extension Problems for Groups of Diffeomorphisms
 January 31, 2013 PSL(2,R) Representations of a Rank 2 Free Group — Alden Walker The character variety of $\textrm{PSL}(2,\mathbb{R})$ representations of a free group is subset of $\mathbb{R}^3$. I’ll describe a classification (not made up by me) of the representations as realizations of the free group as the fundamental group of a hyperbolic pair of pants or once-punctured torus. This will be all about playing with $2 \times 2$ matrices. Reference: Section 4 of Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces
 February 7, 2013 Thurston’s Model Geometries — Andrew Geng In the 1970s, Thurston classified the three-dimensional homogeneous model geometries—simply connected manifolds with transitive isometry groups and a couple of other niceness conditions. Desire to classify everything under the sun aside, why do we even care? And what’s so special (or not) about three dimensions? I’ll sketch Thurston’s proof and highlight some examples that show what changes when the dimension or the definitions change.
 February 12, 2013 Note: There are FFSS talks on Tuesday and Thursday this week. This talk starts at 4:15pm in the usual room, E308. Surfaces in 4–Space — Joshua Batson I will talk about surfaces embedded and immersed in $4$–space, how to visualize them and their normal bundles, and how to build branched covers along them. When you cut a $4$–manifold containing a surface, you get a $3$–manifold with a knot in it — the study of the (knot, surface) pair goes back to Wirtinger (1905). For orientable surfaces bounding torus knots, this is the relative version of the Thom conjecture, resolved in the 90’s using Floer homology. My research concerns nonorientable surfaces, where all the old theorems fail, and I’ll talk about some new conjectures and techniques. Preprint: Nonorientable four-ball genus can be arbitrarily large
 February 14, 2013 Polynomial Invariants of Finite Reflection Groups — Jenny Wilson It is well-known that the algebra of symmetric polynomials in $n$ variables is generated by (for example) the elementary symmetric polynomials, $n$ algebraically independent homogeneous generators. A classical theorem of Chevalley states that this phenomenon holds in general for the algebra of invariant polynomials associated to a finite reflection group — and the degrees of the homogeneous generators have some unexpected combinatorial properties. In this talk, I will briefly review the basics of Coxeter groups, and describe their polynomial invariants. I will define coinvariant algebras, and their relation to the cohomology of flag varieties. Time permitting, I will outline a proof of Chevalley’s theorem. Notes in pdf: Polynomial Invariants of Finite Reflection Groups
 February 21, 2013 Slim Unicorns and Hyperbolicity of Curve Graphs — Sebastian Hensel The curve graph is a graph which encodes the intersection pattern of simple closed curves on a surface. This graph has proven extremely useful in the study of Teichmüller theory, $3$–manifolds and mapping class groups. I will define the curve graph and show some of its applications. Finally, I will describe joint work with Piotr Przytycki and Richard Webb giving a new short proof of hyperbolicity of curve and arc graphs using unicorns. Notes in pdf: Slim Unicorns and Hyperbolicity of Curve Graphs
 February 28, 2013 Surface Bundles Over Surfaces with Multiple Fiberings — Nick Salter The theory of the Thurston norm shows that it is common for a given three-manifold to fiber in infinitely many ways as a surface bundle over the circle. Unfortunately, this ultimately seems to be a story about manifolds that fiber over the circle, rather than about surface bundles. In four dimensions, the situation is much more rigid, and it is not even known if there are examples of surface bundles (of nonzero Euler characteristic) that fiber in three ways! In this talk, I will survey what little we do know about surface bundles over surfaces with multiple fiberings. This will lead into a quest to find surface subgroups of mapping class groups, and I will describe a few constructions. Notes in pdf: Surface Bundles Over Surfaces with Multiple Fiberings
 March 7, 2013 Note: Ilya’s talk on Tuesday has been moved to Thursday. Alex’s talk on Thursday has been canceled. The Horofunction Boundary of the Teichmuller Metric — Ilya Gekhtman The horofunction (or Busemann) boundary of a metric space is a way of compactifying it by its busemann functions. For negatively curved manifolds, it coincides with the Gromov boundary of equivalence classes of asymptotic geodesic rays. Among other things, it is formally the right setting for Patterson–Sullivan theory, even if your space is insufficiently (or not at all) hyperbolic. Miyachi and independently Liu and Siu proved that the horofunction compactification of Teichmuller space coincides with the so called Gardiner–Masur boundary obtained by embedding it into $\mathrm{P}\mathbb{R}^S$, where $S$ is the set of simple closed curves, via extremal length and looking at the closure. (If instead we used hyperbolic length we would get the Thurston boundary PMF). For Teichmuller space, the horofunction compactification differs from both the horrible Gromov compactification and the visual and Thurston compactifications, which can in slightly different ways be identified with PMF. It contains PMF as a very small closed subset. An advantage of the horofunction compactification is that every geodesic ray converges, and those with uniquely ergodic vertical foliations converge to their projective classes. Unfortunately, not every point on the boundary can be approached by geodesic rays.
 March 13, 2013 Note: There are FFSS talks on Wednesday and Thursday this week. This talk will take place at 4pm, in the usual room, E308. Topology of Loop Spaces — Fedor Manin Loop spaces often pop up when you study simply connected spaces. In this talk I will start by giving a combinatorial construction of a space which obviously injects into the loop spaces of most spaces one cares about. If you believe that this injection is a homotopy equivalence, that allows you to find the homology of the loop space. I will then prove this equivalence in as much detail and rigor as time permits. No motivation will be provided, so bring your own.
 March 14, 2013 Some Groups of Mapping Classes not Realized by Diffeomorphisms — Bena Tshishiku In the theory of surface bundles, it is an open question whether every representation of a surface group to Mod($S_g$) lifts to Diff($S_g$). Recently, Bestvina–Church–Souto showed that the point pushing subgroup of the punctured mapping class group Mod($S_g,*$) does not lift to Diff($S_g,*$). I will explain this result and how it can be used to show that $\text{Diff}(S_g)\rightarrow\text{Mod}(S_g)$ does not split. Motivation will be provided.