# Spring 2013

 April 4, 2013 Classifying spaces of diffeomorphism groups: abstract nonsense in a concrete sensible way — Katie Mann I’ll remind you about classifying spaces and why you care about them, then sketch out the geometric ideas behind the proofs of two theorems: 1) (Hatcher) If $M$ is an irreducible $3-$manifold that is Haken, then components of Diff($M$) are contractible (unless $M$ is seifert fibered, and then you get some $S^1$s). 2) (Hendriks-Laudenbach) For an arbitrary orientable $3-$manifold $M$, the classifying space BDiff($M$) has the homotopy type of a finite CW complex. The ideas for doing the proof of #2 in a concrete hands-on way come from some unpublished work of Hatcher.
 April 11, 2013 Abstract commensurators of lattices in Lie groups — Daniel Studenmund The abstract commensurator of a group $G$ is the group of all isomorphisms between finite index subgroups of $G$ up to a natural equivalence relation. Commensurators of lattices in semisimple Lie groups are well understood, using strong rigidity results of Mostow, Prasad, and Margulis. We will describe commensurators of lattices in solvable groups, where strong rigidity fails. If time permits, we will extend these results to lattices in certain groups that are neither solvable nor semisimple.
 April 18, 2013 Stable reduction and compact curves in M_g — Joshua Batson The easiest way to get a family of Riemann surfaces is to write down some curve in the plane, like $y^2 = x^3 + t x + t$. As $t$ varies, we get a family of elliptic curves, ie, a map from our parameter space $\mathbb{C} \backslash \{ 0\}$ to the moduli space $M_1$. Even though setting $t=0$ gives cuspidal curve in the plane, there is a way of replacing the cusp with a nonsingular curve to get a holomorphic map from all of $\mathbb{C}$ to $M_1$. We will discuss similar shenanigans relating families of possibly singular plane quartics and holomorphic curves in $M_3$, and my hopeful plan is to give an explicitly parametrized compact curve in $M_3$.
 April 25, 2013 Entropy of holomorphic maps — Simion Filip Entropy is an invariant of a dynamical system which measures how chaotic is the behavior. It’s often hard to compute, but Gromov (plus Yomdin and others) discovered that for “polynomial” maps, it can be computed using essentially “linear algebra”. I will try to explain why this is the case. Caveat emptor: In general, there are fewer algebraic maps than you might think. How many self-maps does a Riemann surface of genus $g>1$ have?
 April 30, 2013 Note that this week’s FFSS talk is on Tuesday. The talk will start at 3pm in the usual room, E308. A taxonomy of filling functions — Fedya Manin The Dehn function of a discrete group $\Gamma$ describes how hard it is to solve the word problem in $\Gamma$, that is, how many relations it takes to reduce a trivial word of length $k$. This can be reformulated geometrically as the difficulty of filling a loop with a disk in a Cayley complex. In higher dimensions, similar definitions are possible, but several aspects of the definition have multiple possible interpretations, and many of these lead to different functions. Such definitions have been explored by Alonso-Wang-Price, Bridson, Brady-Bridson, Young, and Groft among others. In this talk, I will describe their definitions together with certain new ones, as well as relationships between the functions thus defined. I will also talk about connections to bounded cohomology.

 There is no seminar on May 2, because of the Billingsley Lecture and reception.

 May 7, 2013 Note there are FFSS talks on both Tuesday and Thursday this week. Tuesday’s talk will be at the usual time and place, 4:15pm in E308. Elliptical actions on Teichmüller space — Matthew Durham Kerckhoff’s solution to the Nielsen realization problem showed that the action of any finite subgroup of the mapping class group on Teichmüller space has a fixed point. The set of fixed points is a totally geodesic submanifold. We study the coarse geometry of the set of points which have bounded diameter orbits in the Teichmüller metric. We show that each such almost-fixed point is within a uniformly bounded distance of the fixed point set, but that the set of almost-fixed points is not quasiconvex. In this talk, I will discuss the machinery and ideas used in the proofs of these theorems.
 May 9, 2013 The co-Hopf property for mapping class groups — Nick Salter There is a well-developed analogy between the mapping class group and lattices in Lie groups. In this talk, I will present one aspect of this, and establish the co-Hopf property for mapping class groups, following the original paper of Ivanov and McCarthy. The statement of their theorem is interesting in its own right — it asserts that injections between mapping class groups of “similarly sized” surfaces are realized by embeddings of the surfaces, which has a certain flavor of “super-rigidity” to it.
 May 16, 2013 Cohomology of braid groups and configuration spaces of manifolds — Jenny Wilson The (ordered) configuration space $\mathscr{F}(X, n)$ of a topological space $X$ is the space of all ordered $n$-tuples of distinct points in $X$ — an object of extensive study in algebraic topology and geometry. It is a classical result that when $X$ is the plane $\mathbb{C}$, the space $\mathscr{F}(\mathbb{C}, n)$ is a $K(P_n, 1)$ for the pure braid group $P_n$. I will describe Arnold’s 1968 computation of the cohomology of $P_n$. In 1993, Totaro studied the cohomology of the configuration space of a closed oriented manifold, realizing these cohomology groups as the limit of a Leray spectral sequence. I will outline his results. References: Arnold, The cohomology ring of the group of colored braids Totaro, Configuration spaces of algebraic varieties Notes in pdf: Cohomology of braid groups and configuration spaces
 May 28, 2013 Some results that hold on every flat surface — Jon Chaika Recently Eskin–Mirzakhani–Mohammadi have proven a number of powerful results about the $\mathrm{SL}_2(\mathbb{R})$ orbits of abelian differentials and the $\mathrm{SL}_2(\mathbb{R})$ ergodic measures on the stratum. We discuss some results motivated and enabled by this work. One result is that for every abelian differential there is a measure on the stratum, such that after rotating in almost every direction, the geodesic flow equidistributes for this measure on the stratum. Another result is that for any surface the conclusion of Oseledets multiplicative ergodic theorem holds for the Kontsevich–Zorich cocycle. This has an application, being explored by others, to the windtree model. This is joint work with Alex Eskin.
 May 30, 2013 Braids, rings, wickets, and outer automorphisms — Sebastian Hensel We all know (and love) braid groups and configuration spaces of points in the plane, but what about configurations of other things in other things? For example, what do we know (and why should we care) about configuration spaces of rings in three-space, or wickets (arches) in the upper half-space? I use these questions as an excuse to talk about some cute differential topology and a circle of groups which are similar, related, and yet subtly different from braid groups. In the end I probably can’t resist talking about subgroup distortion…(but only for a tiny bit, I promise!)
 June 6, 2013 Note: This talk will start at 3 pm. Arithmetic Quantum Unique Ergodicity — Wouter van Limbeek For a closed Riemannian manifold $M$, there is a strong connection between the properties of the Laplacian and the geometry of $M$. Rudnick and Sarnack conjectured equidistribution for eigenfunctions of the Laplacian with large eigenvalues if $M$ has negative curvature, known as the “quantum unique ergodicity conjecture”. The conjecture was proved for arithmetic hyperbolic surfaces by Lindenstrauss using measure rigidity results and the symmetry in the arithmetic situation coming from Hecke operators. I will describe the background of this conjecture and some ideas of Lindenstrauss’s proof.