# Geometry/Topology Seminar

Thursdays at 3:00PM, in 308 Eckhart.
if you have problems viewing the abstracts click on the link Geometry/Topology Seminar Web Page

## FALL 2003

Sep 25,
Uri Bader , Technion, Haifa, Israel
Normal subgroup Theorem   Abstract

Oct 9,
Fred Cohen , University of Rochester
Braid groups, the topology of configuration spaces, and homotopy groups   Abstract

Oct 14, Eckart 202, (Special date and place)
Danny Calegari California Institute of Technology
analysis and bounded cohomology of groups acting on the plane   Abstract

Oct 16,
Eli Glasner , Tel Aviv University
Some applications of the Ramsey and dual Ramsey theorems in topological dynamics   Abstract

Oct 23,
Chris Leininger , Columbia University
On groups generated by two positive multi-twists.   Abstract

Oct 30,
Igor Pak , MIT
Geodesic flow on convex polyhedra and nonoverlapping unfolding   Abstract

Tuesday Nov 4, (Special Day and Place) 3:00 pm, Eckart 203, (Joint seminar with Algebraic Topology),
Erik Pedersen , Binghamton
Smoothing loop spaces   Abstract

Nov 6,
Richard Hain , Duke University (visiting UC)
The symplectic cohomology of the genus 2 mapping class groups

Nov 11 Tuesday, Eckart 202 (Special Date and Place),
Jason Behrstock
Some asymptotic geometry of the mapping class group   Abstract

Nov 13, (Time Change) 1:30pm
Maryam Mirzakhani , Harvard University
Simple geodesics and intersection theory on the moduli space of curves

Nov 19, (Special Day and Time and Place) 3:30pm Eckart 312
David Fisher , CUNY Lehman College, New York
Local Rigidity of Groups Actions
 Abstract: In joint work with G.A.Margulis, we have proven  many  results showing that smooth actions of "large groups" on compact manifolds are locally rigid.  By local rigidity we mean that any perturbation of the action is conjugate back to the original action by a diffeomorphism.   The exact meaning of this depends on which topology ($C^k, C^{\infty}$) one uses to define perturbation and the smoothness of the resulting conjugacy.  We have recently dramatically improved our results in this direction and can now prove many of our results for perturbations that are only $C^2$ or $C^3$ close to the original action and also produce $C^{\infty}$ conjugacies as long as the perturbation is  $C^{\infty}$ action.   In order to be able to discuss some of the ideas in the proofs and some relations to KAM theory, after a brief introduction, I will concentrate on the following: Theorem.  Let G be a group with property T of Kazhdan.  Then any isometric action of G on a compact Riemannian manifold is locally rigid.

Nov 20,
Vadim Kaloshin , California Institute of Technology
On Newhouse phenomenon of infinitely many coexisting sinks
 Abstract: Consider the space of $C^r$ diffeomorphisms (smooth invertible selfmaps) of a compact surface $M$ (e.g. $S^2$ or $T^2$) Diff$^r(M)$ with $r\geq 2$. A sink of $f:M \to M$ is a periodic point $x \in M$ which attract all points from its neighbourhood (as in your kitchen). Points attracted to $x$ called basin of attraction of $x$. In 60-th Thom conjectured that a generic diffeomorphism can not have infinitely many coexisting sinks. Indeed, each sink has an open basin of attraction and infinitely many of those seems too much. In 70-th Newhouse constructed an open set of diffeomophisms $N \subset \textup{Diff}^r(M)$ such that generic diffeomorphism in $N$ does have infinitely many coexisting sinks. It is an amazing phenomenon, called Newhouse phenomenon. It disproves Thom's conjecture and significant obstacle to describe ergodic properties of surface diffeomorphisms. We shall discuss this phenomenon and closely related results of Benedicks-Carleson, Mora-Viana, Wang-Young, Morreira-Yoccoz. The main result indicates in some sense this phenomenon has "probability zero". This is a particular case of so-called Palis conjecture. This is a joint work with A. Gorodetsky

Dec 4
Emmanuel Breuillard , Yale University
A topological Tits alternative.
 Abstract: I will describe a joint work with T. Gelander. In his celebrated 1972 paper, J. Tits proved the following dichotomy for linear groups: Tits Alternative: Let K be a field and G a subgroup of GL(n,K), in case char(K) > 0 assume further that G is finitely generated. Then either G contains a solvable subgroup of finite index, or G contains a non commutative free subgroup. We proved a topological version of this theorem. Suppose k is a local field, that is R, C, a finite extension of a p adic field or a field of formal power series in one variable over a finite field. Then the topology of k induces a topology on GL(n,k) and on any subgroup of it. Theorem: Let G be any subgroup of GL(n,k). Then either G contains a relatively open solvable subgroup or G contains a relatively dense free subgroup. For example, the group SL(n,Q) contains free subgroup of rank 2 which is dense with respect to the real topology, and similarly for any prime p it contains a dense free subgroup of rank r(p) which is dense with respect to the p-adic topology. For k=R our theorem answers a question of Carriere and Ghys and provides a short proof for a conjecture of Connes and Sullivan on amenable actions which was first proved by Zimmer by means of super rigidity methods. For k non Archimedean our theorem implies a conjecture of Dixon, Pyber, Seress and Shalev on profinite groups, as well as a conjecture of Shalev on coset identity in pro p groups (in the analytic case). We also apply our theorem to prove a conjecture of Carriere, that the volume growth of the leaves in any Riemannian foliation on a compact manifold is either polynomial or exponential.

Friday Dec 5, 1:30pm-3pm Eckart 308(Special Date, Time and Place)
Richard Hain , Duke
Torelli Groups and Enumerative Geometry of Moduli of Curves Part I
 Abstract: In these talks I will explain how the structure of mapping class and Torelli groups leads to previously unknown expressions in cohomology for classes of certain loci in Deligne-Mumford moduli spaces in terms of tautological classes.

Monday Dec 8, 3-6pm Eckart 206 (Special Date, Time an Place)
Richard Hain , Duke
Torelli Groups and Enumerative Geometry of Moduli of Curves Part II
 Abstract: In these talks I will explain how the structure of mapping class and Torelli groups leads to previously unknown expressions in cohomology for classes of certain loci in Deligne-Mumford moduli spaces in terms of tautological classes.

## Winter 2004

Jan 15,
Lisa Carbone , Rutgers University
TBA

Jan 22,
Curt McMullen , Harvard
Dynamics of SL_2(R) over moduli space

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For questions, contact
• Benson Farb, farb at math.uchicago.edu
• Alex Eskin, eskin at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu
• Jeff Brock, brock at math.uchicago.edu
• Andrzej Zuk, zuk at math.uchicago.edu
• Chris Connell, cconnell at math.uchicago.edu
• Laura DeMarco, demarco at math.uchicago.edu
• Dan Grossman, dan at math.uchicago.edu
• Chris Hruska, chruska at math.uchicago.edu
• Nicolas Monod, monod at math.uchicago.edu
• Roman Muchnik, roma at math.uchicago.edu.