For current seminar information click HERE.

Caltech Geometry and Topology Seminar 2003-2004

Time: Fri 3pm (Except as noted)

Place: Sloan 257 (Except as noted)

Fall Quarter

Winter Quarter

Spring Quarter

Date	Speaker	Organization	Title	Abstract
2 April	David Fisher	CUNY	Local Rigidity, Property T and KAM theory	I will discuss joint work with G.A.Margulis, in which we have proved very general results concerning local rigidity of group actions. An action is locally rigid if any perturbation is conjugate back to the original action by a diffeomorphism. Particular focus will be on recent dramatic improvements on the regularity of the conjugacy. Though this work is part of a series of results concerning rigidity of quasi-affine actions of higher rank lattices, in order to be able to describe the improvements in regularity and some connnections to KAM theory, I will focus on the following theorem: Theorem: Let G be a group with property T of Kazhdan and M a compact manifold. Let A be an isometric action of G on M. Then any other C^{\infty} action of G on M which is sufficiently close to A in the C^{\infty} topology is conjugate to A by a C^{\infty} diffeomorphism f. Though the proof borrows ideas from KAM theory, one cannot in fact prove the general theorem by the KAM method. I will explain why this is true when A is the trivial action. If time permits, I will describe some work in progress, which is an attempt to prove local rigidity theorems for "smaller" groups using a KAM method.
9 April	NO SEMINAR
16 April	Danny Calegari	Caltech	The Euler class of groups acting on the plane	Which 4-manifolds can arise as the total space of a complete foliated R^2 bundle over a surface S? Equivalently, what is the homological classification of actions of a surface group by homeomorphisms on R^2? We concentrate on oriented surfaces and orientation-preserving homeomorphisms, and give a complete homological classification in every degree of smoothness. For surfaces of higher genus, every homological possibility is realized, for any degree of smoothness. Similarly, for surfaces of genus 1, every homological possibility is realized for C^0 actions. But for C^1 or smoother actions, a surprising rigidity phenomenon manifests, and the actions are all homologically trivial. This talk should be of interest to people working in Dynamics, Geometry and Topology, and will be accessible to first year graduate students.
23 April	Darren Long	UC Santa Barbara	Retractions over groups
27 April (Tuesday) 12pm (note time and date change)	Joerg Teschner	Freie Universität Berlin	An analog of a modular functor from the quantization of Teichmüller spaces	The program of the quantization of Teichmüller spaces is motivated by certain problems in mathematical physics, in particular by certain relations with conformal field theory. We plan to give an introduction to this program from a mathematical perspective. The main aim is to construct certain infinite-dimensional representations of the mapping class group in a way that is compatible with pants decompositions of the underlying Riemann surface. We will outline recent work showing that the previously given constructions indeed yield a mapping class group representation with the desired properties.
30 April	NO SEMINAR
7 May	Danny Calegari	Caltech	Quasigeodesic flows and universal circles	The most well-known examples of quasigeodesic flows on hyperbolic 3-manifolds are pseudo-Anosov flows, which are typically constructed dually to codimension one objects like taut foliations or sutured manifold hierarchies. We show that all quasigeodesic flows on hyperbolic 3-manifolds give rise to an abstract dynamical package with many features in common with pseudo-Anosov flows, including a universal circle, and a pair of invariant laminations. As a corollary, we show that the Weeks manifold does not admit a quasigeodesic flow. In the case that a quasigeodesic flow is transverse to a taut foliation, we discuss the relationship between the dynamical packages arising from the foliation and from the flow.
14 May	Daniel Groves	Caltech	Limits of CAT(0) groups
21 May	Michael Handel	CUNY	Periodic Points of Hamiltonian Surface Diffeomorphisms	The focus of the talk will be the following theorem, which is joint work with John Franks. Every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for $S^2$ provided the diffeomorphism has at least three fixed points.
22 May	N+1st Southern California Topology Conference
28 May	Yair Glasner	UIC	On infinite primitive permutation groups	I will describe three results. The methods are rather different but the motivation behind all of them is to understand infinite primitive permutation groups. 1. On primitive free groups. Joint with Miklos Abert. We show that a finitely generated free group F admits a proper subgroup that maps onto every proper quotient of F. The method we use comes from our study of random group actions on trees. 2. On primitive amalgamated products. After Burger and Mozes. When is the group generated by two subgroups A,B < \Gamma isomorphic to the amalgamated free product? 3. On primitive linear groups. Joint with Tsachik Gelander. Towards a full classification of primitive linear groups. After Margulis and Soifer. We characterize the linear finitely generated primitive groups in terms of their Zariski closure.
4 June	Dave Witte Morris	University of Lethbridge	Some arithmetic groups that cannot be left ordered	It is known that finite-index subgroups of the arithmetic group SL(3,Z) cannot be left ordered. (In geometric terms, this means that such subgroups have no interesting actions on the real line.) This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot be left ordered. This problem remains open, but joint work with Lucy Lifschitz establishes a relation with the property of bounded generation and gives new examples that cannot be left ordered. This includes all finite-index subgroups of SL(2,Z[\sqrt(r)]), where r > 1 is square free.