Geometry/Topology Seminar

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

Winter 2005

January 11,
Tara Brendle , Cornell
Commensurations of the Johnson kernel
Abstract: We show that Comm(K) = Aut(K) = Mod(S), where K is the subgroup of the mapping class group Mod(S) generated by twists about separating simple closed curves. In particular, this verifies a conjecture of Farb. (Joint work with Dan Margalit.)

January 13,
Herbert Abels , Bielefeld
Crystollagraphic and properly discontinuous affine groups

January 18,
Herbert Abels ,Bielefeld
Geometry of linear groups, invariant metrics on reductive groups

January 20,
Chris Leininger , Columbia University
Iterated H.N.N. extensions and amalgamated products in the mapping class group

January 25,
Paul Seidel , University of Chicago
What's the problem with symplectic four-manifolds?
Abstract: This will be a warmup talk to Auroux' talks on Wed 26th and Thu 27th's. Symplectic topology in dimension four is a field where, in spite of massive recent advances, open questions abound. I will explain one of the most famous and specific of these open questions; Donaldson's point of view concerning it; and recent results of Auroux-Munoz-Presas, which hopefully bring us one step further along the way to a solution.

Rescheduled indefinetely.
Dan Margalit , University of Utah
The abstract commensurator of the braid group
Abstract: Let Bn be the braid group on n strands. In joint work with Bob Bell, we describe all injections of finite index subgroups of Bn into Bn. In particular, this allows us to calculate the automorphism group of the pure braid group PBn. Certain elements of Aut(PBn) represent new examples of elements of the abstract commensurator of Bn. In joint work with Chris Leininger, which builds on these ideas, we are then able to compute the abstract commensurator of Bn explicitly.

January 26, 1:30pm, Eckart 202
Denis Auroux , MIT
Symplectic 4-manifolds, mapping class groups, and braid groups
Abstract: Donaldson has shown that every compact symplectic 4-manifold can be represented as a Lefschetz pencil (i.e., up to blowups, a fibration over the 2-sphere with at most nodal fibers). This leads to a description of symplectic 4-manifolds by positive factorizations in mapping class groups. In another related approach, symplectic 4-manifolds can be realized as branched covers of the complex projective plane (with a branch curve presenting cusp and node singularities). This leads to a description by factorizations in braid groups. This talk will provide an elementary introduction to these constructions, and a discussion of how they make it possible to reformulate various questions about the classification of symplectic 4-manifolds as (hard) combinatorial group theory problems.

January 27, 1:30pm, Eckart 202 (Time and room Change)
Denis Auroux , MIT
Lefschetz fibrations, plane curves, surgeries, and fiber sums
Abstract: The classification of symplectic 4-manifolds is closely related to that of Lefschetz fibrations and to that of plane curves with node and cusp singularities. We discuss recent results on these problems, in three main directions: (1) isotopy and regular homotopy results for plane curves; (2) surgeries along Lagrangian annuli and tori; (3) stabilization by fiber sums. We will in particular explain how classification results for fiber sums are closely related to Garside-type properties of braid groups and mapping class groups.

February 1,
Chris Hruska , University of Chicago
Commensurability invariants of nonuniform tree lattices (joint with Benson Farb)
Abstract: If X is a locally finite tree, its group of automorphisms G=Aut(X) is a locally compact group. A lattice in G is a discrete subgroup with cofinite Haar measure. With the right normalization of Haar measure, there is a simple combinatorial formula for the Haar measure, or ``covolume'' of a lattice. A study of these ``tree lattices'' generalizes the study of lattices in Lie groups over a nonarchimedean local field, and provides a remarkably rich theory (see the recent book by Bass-Lubotzky). One of the basic problems about a locally compact group is to classify its lattices up to commensurability. Outside the setting of linear groups, commensurability invariants have been hard to come by. We introduce two new commensurability invariants, and construct lattices realizing every possible choice of these invariants. In particular, we construct uncountably many noncommensurable lattices with any given covolume.

February 10,
Nathalie Wahl, University of Chicago
Curve complexes and mapping class groups in dimension 3
Abstract:Complexes of curves are a very fruitfull tool for studying mapping class groups of surfaces. In a similar fashion, one can use complexes of embedded spheres and discs to study mapping class groups of 3-manifolds, and along the way the automorphism groups of free groups. The main theorem is a stability theorem for mapping class groups of 3-manifolds, which we also interpret in terms of automorphisms groups of free groups with boundaries (joint work with Allen Hatcher).

February 24,
Uri Bader, University of Chicago
Normal sub-group theorem for lattices in products
Abstract:A famous theorem of Margulis asserts that any normal sub-group of a higher rank lattice is either finite or of finite index. We will discuss similar result valid in the setting of a lattice in a product of groups. Examples, apart of Lie-groups, consisting of automorphisms of trees and buildings. Our tools are ergodic theoretic. In particular we make use of the notion of the "Poisson boundary" of a group. We will not assume any familiarity with that object. The lecture will be based on a joint work with Yehuda Shalom.

March 1,
Alex Gorodnik , CalTech
Equidistribution on homogeneous spaces
Abstract:We investigate distribution of orbits of lattices acting on homogeneous spaces. In particular, we show that for natural actions of a lattice (in a semisimple Lie group) on finite-volume spaces, dense orbits are equidistributed. We also obtain equidistribution results for infinite-volume homogeneous spaces and deduce an analog of the quantitative Oppenheim conjecture (proved by Eskin, Margulis, Mozes) for Gramm matrices evaluated at integral frames. This is a joint work with Barak Weiss.

March 3,
Laura DeMarco , University of Chicago
Dynamics at the boundary of moduli space
Abstract:The moduli space M(d) is the space of all holomorphic self-maps of the Riemann sphere, modulo the action by conjugation of the group of Mobius transformations. In this talk, I will describe the limiting dynamics at the boundary of M(d). The approach involves algebraic geometry (geometric invariant theory), hyperbolic geometry (the barycenter construction), and of course holomorphic dynamics. I won't assume any background.

March 7, Monday 1:30pm Eckart 202 (Special Time, Place and Date)
Irwin Kra, Stony Brook
Some analytic aspects of Teichmuller theory

March 10,
John Crisp , Univ. de Bourgogne
Automorphisms and abstract commensurators of Artin groups