Geometry/Topology Seminar
Fall 2008
Thursdays (and sometimes Tuesdays) 23pm, in
Eckhart 308

 Thursday September 18 at 2pm in Ry358
 Mikhail Belolipetsky, Durham University
 Class field towers, Pisot numbers, and counting manifolds

Abstract: I am going to talk about some recent
results of a joint work with A. Lubotzky.

 Thursday October 2 at 2pm in E308
 Jan Essert, Universitaet Meunster

 Thursday October 9 at 2pm in E308
 Karsten Grove, University of Notre Dame
 An exotic T_{1}S^{4} with positive curvature?

Abstract: I will explain how the study of positively
curved manifolds with large isometry group has lead to the
construction of a new example. The example is a
cohomogeneity one manifold with the same homeomorphism type
as the unit tangent bundle of the 4sphere. This
is joint work with Luigi Verdiani and Wolfgang Ziller.

 Thursday October 16 at 2pm in E308
 Ralf Spatzier, University of Michigan
 Higher Rank Abelian Anosov Actions on Tori

Abstract: I will discuss recent work on the
classification of Anosov actions by higher rank abelian
groups on tori, and in particular recent joint work with
Boris Kalinin and David Fisher.

 Thursday October 23 at 2pm in E308
 Mohammed Abouzaid, MIT
 Topological models for Lagrangian Floer Homology

Abstract: A general philosophy in the categorical
approach to symplectic topology says that invariants of a
cotangent bundle should be expressible in terms of the base.
I will describe a geometric construction which expresses one
such invariant involving Lagrangians which are wellbehaved
at infinity as a concrete collection of modules over the
chains of the based loop space, thought of as an algebra
with respect to concatenation. Time permitting, I will
explain a conjectural generalization to plumbings which
turns out to be a theorem for punctured surfaces.

 Thursday October 30 at 2pm in E308
 Benson Farb, University of Chicago
 The GrothendieckKatz pcurvature Conjecture
(or, my brush with arithmetic geometry)

Abstract: In this talk I'll explain the
GrothendieckKatz pcurvature conjecture. I'll
then describe some recent progress, joint work with Mark
Kisin, where we prove the conjecture for many locally
symmetric varieties. The new ingredient is the use of
Margulis's deep rigidity theorems, which are combined with
previously known results of Katz and Andre.

 Thursday November 6 at 2pm in E308
 Pierre Py, University of Chicago
 Quasimorphisms continuous for the C^{0}topology on groups of
areapreserving diffeomorphisms of surfaces

Abstract: I will recall the notion of a
quasimorphism on a group and describe a few examples of
quasimorphisms defined on groups of Hamiltonian
diffeomorphisms of surfaces. Then we will try to answer the
following question: which quasimorphisms on these groups
are continuous for the C^{0}topology?
This question is related to the problem of the simplicity of
the group of compactly supported areapreserving
homeomorphisms of the disc. This is based on a joint work
with M. Entov and L. Polterovich.

 Thursday November 13 at 2pm in E308
 Christopher Leininger, University of Illinois at UrbanaChampaign
 Pure braid relations

 Special Day and Room
 Wednesday November 19 at 2pm in Ry358
 Yitwah Cheung, San Francisco State University
 Special divergent trajectories for a homogeneous flow

Abstract: Let L be a unimodular lattice
in R^{d+1} and consider the
evolution L_{t} of this lattice under the
action of
diag(e^{t},...,e^{t},e^{dt}).
The kth successive minimum of a lattice
L, denoted \lambda_{k}(L), is
the smallest possible length for the longest vector in a
linearly independent subset of L of order
k. When k=1, this is simply the
shortest nonzero vector in L. Clearly,
\lambda_{1}(L) \leq ... \leq
\lambda_{d+1}(L). It's also not hard to see
that the product of the successive minima is universally
bounded above and below by some positive constants. Schmidt
conjectured that for any k=1,...,d1 there exist
L such that
\lambda_{k}(L_{t}) tends to zero
and \lambda_{k+2}(L_{t}) tends to
infinity as t tends to infinity (with
\lambda_{k+1}(L_{t}) oscillating
between the two). We construct examples satisfying the
Schmidt conjecture that have the “simplest
combinatorics” in some suitable sense that will be
made precise in the talk. This is joint work with Barak
Weiss. We note that earlier examples satisfying Schmidt's
conjecture have been constructed by Moshchevitin.

 Thursday November 20 at 2pm in E308
 Martin Bridgeman, Boston College
 Positivedefiniteness of the WeilPetersson
extension on quasifuchsian space

Abstract: We consider a natural twoform
G on quasifuchsian space that extends the
WeilPetersson metric on Teichmüller space. We describe
completely the positive definite locus of G,
showing that it is a positive definite metric off the
fuchsian diagonal of quasifuchsian space. We show that
G is equal to the pullback of the pressure metric
from dynamics. We use the properties of G to
prove that critical points of the Hausdorff dimension
function on quasifuchsian space must have Hessian which is
positive definite on at least a halfdimensional subspace.
For questions, contact