Geometry/Topology Seminar
Spring 2010
Thursdays (and sometimes Tuesdays) 23pm, in
Eckhart 308

 Tuesday March 30 at 2pm in E308
 Ana Rechtman, University of Illinois at Chicago and Conacyt
 Existence of periodic orbits of geodesible vector fields on closed 3manifolds

Abstract: A vector field on a closed manifold M is
geodesible if there is a Riemannian metric making its orbits
geodesics. After discussing some examples of geodesible
vector fields and results on the existence of periodic
orbits for vector fields on closed 3manifolds, I will
sketch the proof of the existence of a periodic orbit when M
is either diffeomorphic to S^{3} or has
nontrivial \pi_{2} and X is
either real analytic or C^{∞} and
preserves a volume. On 3manifolds, the class of geodesible
vector fields contains Reeb vector fields, defined by a
contact form, and vector fields that admit a global cross
section.

 Thursday April 1 at 2pm in E308
 Frank Calegari, Northwestern University
 Modp homology growth in 3manifolds

Abstract: There exist 3manifolds X with infinite
fundamental group whose homology is that of the 3sphere. If
X is a hyperbolic 3manifold, however, then Lubotzky
explained how to find covers Y of X such that the homology
H_{1}(Y,Z/pZ) becomes arbitrarily large.
Already this is enough to prove that lattices in
SL_{2}(C) do not satisfy the congruence
subgroup property. In this talk, I will study the growth of
H_{1}(Y,Z/pZ) in some special towers of
covers of X corresponding to padic analytic groups. The
special case when G is the additive group of padic numbers
will be related to the classical Alexander polynomial. The
main theorem will be an exact asymptotic estimate for the
Z/pZrank of cohomology in these towers. This is joint work
with Matthew Emerton.

 Tuesday April 6 at 2pm in E308
 G. Ben Simon, E.T.H. Zurich
 Causal Representations of Surface Groups

Abstract: I will give a new characterization of
classical Teichmuller spaces for closed surfaces. This new
description will be the basis of a generalizations (we call
Causal Representations and is due to Hartnick and myself) of
the classical Teichmuller space. Further, causal
representations description carries a strong geometrical
ingredient. If time will permit I will describe roughly the
main steps in the proof of (one) the key statement(s). Joint
work with : M.Burger, T.Hartnick, A.Iozzi, A.Wienhard.

 Thursday April 8 at 2pm in E308
 Kevin Wortman, University of Utah
 Nonnonpositive curvature of some noncocompact arithmetic groups

Abstract: LeuzingerPittet conjectured that any
irreducible, noncocompact, arithmetic subgroup of a real
semisimple Lie group G has an exponential lower bound to its
isoperimetric inequality in the dimension that is 1 less
than the Rrank of G. Thus, arithmetic groups in higher rank
Lie groups would not admit a nonpositively curved geometry
in any reasonable sense. I'll show a proof of the conjecture
for the case of arithmetic groups whose relative Qtype is
A_{n}, B_{n}, C_{n}, D_{n},
E_{6}, or E_{7}.

 Tuesday April 13 at 2pm in E308
 David Constantine, University of Chicago
 Compact forms of homogeneous spaces and group actions

Abstract: Given a homogeneous space J\H,
is there a discrete subgroup \Gamma in
H such that J\H/\Gamma is a compact
manifold? This is the compact forms question and it is
expected that, apart from a few simple cases, compact forms
are rather rare. I will speak on an approach to the problem
using dynamics  specifically the action of the centralizer
of J. If this group is higherrank and
semisimple, Zimmer noticed that cocycle superrigidity could
be used to prove no compact form exists. In this talk I
build on this work and use tools from hyperbolic dynamics,
Ratner's theorems on unipotent flows to eliminate some
fairly strong algebraic conditions that Zimmer and his
coauthors must assume.

 Double Header  Part I
 Thursday April 22 at 2pm in E308
 John Mackay, University of Illinois at UrbanaChampaign
 Boundaries of hyperbolic groups and conformal dimension

Abstract: The boundary at infinity of a Gromov
hyperbolic group carries a canonical topological structure;
in fact, it has a canonical metric defined up to
quasisymmetric maps. I will survey some of the links between
the algebraic structure of the group and a metric invariant
of the boundary: Pansu's conformal dimension. We will focus
on cases where the boundary has topological dimension zero
or one.

 Double Header  Part II
 Thursday April 22 at 3:15pm in E308
 Julien Paupert, University of Utah
 New nonarithmetic lattices in SU(2,1)

Abstract: The first examples of nonarithmetic
complex hyperbolic lattices were constructed by Mostow in
1980. These examples are generalized triangle groups
generated by complex reflections of orders 3, 4 or 5. We
will see how to parametrize such triangle groups and how to
identify which of them are lattices. We will then survey an
ongoing project with Deraux and Parker whose goal is to use
this idea to construct (many) new nonarithmetic lattices in
SU(2,1).

 Special seminar
 Friday April 23 at 2pm in E207
 Albert Fathi, ENS Lyon
 A weak KAM approach to a theorem of Sullivan

Abstract: In his 1976 Paper in Inv. Math. Sullivan
has obtained a huge amount of results. We will discuss one
of them: If on a compact manifold M we give a continuous
family of closed convex cones in the tangent space, give a
necessary and sufficient conditions to find a closed form
which is positive a each point in the cone. Our goal after
explaining the problem from scratch is to describe an
application of weak KAM methods already used in joint work
with A. Siconolfi to give a proof of that theorem.

 Thursday April 29 at 2pm in E308
 Ben McReynolds, University of Chicago
 Geometric spectra

Abstract: I will discuss some joint work with Alan
Reid on recent generalizations of some classical results in
spectral geometry. For context, I will give an elementary
review of classical spectral geometry. This will cover the
geodesic length spectrum and eigenvalue spectrum for a
closed Riemannian manifold. I will discuss rigidity type
results and methods for producing counterexamples to full
spectral rigidity in this classical setting. I will then
discuss generalized versions of these spectra that arise
from basic geometric problems. For these geometric spectra,
I will discuss rigidity type results and methods for
producing counterexamples full spectral rigidity. An
emphasis will be placed on examples and making the material
accessible to a general geometrically minded audience.

 Thursday May 6 at 2pm in E308
 Simon Brendle, Stanford University
 Blowup phenomena for the Yamabe equation

Abstract: The Yamabe problem asserts that any
Riemannian metric on a compact manifold can be conformally
deformed to one of constant scalar curvature. However, this
metric is not, in general, unique, and there are examples of
manifolds that admit many metrics of constant scalar
curvature in a given conformal class. It was conjectured by
R. Schoen in the 1980s (and later by Aubin) that the set of
all metrics of constant scalar curvature 1 in a given
conformal class is compact, except if the underlying
manifold is conformally equivalent to the sphere
S^{n} equipped with its standard metric.
I will discuss counterexamples to this conjecture in
dimension 52 and higher. I will also describe joint work
with F. Marques, which extends these counterexamples to
dimension 25 and higher. The condition n \geq 25
turns out to be optimal.

 Thursday May 13 at 2pm in E308
 Peter Storm, Jane Street Capital
 Infinitesimal rigidity of hyperbolic manifolds with totally geodesic boundary

Abstract: Using the Bochner technique, Steve
Kerckhoff and I recently proved the following theorem. Let M
be a compact hyperbolic manifold with totally geodesic
boundary. If M has dimension at least four, then the
holonomy representation of M is infinitesimally rigid. This
is an infinite volume analog of the CalabiWeil rigidity
theorem. I will explain some of the background and ideas
used in the proof.

 Friday May 14 at 2pm in E207
 Iosif Polterovich, Universite de Montreal
 Spectral function on a Riemannian manifold: dynamical features and
average growth

Abstract: The talk focuses on asymptotic properties
of the spectral function of the Laplacian on a Riemannian
manifold. In particular, I will show that, roughly speaking,
the dimension of the manifold determines the average growth
of the spectral function. At the same time, its pointwise
behavior is closely related to dynamics of the geodesic
flow. Some open problems concerning almost periodicity of
the spectral function will be discussed. The talk is based
on a joint work with H. Lapointe and Y. Safarov.

 Tuesday May 18 at 2pm in E308
 Roland Roeder, Indiana University Purdue University Indianapolis
 Expanding Blaschke Products for the LeeYang zeros on
the Diamond Hierarchical Lattice

Abstract: In a classical work, Lee and Yang proved
that zeros of certain polynomials (partition functions of
Ising models) always lie on the unit circle. Distribution of
these zeros control phase transitions in the model. We study
this distribution for a special “MigdalKadanoff
hierarchical lattice”. In this case, it can be
described in terms of the dynamics of an explicit rational
function in two variables. More specifically, we prove that
the renormalization operator is partially hyperbolic and has
a unique central foliation. The limiting distribution of
LeeYang zeros is described by a holonomy invariant measure
on this foliation. These results follow from a general
principal of expressing the LeeYang zeros for a
hierarchical lattice in terms of expanding Blaschke products
allowing for generalization to many other hierarchical
lattices. This is joint work with Pavel Bleher and Mikhail
Lyubich.

 Thursday May 27 at 2pm in E308
 David Fisher, University of Indiana
 A Hodgede Rham theorem for group cohomology and applications

Abstract: I will develop a Hodge  de Rham type
theorem for the cohomology of a (locally compact group) with
coefficients in a representation by bounded operators on
Hilbert spaces. A main goal will be to generalize various
results of Mok, KorevaarSchoen and Shalom relating
vanishing of cohomology in families of representations to
vanishing of reduced cohomology. I will give applications to
the cohomology of semisimple Lie groups and their lattices
and also to local rigidity of group actions.

 Thursday June 10 at 2pm in E308
 Dave Morris, University of Lethbridge
 Survey of invariant orders on lattice subgroups

Abstract: At present, there are more questions than
answers about the existence of an invariant order on a
lattice in a semisimple Lie group. We will discuss four
different versions of the problem: the order may be required
only to be invariant under multiplication on one side, or on
both sides, and the order may or may not be required to be
total, rather than only partial.
For questions, contact