Geometry/Topology Seminar
Spring 2011
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Tuesday March 29 at 3pm in E206
 Gabor Elek, Renyi Institute
 Sofic groups, Connes Embedding, and L2 Betti numbers

Abstract: L2Betti numbers of manifolds were
invented by Atiyah. If the fundamental group is torsionfree
then these numbers are conjectured be integers. In the talk
I will speak about how can one recover this numbers by sofic
representations and try to give some reasoning why are they
integers.

 Thursday March 31 at 3pm in E308
 Yitwah Cheung, San Francisco State University
 Hausdorff dimension of the set of singular vectors

Abstract: Many problems in Diophantine approximation
can be addressed by analyzing the dynamics on the space of
lattices. For example, badly approximable vectors can be
understood in terms of bounded trajectories of a certain
diagonal flow. The notion of a singular vector is in a
natural way dual to that of a badly approximable vector. In
terms of the dynamics on lattices, they correspond to
divergent trajectories of the flow. In this talk, I will
describe the proof a recent result joint with Nicolas
Chevallier establishing that the Hausdorff dimension of the
set of singular vectors in R^{d} is
d^{2}/(d+1) for any d>1.
Essentially, this is equivalent to the statement that the
set of divergent trajectories of the flow by
diag(e^{t},...,e^{t},e^{dt})
acting on G/\Gamma where G=SL(d+1,R)
and \Gamma=SL(d+1,Z) has Hausdorff dimension
equal to dim G  d/(d+1). The general approach is
consistent with the strategy employed in the case
d=2 (Ann. Math. 173, (2011), 127167), which uses
a piecewise linear description of the flow trajectories to
obtain an encoding in terms of best approximants. I will
focus on the new ideas needed in the case d>2,
such as a sphere packing of H^{d+1} that
generalizes the packing of the upper half plane by disks
based at rational points p/q of diameter
1/q^{2}.

 Tuesday April 5 at 3pm in E308
 Luis Diogo, Stanford University
 Floer homology and holomorphic curves

Abstract: Floer homology is a powerful tool in
symplectic geometry. It is often hard to compute it
explicitly, because it involves counting solutions of a
perturbed CauchyRiemann equation. I will explain an
approach to showing how Floer homology can instead be
described (under certain hypotheses) by counting solutions
of (unperturbed) CauchyRiemann equations, as well as
gradient flow lines of auxiliary Morse functions. The
advantage is that this information can sometimes be computed
explicitly. This framework seems adequate for studying
spectral invariants, symplectic homology and Floertype
operations. This is part of a joint project with S. Borman,
Y. Eliashberg, S. Lisi and L. Polterovich.

 Thursday April 7 at 3pm in E308
 Howard Masur, University of Chicago
 Ergodicity of the WeilPetersson geodesic flow

Abstract: This is joint work with Keith Burns and
Amie Wilkinson. Let \Sigma be a surface of genus
g with n punctures. We assume
3g3+n>0. Associated to \Sigma is
the Teichmuller space. This is the space of hyperbolic
metrics one can put on \Sigma, up to isotopy. The
mapping class group acts on the Teichmuller space with
quotient, the Riemann moduli space \cal
M(\Sigma). There are a number of interesting metrics
on \cal M(\Sigma); one of which is the
WeilPetersson metric. It is a Riemannian metric of negative
curvature and finite volume but it is not complete. In this
talk I will discuss the background on this metric and the
following theorem. \ Theorem: The WeilPetersson geodesic
flow is ergodic on \cal M(\Sigma).

 Tuesday April 12 at 3pm in E308
 Sanghyun Kim, KAIST/Tufts
 Hyperbolic Surface Subgroups of Doubles

Abstract: Let us consider the presentation complex
X of an arbitrary 2generator presentation. We
puncture each 2cell and double along the holes to get
Y, called the double of X. We show
that the fundamental group D of Y
contains a closed hyperbolic surface group or D
splits as a free product (and hence reduces to a simpler
case). With more generators, we also resolve the case when
the relator set is minimal and kregular.

 Thursday April 14 at 3pm in E308
 Michael Brandenbursky, Vanderbilt University
 Finite type invariants obtained by counting surfaces

Abstract: A Gauss diagram is a simple, combinatorial
way to present a knot. It is known that any Vassiliev
invariant may be obtained from a Gauss diagram formula that
involves counting (with signs and multiplicities)
subdiagrams of certain combinatorial types. These formulas
generalize the calculation of a linking number by counting
signs of crossings in a link diagram. Until recently,
explicit formulas of this type were known only for few
invariants of low degrees. I will present simple formulas
for an infinite family of invariants arising from the
HOMFLYPT polynomial. I will also discuss an interesting
interpretation of these formulas in terms of counting
surfaces of a certain genus and number of boundary
components in a Gauss diagram. This is a joint work with M.
Polyak.

 Tuesday April 19 at 3pm in E308
 YongGeun Oh, University of Wisconsin  Madison
 Floer homology and continuous Hamiltonian dynamics

Abstract: Alexander isotopy on the ndisc exists in
almost all the known categories of existing topology; e.g.,
diffeomorphism, homeomorphism, symplectic diffeomorphism and
symplectic homeomorphism and others. In this talk, we will
explain our recent result that Alexander isotopy exists in
the category of Hamiltonian homeomorphisms introduced by
Mueller and the speaker a few years ago. As a consequence,
this implies that the group of area preserving
homeomorphisms of the 2disc (compactly supported in the
interior) is not simple. The proof uses the Floer homology
theory in full throttle. We will try to give some overview
of the proof in this talk.

 Thursday April 21 at 3pm in E308
 Ralf Spatzier, University of Michigan
 Exponential mixing and global rigidity of higher rank
Anosov actions

Abstract: I will discuss recent joint work with
Fisher and Kalinin on global rigidity of higher rank Anosov
actions on tori and nilmanifolds. In the nil case, this uses
joint work with Gorodnik on exponential mixing of
automorphisms of nilmanifolds.

 Tuesday April 26 at 3pm in E308
 Niclas Monod, Ecole Polytechnique Federale de Lausanne
 Littlewood and Large Forests

Abstract: Motivated by a classical result of
Sz.Nagy in functional analysis, Dixmier asked in 1950 which
group representations can be made unitary. This question is
still open, but I will report on some progress obtained with
Epstein and Ozawa. We approach the question with ideas
borrowed from XIXth century electricity theory as well as
from contemporary percolation theory. As a result, we obtain
notably nonunitarizable representations for Burnside groups
and a new characterization of amenable groups. (The talk
will be expository.)

 Thursday April 28 at 3pm in E308
 David Fisher, Indiana University
 The geometry of rank one solvable Lie groups

Abstract: I will describe some aspects of the large
scale geometry of rank one solvable Lie groups. These are
exactly those solvable Lie groups that can be written as
G=R \ltimes N where N is simply connected and
nilpotent and is the exponentially distorted in
G. The main results concern the structure of
certain discretizations of large “boxes" in
G and show that the resulting graphs are
homogeneous in certain very strong senses. In the special
case where N is abelian, these graphs are
essentially complete bipartite, a fact which play a key role
in work of EskinFisherWhyte and Peng on QI rigidity of
certain classes of solvable groups. The results discussed in
this talk are (one of) the necessary ingredients for
generalizing that work to cover all polycyclic groups. This
is joint work with Irine Peng and is motivated by some joint
work with Eskin and Peng.

 Thursday May 5 at 3pm in E308
 Charles Pugh, UC Berkeley
 Smoothing a submanifold

Abstract: The CairnsWhitehead Theorem states that a
Lipschitz mdimensional submanifold M
of (m+k)space has a compatible smooth structure
if there exists a continuous field of kplanes
transverse to M. I will discuss a proof of this
using methods of dynamical systems. I will also discuss a
“numerical barrier to smoothability of a Lipschitz
submanifold.”

 Thursday May 12 at 3pm in E308
 Romain Tessera, CNRS at Ecole Normale Superieure Lyon
 Filling big loops in homogeneous Riemannian manifolds

Abstract: In a simply connected Lie group, a loop of
length n can always be filled with a disc of area
exp(n). But if for instance the group is
nilpotent, then we can find discs of polynomial area. In
semisimple Lie groups, it is well known that we can even
obtain quadratic area. But in the wild world of solvable Lie
groups, anything between linear and exponential is a priori
possible. In this talk, I will identify the phenomena
leading to exponential filling area, and prove that when the
area is not exponential, then it is polynomial. The
characterization of Lie groups with exponential filling area
is given by an explicitly computable algebraic criterion on
the Lie algebra. This is based on a work with Yves
Cornulier.

 Tuesday May 31 at 3pm in E308
 Peter Albers, Purdue University
 A variational approach to Givental's nonlinear Maslov index

Abstract: Givental's nonlinear Maslov index for real
projective spaces had deep and diverse applications in
symplectic geometry. We consider a variant of Rabinowitz
Floer homology in order to define a homological count of
discriminant points for paths of contactomorphisms on
(symplectically fillable) contact manifolds. The growth rate
of this count can be seen as an analogue of Givental's
nonlinear Maslov index. As an application we prove a
BottSamelson type obstruction theorem for positive loops of
contactomorphisms. This is joint work with Urs Frauenfelder.
The amount of technicalities will be kept to a minimum.

 Thursday June 2 at 3pm in E308
 Melody Chan, UC Berkeley
 Combinatorics of the Tropical Torelli Map

Abstract: The Torelli map, taking an algebraic curve
to its Jacobian, has a tropical analogue, developed in
recent work by Brannetti, Melo, and Viviani. I will discuss
the tropical Torelli map, with a focus on combinatorics and
computations in low genus. Along the way, I will construct
moduli spaces of metric graphs and of positive semidefinite
forms, corresponding to tropical M_{g}
and A_{g}, respectively.
For questions, contact