Geometry/Topology Seminar
Fall 2010
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday September 23 at 3pm in E308
 Anton Zorich, Rennes University
 Lyapunov exponents of the Teichmuller flow,
determinant of Laplacian,
and volumes of moduli spaces
of quadratic differentials

Abstract: TBA

 Thursday September 30 at 3pm in E308
 Karen Vogtmann, Cornell University
 Lie trees and Outer space

Abstract: I will describe both the simplicial
structure of Outer space and Kontsevich's Lie graph
homology, then show how the two are related and how this
relation can be used to find nontrivial cycles in the moduli
space of graphs. Morita originally constructed such cycles
using a map from the Lie algebra of "Lie trees" into its
abelianization. We show that the abelianization is strictly
larger than the image of this map. We then use our new
elements of the abelianization to construct new cycles.
These include a heretofore mysterious unstable cohomology
class in Aut(F_{5}) which was found in
2002 by F. Gerlits. This is joint work with Jim Conant and
Martin Kassabov.

 Thursday October 7 at 3pm in E308
 Fanny Kassel, University of Chicago
 Deformation of compact quotients of homogeneous spaces

Abstract: Many mathematicians have worked on the
problem of determining which homogeneous spaces G/H admit
proper and cocompact actions by discrete groups Gamma. This
question is highly nontrivial when H is noncompact, and
still far from being solved. I will consider homogeneous
spaces G/H that do admit such actions and examine the
deformation of the compact quotients Gamma\G/H. I will prove
that for most known examples with G and H reductive, the
proper discontinuity of the action is preserved under any
small deformation of Gamma in G. For G/H=SO(2,2)/SO(1,2),
this is related to the existence of Thurston's asymmetric
distance on Teichmüller space. I will also address similar
questions in the setting of padic homogeneous spaces.

 Thursday October 14 at 3pm in E308
 Yves de Cornulier, CNRS and Rennes University
 Large scale geometry of Lie groups and cone equivalences

Abstract: We study the class of Lie groups up to
quasiisometry, and introduce the coarser notion of
conebilipschitz equivalent groups.

 Special Colloquium
 Tuesday October 19 at 3pm in E206
 Emmanuel Giroux, CNRS and Ecole Normale Superieure de Lyon
 Contact geometry, some problems and perspectives

Abstract: Contact geometry is nowadays recognized as
part of symplectic geometry, but while the latter has
physical roots, the former arose from the mathematical study
of differential equations and the socalled contact
transformations. We will briefly review this historical
origin before discussing a number of questions concerning
contact transformations that the development of modern
contact geometry raises.

 Thursday October 21 at 3pm in E308
 Julie Deserti, University Paris 7 and Indiana University
 Zimmer's program and Cremona group

Abstract: In the spirit of Zimmer's program we
describe embeddings from SL(n,Z) into the group of
birational maps of the complex projective space (also called
Cremona group). We use the structure of SL(n,Z) and some
properties on the birational maps.

 Thursday October 28 at 3pm in E308
 Jonathan Chaika, University of Chicago
 Some tools for understanding interval exchange transformations

Abstract: This talk will provide an introduction to
interval exchange transformations focusing on the fact that
they are generalizations of rotations of the circle. We will
address an analogue of the Gauss map, the fact that they
have linear block growth and related properties.

 Tuesday November 2 at 3pm in E308
 Walter Neumann, Columbia University
 Quasiisometric classification of 3manifold groups

Abstract: (Joint work with Jason Behrstock). The
classification of 3manifold groups up to quasiisometry is
now almost complete, the main unresolved case being that of
an irreducible nongeometric manifold whose geometric
decomposition consists exclusively of arithmetic hyperbolic
pieces. However, other problems remain, for example when all
pieces are hyperbolic but at least one is not arithmetic
then, using deep work of Dani Wise, we find that
quasiisometry is often equivalent with commensurability,
but we do not know if it always is. I will give an overview
of this classification of some of these remaining problems.

 Thursday November 11 at 3pm in E308
 Karin Melnick, University of Maryland
 Normal forms for conformal vector fields on Lorentzian manifolds

Abstract: Isometries of a Riemannian or
pseudoRiemannian manifold fixing a point are conjugate to
their differential via the exponential map. No such
linearization exists in general for conformal
transformations fixing a point. The main theorem of this
talk asserts that on a realanalytic Lorentzian manifold M,
any conformal vector field vanishing at a point has
linearizable flow, or M is conformally flat. This result
leads to a normal form for any such vector field near its
singularity. (joint work with Charles Frances)

 Tuesday November 16 at 3pm in E308
 David Ralston, Ohio State University
 Growth Rates and OneSided Boundedness of Discrepancy Sums

Abstract: We will turn our attention to a thoroughly
studied system: given a starting point x and an
irrational rotation \theta, consider the sequence
of points on the circle given by x rotated by
\theta zero times, once, twice, through
n times, and compare the number of times the
point has landed in one half versus the other. The
difference between the two is called the \textit{discrepancy
sum} associated to the pair (x,\theta). Various
probabilistic statements about the sequence of such sums (as
n=0,1,2,\ldots) date far back, in some sense as
far back as Gauss (although more modern ergodic theoretic
techniques simplify these older results). The problem of
describing this sequence for a \textit{specific} pair,
however, is fairly difficult; although there is nothing
random in this sequence, it mimics the behavior of a random
walk in so many respects that another term used is
\textit{deterministic random walk}. Using very simple
techniques, however, we will be able to construct specific
starting data which exhibits exotic behavior. In particular,
we will construct such sequences which grow at unusual rates
or are bounded on one side.

 Thursday November 18 at 2pm in E202
 Kasra Rafi, Oklahoma University
 Rank of Teichmuller space

Abstract: Let S be surface of hyperbolic type and
T(S) be the Teichmuller space of S. We would like to study
the group of quasiisometries of T(S). As a first step we
determine the rank of T(S), that is, the largest dimension N
where a large box B in R^{N} can be embedded
quasiisometrically into T(S). The main tool we use is the
coarse differentiation lemma. This states that, any
quasiisometric embedding, in the correct scale, is nearly
an affine map.

 Thursday November 18 at 3pm in E308
 Moira Chas, SUNY Stony Brook
 Combinatorial length, geometric length and selfintersection of curves
on surfaces.

Abstract: Consider an orientable surface S with
boundary and a free homotopy class C of closed oriented
curves in that surface. The combinatorial length of C is the
minimum number of letters required for a description of C in
terms of a set of standard generators of the fundamental
group of S. The selfintersection of C is the minimum number
of times in which a representative of C crosses itself. If
the surface is endowed with a hyperbolic metric, then one
can also definethe geometry length of C, as the length of
the unique geodesic representative in C. Several relations
between combinatorial length, geometric length and
selfintersection number will be discussed in the first part
of the talk. In the second part of the talk we will discuss
the definition the GoldmanTuraev Lie bialgebra and how this
algebraic structure relates to the intersection and
selfintersection number of curves on an surface. Parts of
this work are joint with Fabiana Krongold, Steve Lalley and
Anthony Phillips.

 Tuesday November 23 at 3pm in E308
 Frol Zapolsky, IHES
 A comparison of symplectic homogenization and Calabi quasistates

Abstract: Symplectic homogenization has been defined
by C. Viterbo as a certain limit of Hamiltonians of the form
H(kq,p) for natural k tending to infinity, on the cotangent
bundle of a torus. It turns out that the map sending a
Hamiltonian to its homogenized version has properties
similar to those possessed by certain nonlinear
functionals, due to M. Entov and L. Polterovich, on the
space of continuous functions of some symplectic manifolds
(for example complex projective spaces), called Calabi
quasistates. My goal is to compare symplectic
homogenization with Calabi quasistates. I would also like
to present a generalization of the construction by Viterbo
to cotangent bundles of more general manifolds, and to
indicate some applications. This is joint project with
Alexandra Monzner and Nicolas Vichery.

 Thursday December 2 at 3pm in E308
 Pierre PY, University of Chicago
 Kahler groups, real hyperbolic spaces and the Cremona group

Abstract: Starting from a classical theorem of
Carlson and Toledo, we will discuss actions of fundamental
groups of compact Kahler manifolds on finite or infinite
dimensional real hyperbolic spaces. We will see that such
actions almost always (but not always) come from surface
groups. We then give an application to the study of the
Cremona group. The talk will take us from the infinite
dimensional representation theory of PSL(2,R) to algebraic
geometry. This is a joint work with Thomas Delzant.

 Tuesday December 7 at 3pm in E308
 Yael Karshon, University of Toronto
 Symplectic blowups of the complex projective plane,
and counting torus actions

Abstract: In how many different ways can a twotorus
act on a given simply connected symplectic fourmanifold? If
the second Betti number is one or two, this was known. If
the second Betti number is three or more, to reduce the
question to combinatorics, we describe the manifold as a
symplectic blowup in a way that is compatible with all the
torus actions simultaneously. For this we use the theory of
pseudoholomorphic curves. This is joint work with Liat
Kessler and Martin Pinsonnault.

 Thursday December 9 at 3pm in The Barn
 Vladimir Markovic, University of Warwick
 Counting incompressible surfaces in a closed hyperbolic 3manifold

Abstract: (Joint work with J. Kahn). Let M denote a
closed hyperbolic 3manifold, and let s(M,g) denote the
number of equivalence classes of incopressible subsurfaces
of M, of genus at most g. Builiding on our previous work on
the surface subgroup theorem and improving on the work of J.
Masters, we show that for some constants c≤ d,
the inequalities (cg)^{2g} ≤ s(M,g) ≤
(dg)^{2g} hold for every large g. In
particular, we will see that the exponent 2g that appears in
the above inequalities is very closely related to the fact
that the rank of the first homology a closed surface of
genus g is equal to 2g.
For questions, contact