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

 Thursday January 7 at 2pm in E308
 A. Mohammadi, University of Chicago
 Inhomogeneous quadratic forms

Abstract: We will address a recent joint work with
G. Margulis on a quantitative version of the Oppenheimer
conjecture for inhomogeneous quadratic forms. This
generalizes the previous works of Eskin, Margulis and Mozes
in the homogeneous setting also the work of J. Marklof.

 Thursday January 14 at 2pm in E308
 J. DeBlois, University of Illinois at Chicago
 Rank and rank gradient of 3manifold groups that split

Abstract: When a finitely generated group decomposes
as a graph of groups, there are theorems (many due to R.
Weidmann) relating its rank  the minimal number of
generators  to the rank of the corresponding vertex and
edge groups, or to the combinatorics of the underlying
graph. I will discuss applications of such theorems to
questions regarding the rank gradient of families of
subgroups, which measures how quickly rank grows among their
members. Then I will speculate about the possibility of
improving Weidmann's estimates in some cases using geometric
methods due to Souto and others.

 Thursday January 21 at 2pm in E308
 Michael Farber, Durham University
 Stochastic algebraic topology and robotics

Abstract: I will describe solutions to several
problems of mixed probabilistictopological nature which are
inspired by applications in topological robotics. These
problems deal with systems depending on a large number of
random parameters, n\to ∞. Our results predict the
values of various topological characteristics of
configuration spaces of such systems.

 Thursday January 28 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.

 Thursday February 4 at 2pm in E308
 Alexander Bufetov, Rice University and Steklov Institute
 Limit theorems for translation flows

Abstract: Consider a compact oriented surface of
genus at least two endowed with a holomorphic oneform. The
real and the imaginary parts of the oneform define two
foliations on the surface, and each foliation defines an
areapreserving translation flow. By a Theorem of H.Masur
and W.Veech, for a generic surface these flows are ergodic.
The talk will be devoted to the speed of convergence in the
ergodic theorem for translation flows. The main result,
which extends earlier work of A.Zorich and G.Forni, is a
multiplicative asymptotic expansion for time averages of
Lipschitz functions. The argument, close in spirit to that
of G.Forni, proceeds by approximation of ergodic integrals
by special holonomyinvariant Hoelder cocycles on
trajectories of the flows. Generically, the dimension of the
space of holonomyinvariant Hoelder cocycles is equal to the
genus of the surface, and the ergodic integral of a
Lipschitz function can be approximated by such a cocycle up
to terms growing slower than any power of the time. The
renormalization effectuated by the Teichmueller geodesic
flow on the space of holonomyinvariant Hoelder cocycles
allows one also to obtain limit theorems for translation
flows: it is proved that along certain sequences of times
ergodic integrals, normalized to have variance one, converge
in distribution to a nondegenerate compactly supported
measure. The argument uses a symbolic representation of
translation flows as suspension flows over Vershik's
automorphisms, a construction similar to one proposed by
S.Ito.

 Thursday February 11 at 2pm in E308
 Nikolay Dmitrov, CRM and McGill University
 Illumination on Flat Surfaces of Infinite Area

Abstract: The motivation for this talk is related to
the situation when a configuration of two sided mirrors is
given in the plane and a source of light is placed at an
arbitrarily chosen point. In this setting, one of the goals
is to study the regions that are not illuminated by the
source of light. More precisely, it is interesting to look
for "dark" domains with nonempty interior. The mirror
configuration in the plane gives rise to a surface with a
translation structure of infinite area, which corresponds to
a compact Riemann surface with a meromorphic differential
that has only double poles with zero residues. Our main
result is that on flat surfaces of this type that have at
least two poles, one can find "dark" regions isometric to
infinite planar sectors. Moreover, the total angle of such
sectors exceeds certain value related to the number of
poles. Using similar ideas, we can also show the equivalence
between a given property of certain circle maps and the
existence of dark infinite sectors in the plane with
mirrors. If time permits, we would discuss some future work
related to moduli spaces of compact tori with meromorphic
differentials with one pole.

 Thursday February 18 at 2pm in E308
 Hamid Hezari, M.I.T.
 Zeros of eigenfunctions and the dynamics of the classical flow

Abstract: It is conjectured that if the geodesic
flow of a Riemannian manifold is ergodic then the zeros of
LaplaceBeltrami eigenfunctions become uniformly distributed
as the eigenvalue tends to infinity. This is a very
difficult open problem but there are some answers if one
looks at the complex zeros! Steve Zelditch has recently
proved that such limit distribution exists if one considers
the complex zeros of the complexified eigenfunctions. There
is hope that results on the distribution of complex zeros
would imply some results about the real zeros. In this talk
we will mention the recent results in this subject and we
will study the similar problem in the opposite situation
when the classical flow is completely integrable. We will
mainly focus on the eigenfunctions of 1D Schrödinger
operators.

 Tuesday February 23 at 2pm in E308
 Lior Silberman, University of British Colombia
 Groups not acting on manifolds

Abstract: We show that random groups with strong
fixedpoint properties have no nontrivial smooth
volumepreserving actions on compact manifolds.

 Thursday February 25 at 2pm in E308
 Jeffrey Brock, Brown University
 Closed WeilPetersson geodesics in moduli space: asymptotic growth, entropy, and
length spectra

Abstract: The lengths of closed geodesics on a
Riemannian manifold carry important information about its
geometry. In this talk I will survey some curious facts
about the length spectrum of the Riemann moduli space with
its WeilPetersson metric that sit in contradistinction with
their counterparts in the Teichmüller metric. This is joint
work with Howard Masur and Yair Minsky.

 Wednesday March 3 at 1:30pm in E202
 Jordan Ellenberg, University of Wisconsin, Madison
 Stable cohomology for Hurwitz spaces and arithmetic applications

Abstract: Hurwitz spaces are moduli spaces of finite
branched covers of P1. We will discuss the stabilization of
the cohomology of these spaces (or, what is the same, for
certain congruence subgroups of Artin braid groups) as the
number of branch points (resp. number of strands) grows,
with the Galois group of the cover being fixed; this can be
thought of as a "Harer theorem" for this family of moduli
spaces. It turns out that the function field analogues of
many popular conjectures in analytic number theory (due to
CohenLenstra, Bhargava, etc.) reduce to topological
questions about Hurwitz spaces. We will discuss the
arithmetic consequences of the stabilization theorem, and of
a geometrically natural conjecture about the stable
cohomology classes of Hurwitz spaces. (joint work with
Akshay Venkatesh and Craig Westerland)

 Tuesday March 16 at 2pm in E308
 Dick Canary, University of Michigan
 Moduli spaces of hyperbolic 3manifolds

Abstract: It is common to study the moduli space of
closed hyperbolic surfaces of a fixed genus. In 3
dimensions, one usually studies the space AH(M) of (marked)
hyperbolic 3manifolds homotopy equivalent to a fixed
compact 3manifold M. One may regard AH(M) as the
3dimensional analogue of the Teichmuller space of marked
hyperbolic surfaces of a fixed genus. In this talk, we
discuss the moduli space of hyperbolic 3manifolds homotopy
equivalent to M, which is the quotient of AH(M) by the
action of the outer automorphism group of the fundamental
group of M. We will describe results on the topology of the
moduli space (which can be quite pathological) and the
dynamics of the action of the outer automorphism group on
AH(M) and more generally on the PSL(2,C)character variety
of M. This talk describes joint work with Pete Storm.
For questions, contact