Geometry/Topology Seminar
Fall 2009
Thursdays (and sometimes Tuesdays) 23pm, in
Eckhart 308

 Thursday September 24 at 2pm in E308
 A. Zorich, Rennes
 Origami, cyclic coverings, and Lyapunov exponents of the Hodge bundle

Abstract: In recent papers of I. Bouw and M.
Moeller, and G. Forni and C. Matheus it was discovered that
cyclic coverings over a projective line branched at four
points produce a collection of closed Teichmuller discs with
very interesting properties. In particular, this allowed
Bouw and Moeller to construct new Veech surfaces and G.
Forni and C. Matheus to construct arithmetic Teichmuller
discs with completely degenerate Lyapunov spectrum
("Eierlegende Wollmilchsaus"). We show that for any
arithmetic Teichmuller disc corresponding to cyclic
coverings of a fixed combinatorial type over a projective
line branched at four points one can explicitly compute the
spectrum of Lyapunov exponents of the Hodge bundle along the
Teichmuller flow. This is part of a joint work with Alex
Eskin and Maxim Kontsevich.

 Thursday October 1 at 2pm in E308
 Kasra Rafi, University of Oklahoma
 The Teichmuller diameter of the thick part of moduli space

Abstract: We study the shape of the moduli space of
a surface of finite type. In particular, we compute the
asymptotic behavior of the Teich diameter of the thick part
of the moduli space. For a surface S of genus g with b
boundary components define the complexity of S to be 3g3+b.
We show that the diameter grows like logarithm of the
complexity. This talk is a preliminary report of work in
progress. (Joint with Jing Tao.)

 Thursday October 8 at 2pm in E308
 Michael Brandenbursky, Technion (Haifa)
 Knot theory and quasimorphisms

Abstract: Quasimorphisms on a group are realvalued
functions which satisfy the homomorphism equation "up to a
bounded error". They are known to be a helpful tool in the
study of the algebraic structure of nonAbelian groups. I
will discuss a construction relating a) certain knot and
link invariants, in particular, the ones that come from the
knot Floer homology and a Khovanovtype homology; b) braid
groups; c) the dynamics of areapreserving diffeomorphisms
of a twodimensional disc; d) quasimorphisms on the group
of all such compactly supported diffeomorphisms of the disc.

 Tuesday October 13 at 2pm in E308
 Thomas Koberda, Harvard University
 Representations of mapping class groups and residual properties of
3manifold groups

Abstract: I will talk about homological
representations of mapping class groups, namely ones which
arise from actions on covering spaces. I will prove that
these are asymptotically faithful and indicate how the
NielsenThurston classification can be obtained from these
representations. I will then discuss how mapping tori of
mapping classes can be used to analyze the image of these
representations. As a corollary, I will exhibit a class of
compact 3manifolds whose fundamental groups are, for every
prime p, virtually residually finite p.

 Tuesday October 20 at 2pm in E308
 William Lopes, University of Chicago
 SeibergWitten theory for a surface times a circle

Abstract: The SeibergWitten equations from
mathematical physics have become a central tool in
4manifold topology. I will introduce these equations and
the invariants they define, which have led to many examples
of exotic smooth structures on 4manifolds. When X is
decomposed into two components along a 3manifold Y,
solutions to the SeibergWitten equations on X can be
understood in terms of solutions on the two pieces. I will
describe the SeibergWitten Floer homology groups of Y, due
to Kronheimer and Mrowka, which formalize this idea and are
themselves an interesting 3manifold invariant. Finally, I
will describe my work on calculating these groups for the
product of a genus g surface and a circle.

 Tuesday October 27 at 2pm in E308
 Chloe Perin, Hebrew University
 Elementary embeddings in hyperbolic groups

Abstract: A subgroup H of a group G is elementarily
embedded in G if its elements satisfy exactly the same first
order properties over G and over H (so, for example, an
element of H commutes with all the elements of G if and only
if it commutes with all the elements of H). We will consider
elementarily embedded subgroups of free and hyperbolic
surface groups, and more generally of torsionfree
hyperbolic groups. We get a description of these in term of
the very geometric structure of hyperbolic towers defined by
Sela. Hyperbolic towers appear in the answer to several
questions about the firstorder logic of free and hyperbolic
groups solved by Sela.

 Thursday October 29 at 2pm in E308
 Yaron Ostrover, I.A.S.
 Symplectic Measurements and Convex Geometry

Abstract: In this talk I will present a joint
project with S. ArtsteinAvidan and V. Milman in which we
attempt to bridge between Symplectic Geometry on the one
hand and Asymptotic Geometric Analysis on the other. We will
show examples where tools from one field can be imported and
used to tackle questions in the other (and vice verse  if
time permit). No previous background is required.

 Tuesday November 3 at 2pm in E308
 Matthew Stover, University of Texas
 Cusps of locally symmetric spaces

Abstract: If G is a semisimple Lie group,
X its symmetric space, and \Gamma \subset
G a discrete subgroup, then the
\Gammaaction on \partial_{\}infty
X has long been of interest. When \Gamma is
a nonuniform lattice, \Gammaorbits of parabolic
fixed points on \partial_{\}infty X are
called cusps of \Gamma. Cusps are classically
wellunderstood for certain arithmetic groups, e.g.
SL_{2}(O), and are generally
related to ideal class groups. I will describe how to count
cusps for maximal arithmetic subgroups of quasisplit
unitary groups, paying particular attention to
\SU(2,1), where the corresponding locally
symmetric spaces are Picard modular surfaces.

 Thursday November 5 at 2pm in E308
 Eriko Hironaka, Florida State University and Harvard University
 Small dilatation mapping classes from the simplest pseudoAnosov
braid

Abstract: By a recent theorem of Farb, Leininger and
Margalit, the set of 3manifolds `realizing' mapping classes
with small dilatation (compared to Euler characteristic) is
finite. We show that all known minimal dilatation mapping
classes for small genus are realized on the complement of
Rolfsen's 6_{2}^{2} link in
S^{3}, and discuss the plausibility that
minimal dilatation mapping classes for all genus are
realized on this manifold.

 Thursday November 12 at 2pm in E308
 Igor Belegradek, Georgia Institute of Technology
 Moduli spaces and nonunique souls

Abstract: We use surgery and homotopy theoretic
techniques to study the modili space of complete
nonnegatively curved metrics on an open manifold N. A
starting point is that the diffeomorphism type of the soul,
or more generally, the diffeomorphism type of the pair (N,
soul) defines a locally constant function on the moduli
space. We focus on the harder case when nondiffeomorphic
souls have low codimension. One of the most delicate results
is an example of a simplyconnected manifold with
homeomorphic nondiffeomorphic souls of codimension 2.
Previously, examples of homeomorphic nondiffeomorphic
closed simplyconnected nonnegatively curved manifolds have
been only known in dimension 7 thanks to work of
KreckStolz, while we construct such examples in each
dimension 4r1 > 10, and realize them as
codimension two souls. This is joint work with Slawomir
Kwasik and Reinhard Schultz.

 Tuesday November 17 at 2pm in E308
 Andres Navas, University of Santiago de Chile
 Hecke groups and orderability

Abstract: Braid groups are perhaps the most
important examples of leftorderable groups. A remarkable
property discovered by Dubrovina and Dubrovin is that they
may be decomposed as a disjoint union of the form S U
S^{1} U {id}, where S is a finitely generated
semigroup. In this talk I will show that a similar property
holds for certain central extensions of the Hecke groups. As
a byproduct, we will retrieve the Dehornoy's ordering on
B_{3} by elementary and completely new
methods. Several open questions will be addressed.

 Thursday November 19 at 2pm in E308
 Sean Lawton, University of TexasPan American
 Singularities of free group character varieties

Abstract: Let X be the moduli of SL(n,C), SU(n),
GL(n,C), or U(n) valued representations of a rank r free
group. We compute the fundamental group of X and show that
these four moduli otherwise have identical higher homotopy
groups. We then classify the singular stratification of X.
This comes down to showing that the singular locus
corresponds exactly to reducible representations if there
exist singularities at all. Lastly, we show that the moduli
X are generally not topological manifolds, except for a few
examples we explicitly describe.

 Tuesday November 24 at 2pm in E308
 Artem Pulemotov, University of Chicago
 The heat equation and the Ricci flow on manifolds with boundary

Abstract: The first part of the talk will discuss
gradient estimates for the heat equation on a manifold
M with nonempty boundary and a fixed Riemannian
metric. We will mainly focus on LiYautype inequalities.
The second part of the talk will deal with the heat equation
on M in the case where the Riemannian metric on
M evolves under the Ricci flow. After motivating
the problem and explaining the boundary conditions involved,
we will look at how LiYautype inequalities adapt to this
case. Based on joint work with Mihai Bailesteanu and
Xiaodong Cao.

 Thursday December 3 at 2pm in E308
 Andy Putman, M.I.T.
 The Picard Group of the Moduli Space of Curves with Level
Structures

Abstract: The Picard group of an algebraic variety
X is the set of complex line bundles over
X. In this talk, we will describe the Picard
groups of certain finite covers of the moduli space of
curves. The methods we use combine ideas from algebraic
geometry, finite group theory, and algebraic/geometric
topology.
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