Geometry/Topology Seminar
Fall 2009
Thursdays (and sometimes Tuesdays) 2-3pm, in
Eckhart 308
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- Thursday September 24 at 2pm in E308
- A. Zorich, Rennes
- Origami, cyclic coverings, and Lyapunov exponents of the Hodge bundle
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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.
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- Thursday October 1 at 2pm in E308
- Kasra Rafi, University of Oklahoma
- The Teichmuller diameter of the thick part of moduli space
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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 3g-3+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.)
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- Thursday October 8 at 2pm in E308
- Michael Brandenbursky, Technion (Haifa)
- Knot theory and quasi-morphisms
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Abstract: Quasi-morphisms on a group are real-valued
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 non-Abelian 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 Khovanov-type homology; b) braid
groups; c) the dynamics of area-preserving diffeomorphisms
of a two-dimensional disc; d) quasi-morphisms on the group
of all such compactly supported diffeomorphisms of the disc.
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- Tuesday October 13 at 2pm in E308
- Thomas Koberda, Harvard University
- Representations of mapping class groups and residual properties of
3-manifold groups
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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
Nielsen-Thurston 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 3-manifolds whose fundamental groups are, for every
prime p, virtually residually finite p.
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- Tuesday October 20 at 2pm in E308
- William Lopes, University of Chicago
- Seiberg-Witten theory for a surface times a circle
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Abstract: The Seiberg-Witten equations from
mathematical physics have become a central tool in
4-manifold topology. I will introduce these equations and
the invariants they define, which have led to many examples
of exotic smooth structures on 4-manifolds. When X is
decomposed into two components along a 3-manifold Y,
solutions to the Seiberg-Witten equations on X can be
understood in terms of solutions on the two pieces. I will
describe the Seiberg-Witten Floer homology groups of Y, due
to Kronheimer and Mrowka, which formalize this idea and are
themselves an interesting 3-manifold invariant. Finally, I
will describe my work on calculating these groups for the
product of a genus g surface and a circle.
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- Tuesday October 27 at 2pm in E308
- Chloe Perin, Hebrew University
- Elementary embeddings in hyperbolic groups
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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 torsion-free
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 first-order logic of free and hyperbolic
groups solved by Sela.
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- Thursday October 29 at 2pm in E308
- Yaron Ostrover, I.A.S.
- Symplectic Measurements and Convex Geometry
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Abstract: In this talk I will present a joint
project with S. Artstein-Avidan 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.
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- Tuesday November 3 at 2pm in E308
- Matthew Stover, University of Texas
- Cusps of locally symmetric spaces
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Abstract: If G is a semisimple Lie group,
X its symmetric space, and \Gamma \subset
G a discrete subgroup, then the
\Gamma-action on \partial\infty
X has long been of interest. When \Gamma is
a nonuniform lattice, \Gamma-orbits of parabolic
fixed points on \partial\infty X are
called cusps of \Gamma. Cusps are classically
well-understood for certain arithmetic groups, e.g.
SL2(O), and are generally
related to ideal class groups. I will describe how to count
cusps for maximal arithmetic subgroups of quasi-split
unitary groups, paying particular attention to
\SU(2,1), where the corresponding locally
symmetric spaces are Picard modular surfaces.
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- Thursday November 5 at 2pm in E308
- Eriko Hironaka, Florida State University and Harvard University
- Small dilatation mapping classes from the simplest pseudo-Anosov
braid
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Abstract: By a recent theorem of Farb, Leininger and
Margalit, the set of 3-manifolds `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 622 link in
S3, and discuss the plausibility that
minimal dilatation mapping classes for all genus are
realized on this manifold.
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- Thursday November 12 at 2pm in E308
- Igor Belegradek, Georgia Institute of Technology
- Moduli spaces and non-unique souls
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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 non-diffeomorphic
souls have low codimension. One of the most delicate results
is an example of a simply-connected manifold with
homeomorphic non-diffeomorphic souls of codimension 2.
Previously, examples of homeomorphic non-diffeomorphic
closed simply-connected nonnegatively curved manifolds have
been only known in dimension 7 thanks to work of
Kreck-Stolz, while we construct such examples in each
dimension 4r-1 > 10, and realize them as
codimension two souls. This is joint work with Slawomir
Kwasik and Reinhard Schultz.
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- Tuesday November 17 at 2pm in E308
- Andres Navas, University of Santiago de Chile
- Hecke groups and orderability
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Abstract: Braid groups are perhaps the most
important examples of left-orderable 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
B3 by elementary and completely new
methods. Several open questions will be addressed.
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- Thursday November 19 at 2pm in E308
- Sean Lawton, University of Texas-Pan American
- Singularities of free group character varieties
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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.
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- Tuesday November 24 at 2pm in E308
- Artem Pulemotov, University of Chicago
- The heat equation and the Ricci flow on manifolds with boundary
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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 Li-Yau-type 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 Li-Yau-type inequalities adapt to this
case. Based on joint work with Mihai Bailesteanu and
Xiaodong Cao.
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- Thursday December 3 at 2pm in E308
- Andy Putman, M.I.T.
- The Picard Group of the Moduli Space of Curves with Level
Structures
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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.
For questions, contact