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

 Wednesday September 14 at 3pm in E312
 Andres Navas, Universidad de Santiago de Chile
 Nilpotence, growth, and sharp regularity for groups of
1dimensional diffeomorphisms

Abstract: B. Farb and J. Franks proved that every
finitely generated, torsionfree, nilpotent group embeds
into the group of C^{1} diffeomorphisms
of the interval. This is in contrast to the classical
PlanteThurston theorem, which establishes that this is
never the case for C^{2} regularity
unless the group is Abelian. In this talk I will give a
sharp version of the FarbFranks action in what concerns the
Hölder regularity for the derivatives. Somewhat
surprisingly, the proof of the optimality uses ideas of
nonstandard random walks in nilpotent groups that are
interesting by themselves. I will also explain the relation
of this with the fact that the group of diffeomorphisms of
the interval does not contain groups of intermediate growth.

 Friday September 23 at 1pm in E308
 Dick Hain, Duke University
 The Torelli group in genus 3

Abstract: TBA

 Thursday September 29 at 3pm in E308
 Mohammed Abouzaid, MIT
 Symplectic manifolds with enough Lagrangians

Abstract: In complex analytic geometry one can
characterise projective algebraic varieties as those complex
manifolds which have "enough holomorphic line bundles." I
will explain a Floer theoretic notion that corresponds, in
symplectic topology, to a manifold having enough
Lagrangians, and explain some properties that can be derived
assuming this property holds. Time permitting, I will
explain joint work with Fukaya, Oh, Ohta, and Ono which
applies these ideas to prove the Homological Mirror Symmetry
conjecture for toric varieties from these considerations.

 Thursday October 6 at 3pm in E308
 Yaron Ostrover, Tell Aviv
 On the uniqueness of Hofer's geometry

Abstract: One of the remarkable facts regarding the
group of Hamiltonian diffeomorphisms is that it carries an
intrinsic geometry given by a Finsler biinvariant metric.
This metric, which was discovered by Hofer, yields a
geometric intuition for Hamiltonian systems, and can be used
in many ways as a powerful tool in symplectic geometry and
dynamics. In this talk we address the question of the
uniqueness of Hofer's metric. The talk is based on a joint
work with Lev Buhovsky.

 Thursday October 13 at 3pm in E308
 Sebastian Hensel, Universitat Bonn
 Sphere systems and the geometry of Out(F_{n})

Abstract: By a theorem of Laudenbach, the outer
automorphism group Out(F_{n}) of a free
group is a cofinite quotient of the mapping class group of a
suitable 3manifold M_{n}. This point of
view allows to study the geometry of
Out(F_{n}) with methods inspired from
surface mapping class groups. In this talk, we present an
application of this strategy: we show that the natural
inclusion of the mapping class group of a surface of genus g
with one puncture into Out(F_{2g}) is
undistorted. The results are joint work with Ursula
Hamenstädt.

 Thursday October 27 at 3pm in E308
 Kasra Rafi, University of Oklahoma
 Thurston's Lipschitz metric on Teichmüller space

Abstract: In 1980's, Thurston introduced an
asymmetric metric for Teichmüller space which we refer
to as the Lipschitz metric. For two marked hyperbolic
structures X and Y, the distance
between X and Y is defined to be the
logarithm of the infimum of Lipschitz constants of
homeomorphisms from X to Y that are
homotopic to the identity. The geometry of the Lipschitz
metric is very rich, as Thurston shows in his paper.
However, many aspects of it have remained unexamined. There
has been a recent spike in interest in understanding and
expanding Thurston's original ideas. We give a survey of
what is known and some new results.

 Tuesday November 1 at 3pm in E308
 Alvaro Pelayo, Washington University in Saint Louis
 Symplectic, spectral and bifurcation theory for integrable
systems

Abstract: I will start with a brief review of the
basic ideas of symplectic geometry and the theory of finite
dimensional completely integrable systems. Then I will
introduce new symplectic invariants for completely
integrable Hamiltonian systems in low dimensions, and
explain how these invariants determine, up to isomorphisms,
the so called "semitoric integrable systems". Semitoric
integrable systems are integrable systems with two degrees
of freedom on fourmanifolds, for which one component of the
system generates a periodic flow. I will conclude by
discussing some recent developments on symplectic
bifurcation theory for integrable systems, and on the
quantization and inverse spectral theory of quantum
integrable systems. The talk combines complexalgebraic and
symplectic methods with recently developed microlocal
methods. Parts of this talk are based on joint work with
Johannes J. Duistermaat, Tudor S. Ratiu and San Vu Ngoc.

 Thursday November 3 at 23pm in Barn, Ryerson
 Rob Kirby, Berkeley
 The torus trick

Abstract: This is a historical talk about the proof
of the annulus conjecture and theorems on the existence and
uniqueness of triangulations of manifolds. Not much
background is needed because not much was known about
topological manifolds in 1968.

 Monday November 7 at 3:30pm in E207
 Aaron Lauve, Loyola (Chicago)
 Special Combinatorics/Representation Theory Talk: A menagerie of
"coinvariant" spaces: a survey of modern takes on a classical
theorem of ChevalleyShephardTodd

Abstract: We take a brief tour through the history
of the invariant theory of the symmetric group, from Schur
and Weyl, through Chevalley, to some modern masters.
Scheduled stops include Macdonald's positivity conjecture,
the n! conjecture of GarsiaHaiman, and a
menagerie of coinvariant spaces (each indexed by famous
combinatorial gadgets) studied by F. Bergeron and friends.
We close with some recent stability results for diagonal
coinvariants due to Farb, Bergeron, and others.

 Thursday November 10 at 3pm in E308
 Thomas Koberda, Harvard University
 Rightangled Artin subgroups of rightangled Artin groups

Abstract: I will present a systematic way of
classifying all rightangled Artin subgroups of a given
rightangled Artin group. The methods used have a number of
corollaries: for instance, it can be shown that there is an
embedding between a rightangled Artin group on a cycle of
length m to one on a cycle of length n
if and only if m=n+k(n4) for some nonngeative
integer k. I will also give some rigidity
results. This is joint work with Sanghyun Kim.

 Tuesday November 15 at 3pm in E308
 Egor Shelukhin, Tel Aviv University
 Quasimorphisms, moment maps and almost complex structures

Abstract: We describe a general theorem that
produces on equal footing a quasimorphism on (the universal
cover of) any Hermitian Lie group G from the
properties of its action on the symmetric space
X=G/K  and a quasimorphism on (the universal
cover of) the group of Hamiltonian diffeomorphisms of any
closed symplectic manifold  from the properties of its
action on the space of compatible almost complex structures
. These properties include an equivariant moment map and a
uniform bound on the symplectic areas of geodesic triangles.
We then discuss several applications of the resulting
objects and formulate related questions.

 Thursday November 17 at 3pm in E308
 Fanny Kassel, CNRS and Universite Lille 1
 Discrete spectrum of the Laplacian on pseudoRiemannian
locally symmetric spaces

Abstract: The spectrum of the Laplacian on
Riemannian manifolds has been an active area of research for
decades, relating geometry and analysis with arithmetic and
mathematical physics. The case of Riemannian locally
symmetric spaces is particularly rich and important.
Toshiyuki Kobayashi and I have considered similar problems
for pseudoRiemannian locally symmetric spaces. I will
explain our results on the discrete spectrum of the
Laplacian in the case of antide Sitter 3manifolds, i.e.
Lorentz 3manifolds of constant negative curvature; in this
case, spectral theory relies on a good understanding of the
geometry of the manifolds.

 Tuesday November 29 at 3pm in E308
 Michael Brandenbursky, Vanderbilt
 Quasiisometric embeddings into diffeomorphism groups

Abstract: Let M be a smooth compact
connected oriented manifold of dimension at least two
endowed with a volume form. Assuming certain conditions on
the fundamental group of M, we will construct
quasiisometric embeddings of either free Abelian or direct
products of nonAbelian free groups into the group of volume
preserving diffeomorphisms of M equipped with the
L^{p} metric induced by a Riemannian
metric on M. If time permits we will explain a
relation between quasimorphisms and the
L^{p} metrics. This talk is based on the
joint work with Jarek Kedra.

 Wednesday November 30 at 3pm in E308
 Julianna Tymoczko, University of Iowa
 An introduction to flag varieties

Abstract: This is the first of a twopart talk on
the cohomology of flag varieties and related varieties. In
this talk, we introduce flag varieties and some classical
constructions in geometry, topology, and combinatorics that
are used to study the cohomology of flag varieties. We also
describe and discuss group actions on the cohomology of the
flag variety and some of its important subvarieties,
including Schubert varieties and, as time permits, Springer
varieties and Hessenberg varieties.

 Thursday December 1 at 3pm in E308
 Julianna Tymoczko, University of Iowa
 Equivariant cohomology of flag varieties

Abstract: This is the second of a twopart talk on
the cohomology of flag varieties and related varieties. In
this talk, we describe a combinatorial process to identify
torusequivariant cohomology of flag varieties, often
referred to as GKM theory (after
GoreskyKottwitzMacPherson). We describe how classical
constructions from the first talk appear in this context. We
also discuss how GKM theory can be extended from its
original setting to apply to many of the subvarieties
presented in the first talk.

 Tuesday December 6 at 3pm in E308
 Octav Cornea, Universite de Montreal
 Lagrangian cobordism

Abstract: The talk is based on joint work with Paul
Biran (ETH) and discusses the existence and properties of a
functor with target the derived Fukaya category and with
domain a category with objects Lagrangian submanifolds
(under certain constraints) and with morphisms induced by
emebdded Lagrangian cobordisms.

 Thursday December 8 at 3pm in E308
 Jordan Ellenberg, University of Wisconsin
 FImodules and representation stability

Abstract: (joint with Tom Church and Benson Farb)
The dimension of the ith cohomology group of the
configuration space F(M,n) of n
ordered distinct points on a manifold M is a
polynomial in n when n is large
enough; why? The answer has to do with the phenomenon of
"representation stability," as studied in a 2010 paper of
Church and Farb  the sequence of vector spaces
H^{i}(F(M,n),Q) (n=1,2,3,...)
gets larger and larger as n increases, but for
n large enough it is always "the same"
representation of S_{n}. Since the group
being represented is changing with n, it takes
some real work to remove the quotation marks from "the
same"! Representation stability turns out to arise in a
broad array of contexts in geometry, algebra, and
combinatorics. We will discuss the category of FImodules,
in which the notion of representation stability is recast as
a finite generation condition. This change of viewpoint
allows us to prove many structural theorems about
representation stability for representations of the
symmetric group which were unavailable in the first paper;
we will show how a uniform proof allows us to give finite
descriptions of the character of S_{n}
acting on the cohomology of configuration spaces, sections
of line bundles on determinantal varieties, graded pieces of
diagonal coinvariant algebras, the degree i piece
of the tautological Chow ring of M_{g,n},
the BhargavaSatriano algebra, etc. Under favorable
circumstances, the amount of information we need is quite
minimal  for instance, in some circumstances (e.g.
cohomology of configuration space of a manifold with at
least one boundary component) we can show that the mere fact
that a sequence of representations V_{n}
has dimension bounded by some polynomial implies that the
dimension actually is a polynomial, not only for
sufficiently large n but for all nonnegative
n.

 Monday January 23 at 3pm in E308
 Khalid BouRabee, University of Michigan
 TBA

Abstract: TBA

 Tuesday February 21 at 3pm in E308
 Vadim Kaloshin, University of Maryland
 TBA

Abstract: TBA

 Tuesday February 28 at 3pm in E308
 Tatyana Barron, University of Western Ontario
 TBA

Abstract: TBA

 Thursday March 1 at 3pm in E308
 Juan Suoto, University of British Columbia
 TBA

Abstract: TBA
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