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

 Thursday January 13 at 3pm in E308
 Chris Leininger, University of Illinois at UrbanaChampaign
 Quasiisometric embeddings into Teichmuller space

Abstract: It is unknown whether or not there exists
a surface bundle over a surface for which the fundamental
group is Gromov hyperbolic. According to work of FarbMosher
and Hamenstadt, this is known to be equivalent to the
existence of a quasiisometric embedding of the hyperbolic
plane satisfying both a geometric property as well as a
group theoretic property. In this talk I will describe how
to construct quasiisometric embeddings with either one of
these properties (though not both simultaneously). This is
joint work with Matt Clay, Johanna Mangahas and Saul
Schleimer.

 Thursday January 20 at 3pm in E308
 Francois Gueritaud, Lille
 Veering triangulations and positive angle structures

Abstract: Casson and Rivin have proposed a program
to explicitly construct the finitevolume hyperbolic metric
on a 3manifold given as a combinatorial triangulation:
namely, find dihedral angles for the tetrahedra, subject to
certain gluing conditions. These angles are hard to find in
general, and solving a linearized version of the problem
(finding an "angle structure") already has strong
topological implications. Agol recently introduced a class
of "veering" triangulations with pleasant existence and
uniqueness properties. We will prove these triangulations
admit angle structures, and explain some context and
refinements. Joint work with Dave Futer.

 Thursday January 27 at 3pm in E308
 Alden Walker, Caltech
 Isometric endomorphisms of free groups

Abstract: An arbitrary homomorphism between groups
is nonincreasing for stable commutator length, and there are
infinitely many (injective) homomorphisms between free
groups which strictly decrease the stable commutator length
of some elements. However, we show in this paper that a
random homomorphism between free groups is almost surely an
isometry for stable commutator length for every element; in
particular, the unit ball in the scl norm of a free group
admits an enormous number of exotic isometries. Using
similar methods, we show that a random fatgraph in a free
group is extremal (i.e. is an absolute minimizer for
relative Gromov norm) for its boundary; this implies, for
instance, that a random element of a free group with
commutator length at most n has commutator length exactly n
and stable commutator length exactly n1/2. Our methods also
let us construct explicit (and computable) quasimorphisms
which certify these facts. This is joint work with Danny
Calegari.

 Tuesday February 8 at 3pm in E308
 Jayadev Athreya, University of Illinois at UrbanaChampaign
 On the distribution of gaps for saddle connection
directions

Abstract: In joint work with J. Chaika, we prove
results on the distribution of gaps of angles between saddle
connections on flat surfaces. Our techniques draw on the
work of MarklofStrombergsson on the periodic Lorentz gas
and that of EskinMasur on flat surfaces. We describe some
applications to billiards in polygons.

 Thursday February 10 at 3pm in E308
 Alex Furman, University of Illinois at Chicago
 Lattice envelopes

Abstract: In a joint work with Uri Bader and Roman
Sauer we consider the following general problem: Given a
countable group \Gamma describe all its lattice
envelopes, that is all locally compact second countable
groups G that contain \Gamma as a lattice. We
describe the solution to this problem for a large class of
groups \Gamma. The proofs rely on the work of
BreuillardGelander, Margulis' commensurator superrigidity
and normal subgroup theorems, and some quasiisometric
rigidity results.

 Tuesday February 15 at 3pm in E308
 Masatoshi Sato, Osaka University
 The abelianization of the leveld mapping class group

Abstract: The leveld mapping class group is a
normal subgroup of finite index in the mapping class group
of an orientable surface. In this talk, I will describe the
abelianization of this group. For d=2, I will construct a
homomorphism from the level2 mapping class group to the
third spin bordism group of the EilenbergMacLane space of
the Z/2Zcoefficient first homology of the surface. It
induces an injective homomorphism on the abelianization of
this group, and is calculated in terms of the Brown
invariants of pin^{} structures of
embedded surfaces in mapping tori of the surface.

 Thursday February 17 at 3pm in E308
 Megumi Harada, McMaster University
 Equivariant cohomology, GKMcompatibility, and Schubert
calculus for Hessenberg varieties

Abstract: Hessenberg varieties are a class of
subvarieties of the flag variety which appear in many areas,
e.g. in geometric representation theory. (Springer
varieties, for example, are a special case.) In order to
generalize Schubert calculus to Hessenberg varieties, a
first step is to construct computationally convenient module
bases for the (equivariant) cohomology rings of Hessenberg
varieties analogous to the famous Schubert classes which are
a basis for the cohomology of flag varieties.
GoreskyKottwitzMacPherson ("GKM") theory gives a concrete
combinatorial description of the equivariant cohomology of
spaces with torus action which satisfy certain conditions
(usually called the GKM conditions). We propose a framework
for approaching the problem of constructing module bases for
Hessenberg varieties which uses GKM theory. The main
conceptual challenge in this context is that conventional
GKM theory requires a `sufficiently largedimensional torus'
action (to be made precise in the talk), while Hessenberg
varieties generally have only a circle action. To resolve
this, we define the notion of GKMcompatible subspaces of
GKM spaces and give applications in some special cases of
Hessenberg varieties. This is mainly joint work with
Tymoczko; time permitting, I will mention joint work with
Bayegan, and also with Dewitt.

 Tuesday February 22 at 3pm in E308
 Tim Austin, Brown University
 Some recent advances in Multiple Recurrence

Abstract: In 1975 Szemeredi proved the remarkable
combinatorial fact that any subset of the integers having
positive upper density contains arbitrarily long arithmetic
progressions. Shortly afterwards Furstenberg gave a new
proof of Szemerédi's Theorem using a conversion to an
assertion of `multiple recurrence' for
probabilitypreserving systems, which he then proved using
newlydeveloped machinery in ergodic theory. Furstenberg's
work gave rise to a new subdiscipline called `Ergodic Ramsey
Theory', which then found several further combinatorial
applications. More recent work has provided a much more
detailed picture of the structures that underlie this area
of ergodic theory, and offered a clearer insight into the
connections between this field and purely combinatorial
approaches to the same results. I will describe a purely
structural question within ergodic theory that has recently
emerged from these efforts, and whose solution in some
special cases gives a new approach to the multidimensional
generalizations of multiple recurrence and Szemeredi's
Theorem.

 Thursday February 24 at 3pm in E308
 Martin Bridgeman, Boston College
 The orthospectra of finite volume hyperbolic manifolds with
totally geodesic boundary and associated volume identities

Abstract: Given a finite volume hyperbolic
nmanifold M with totally geodesic boundary, an
orthogeodesic of M is a geodesic arc which is
perpendicular to the boundary. For each dimension n, we show
there is a real valued function F_{n}
such that the volume of any M is the sum of
values of F_{n} on the orthospectrum
(length of orthogeodesics). For n=2 the function
F_{2} is the Rogers Lfunction and the
summation identities give dilogarithm identities on the
Moduli space of surfaces.

 Thursday March 3 at 3pm in E308
 Michael Usher, University of Georgia
 Boundary depth and the Hofer norm

Abstract: In 1990, Hofer introduced a remarkable
conjugationinvariant norm on the group of Hamiltonian
diffeomorphisms of a symplectic manifold. Many properties of
this norm remain littleunderstood; in particular it is
still not known whether the group always has infinite
diameter with respect to the norm. I will discuss a proof
that the group has infinite diameter whenever the manifold
satisfies a certain simple dynamical condition. The proof is
based on Hamiltonian Floer theory, and in particular on the
behavior of a Floertheoretic quantity called the boundary
depth of a Hamiltonian diffeomorphism.

 Tuesday March 8 at 3pm in E308
 Dan Cohen, Louisiana State University
 Pure braid groups are not residually free

Abstract: A group G is residually free if
for every nontrivial element x in G,
there is a homomorphism f from G to a
free group for which x is not in the kernel of
f. We show that the Artin pure braid group on at
least four strands is not residually free.

 Thursday March 10 at 3pm in E308
 Matthew Kahle, Institute for Advanced Study
 Isoperimetric and coisoperimetric inequalities for random
simplicial complexes

Abstract: I will discuss the geometry of certain
random 2dimensional simplicial complexes. Babson, Hoffman,
and I found the vanishing threshold for the fundamental
group of these complexes to be much denser than the
vanishing threshold Linial and Meshulam found for homology.
A crucial part of our proof is establishing a linear
isoperimetric inequality: C L(S) > A(S) for
every nullhomotopic loop S and some contant
C. In more recent work I showed that these
complexes also satisfy a "coisoperimetric" inequality :
df > cf for all cochains f,
where df is the coboundary of f, and
c is a constant. This coisoperimetric inequality
provide a higherdimensional analogue of edge expansion for
graphs, and the constant c can be thought of as
analogous to the Cheeger constant.

 Tuesday March 15 at 3pm in E308
 Piotr Przytycki, Institute for Mathematics of the Polish Academy of
Sciences
 The ending lamination space of the fivepunctured sphere is
the Noebeling curve

Abstract: Joint with S.Hensel. The Noebeling curve
is the subspace of R^{3} consisting of
points with at least 2 irrational coordinates. It is the
unique 1dimensional Polish space which is connected,
locally path connected, and universal in dimension 1.
Connectivity and local pathconnectivity for ending
lamination spaces was proved by D. Gabai. Here we give an
argument, in the case of the fivepunctured sphere, for
1dimensionality and universality.

 Thursday March 17 at 3pm in E207
 Caroline Klivans, University of Chicago
 A geometric interpretation of the characteristic polynomial
of reflection arrangements

Abstract: We consider projections of points onto
fundamental chambers of finite real reflection groups. We
prove that for any finite real hyperplane arrangement the
average projection volumes of the maximal cones is given by
the coefficients of the characteristic polynomial of the
arrangement. These results naturally extend those of De
Concini and Procesi, Stembridge, and Denham which establish
the relationship for 0dimensional projections. We will
explain how this work arises in the field of
orderrestricted statistical inference, where projections of
random points play an important role. Joint work with
Matthias Drton and Ed Swartz.

 Tuesday March 29 at 3pm in E206
 Gabor Elek, Renyi Institute
 TBA

Abstract: TBA

 Thursday March 31 at 3pm in E308
 Yitwah Cheung, San Francisco State University
 Hausdorff dimension of the set of singular vectors

Abstract: Many problems in Diophantine approximation
can be addressed by analyzing the dynamics on the space of
lattices. For example, badly approximable vectors can be
understood in terms of bounded trajectories of a certain
diagonal flow. The notion of a singular vector is in a
natural way dual to that of a badly approximable vector. In
terms of the dynamics on lattices, they correspond to
divergent trajectories of the flow. In this talk, I will
describe the proof a recent result joint with Nicolas
Chevallier establishing that the Hausdorff dimension of the
set of singular vectors in R^{d} is
d^{2}/(d+1) for any d>1.
Essentially, this is equivalent to the statement that the
set of divergent trajectories of the flow by
diag(e^{t},...,e^{t},e^{dt})
acting on G/\Gamma where G=SL(d+1,R)
and \Gamma=SL(d+1,Z) has Hausdorff dimension
equal to dim G  d/(d+1). The general approach is
consistent with the strategy employed in the case
d=2 (Ann. Math. 173, (2011), 127167), which uses
a piecewise linear description of the flow trajectories to
obtain an encoding in terms of best approximants. I will
focus on the new ideas needed in the case d>2,
such as a sphere packing of H^{d+1} that
generalizes the packing of the upper half plane by disks
based at rational points p/q of diameter
1/q^{2}.

 Tuesday April 5 at 3pm in E308
 Luis Diogo, Stanford University
 TBA

Abstract: TBA

 Thursday April 7 at 3pm in E308
 Howard Masur, University of Chicago
 Ergodicity of the WeilPetersson geodesic flow

Abstract: This is joint work with Keith Burns and
Amie Wilkinson. Let \Sigma be a surface of genus
g with n punctures. We assume
3g3+n>0. Associated to \Sigma is
the Teichmuller space. This is the space of hyperbolic
metrics one can put on \Sigma, up to isotopy. The
mapping class group acts on the Teichmuller space with
quotient, the Riemann moduli space \cal
M(\Sigma). There are a number of interesting metrics
on \cal M(\Sigma); one of which is the
WeilPetersson metric. It is a Riemannian metric of negative
curvature and finite volume but it is not complete. In this
talk I will discuss the background on this metric and the
following theorem. \ Theorem: The WeilPetersson geodesic
flow is ergodic on \cal M(\Sigma).

 Tuesday April 12 at 3pm in E308
 Sam Kim, Tufts
 TBA

Abstract: TBA

 Thursday April 14 at 3pm in E308
 Michael Brandenbursky, Vanderbilt University
 Finite type invariants obtained by counting surfaces

Abstract: A Gauss diagram is a simple, combinatorial
way to present a knot. It is known that any Vassiliev
invariant may be obtained from a Gauss diagram formula that
involves counting (with signs and multiplicities)
subdiagrams of certain combinatorial types. These formulas
generalize the calculation of a linking number by counting
signs of crossings in a link diagram. Until recently,
explicit formulas of this type were known only for few
invariants of low degrees. I will present simple formulas
for an infinite family of invariants arising from the
HOMFLYPT polynomial. I will also discuss an interesting
interpretation of these formulas in terms of counting
surfaces of a certain genus and number of boundary
components in a Gauss diagram. This is a joint work with M.
Polyak.

 Thursday April 14 at 4pm in TBA
 Ted Chinburg, UPenn
 TBA

Abstract: TBA

 Tuesday April 19 at 3pm in E308
 YongGeun Oh, University of Wisconsin  Madison
 TBA

Abstract: TBA

 Thursday April 21 at 3pm in E308
 Ralf Spatzier, University of Michigan
 TBA

Abstract: TBA

 Thursday April 28 at 3pm in E308
 David Fisher, Indiana University
 TBA

Abstract: TBA

 Thursday May 5 at 3pm in E308
 Charles Pugh, UC Berkeley
 TBA

Abstract: TBA

 Thursday May 12 at 3pm in E308
 Romain Tessera, CNRS at Ecole Normale Superieure Lyon
 TBA

Abstract: TBA

 Thursday June 2 at 3pm in E308
 Melody Chan, UC Berkeley
 TBA

Abstract: TBA
For questions, contact