Geometry/Topology Seminar
Winter 2011
Thursdays (and sometimes Tuesdays) 3-4pm, in
Eckhart 308
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- Thursday January 13 at 3pm in E308
- Chris Leininger, University of Illinois at Urbana-Champaign
- Quasi-isometric embeddings into Teichmuller space
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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 Farb-Mosher
and Hamenstadt, this is known to be equivalent to the
existence of a quasi-isometric 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 quasi-isometric embeddings with either one of
these properties (though not both simultaneously). This is
joint work with Matt Clay, Johanna Mangahas and Saul
Schleimer.
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- Thursday January 20 at 3pm in E308
- Francois Gueritaud, Lille
- Veering triangulations and positive angle structures
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Abstract: Casson and Rivin have proposed a program
to explicitly construct the finite-volume hyperbolic metric
on a 3-manifold 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.
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- Thursday January 27 at 3pm in E308
- Alden Walker, Caltech
- Isometric endomorphisms of free groups
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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 n-1/2. Our methods also
let us construct explicit (and computable) quasimorphisms
which certify these facts. This is joint work with Danny
Calegari.
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- Tuesday February 8 at 3pm in E308
- Jayadev Athreya, University of Illinois at Urbana-Champaign
- On the distribution of gaps for saddle connection
directions
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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 Marklof-Strombergsson on the periodic Lorentz gas
and that of Eskin-Masur on flat surfaces. We describe some
applications to billiards in polygons.
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- Thursday February 10 at 3pm in E308
- Alex Furman, University of Illinois at Chicago
- Lattice envelopes
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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
Breuillard-Gelander, Margulis' commensurator superrigidity
and normal subgroup theorems, and some quasi-isometric
rigidity results.
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- Tuesday February 15 at 3pm in E308
- Masatoshi Sato, Osaka University
- The abelianization of the level-d mapping class group
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Abstract: The level-d 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 level-2 mapping class group to the
third spin bordism group of the Eilenberg-MacLane space of
the Z/2Z-coefficient 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.
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- Thursday February 17 at 3pm in E308
- Megumi Harada, McMaster University
- Equivariant cohomology, GKM-compatibility, and Schubert
calculus for Hessenberg varieties
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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.
Goresky-Kottwitz-MacPherson ("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 large-dimensional 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 GKM-compatible 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.
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- Tuesday February 22 at 3pm in E308
- Tim Austin, Brown University
- Some recent advances in Multiple Recurrence
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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
probability-preserving systems, which he then proved using
newly-developed 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.
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- 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
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Abstract: Given a finite volume hyperbolic
n-manifold 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 Fn
such that the volume of any M is the sum of
values of Fn on the orthospectrum
(length of orthogeodesics). For n=2 the function
F2 is the Rogers L-function and the
summation identities give dilogarithm identities on the
Moduli space of surfaces.
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- Thursday March 3 at 3pm in E308
- Michael Usher, University of Georgia
- Boundary depth and the Hofer norm
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Abstract: In 1990, Hofer introduced a remarkable
conjugation-invariant norm on the group of Hamiltonian
diffeomorphisms of a symplectic manifold. Many properties of
this norm remain little-understood; 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 Floer-theoretic quantity called the boundary
depth of a Hamiltonian diffeomorphism.
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- Tuesday March 8 at 3pm in E308
- Dan Cohen, Louisiana State University
- Pure braid groups are not residually free
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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.
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- Thursday March 10 at 3pm in E308
- Matthew Kahle, Institute for Advanced Study
- Isoperimetric and co-isoperimetric inequalities for random
simplicial complexes
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Abstract: I will discuss the geometry of certain
random 2-dimensional 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 null-homotopic loop S and some contant
C. In more recent work I showed that these
complexes also satisfy a "co-isoperimetric" inequality :
||df|| > c||f|| for all cochains f,
where df is the coboundary of f, and
c is a constant. This co-isoperimetric inequality
provide a higher-dimensional analogue of edge expansion for
graphs, and the constant c can be thought of as
analogous to the Cheeger constant.
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- Tuesday March 15 at 3pm in E308
- Piotr Przytycki, Institute for Mathematics of the Polish Academy of
Sciences
- The ending lamination space of the five-punctured sphere is
the Noebeling curve
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Abstract: Joint with S.Hensel. The Noebeling curve
is the subspace of R3 consisting of
points with at least 2 irrational coordinates. It is the
unique 1-dimensional Polish space which is connected,
locally path connected, and universal in dimension 1.
Connectivity and local path-connectivity for ending
lamination spaces was proved by D. Gabai. Here we give an
argument, in the case of the five-punctured sphere, for
1-dimensionality and universality.
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- Thursday March 17 at 3pm in E207
- Caroline Klivans, University of Chicago
- A geometric interpretation of the characteristic polynomial
of reflection arrangements
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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 0-dimensional projections. We will
explain how this work arises in the field of
order-restricted statistical inference, where projections of
random points play an important role. Joint work with
Matthias Drton and Ed Swartz.
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- Tuesday March 29 at 3pm in E206
- Gabor Elek, Renyi Institute
- TBA
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Abstract: TBA
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- Thursday March 31 at 3pm in E308
- Yitwah Cheung, San Francisco State University
- Hausdorff dimension of the set of singular vectors
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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 Rd is
d2/(d+1) for any d>1.
Essentially, this is equivalent to the statement that the
set of divergent trajectories of the flow by
diag(et,...,et,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), 127-167), 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 Hd+1 that
generalizes the packing of the upper half plane by disks
based at rational points p/q of diameter
1/q2.
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- Tuesday April 5 at 3pm in E308
- Luis Diogo, Stanford University
- TBA
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Abstract: TBA
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- Thursday April 7 at 3pm in E308
- Howard Masur, University of Chicago
- Ergodicity of the Weil-Petersson geodesic flow
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Abstract: This is joint work with Keith Burns and
Amie Wilkinson. Let \Sigma be a surface of genus
g with n punctures. We assume
3g-3+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
Weil-Petersson 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 Weil-Petersson geodesic
flow is ergodic on \cal M(\Sigma).
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- Tuesday April 12 at 3pm in E308
- Sam Kim, Tufts
- TBA
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Abstract: TBA
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- Thursday April 14 at 3pm in E308
- Michael Brandenbursky, Vanderbilt University
- Finite type invariants obtained by counting surfaces
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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
HOMFLY-PT 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.
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- Thursday April 14 at 4pm in TBA
- Ted Chinburg, UPenn
- TBA
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Abstract: TBA
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- Tuesday April 19 at 3pm in E308
- Yong-Geun Oh, University of Wisconsin - Madison
- TBA
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Abstract: TBA
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- Thursday April 21 at 3pm in E308
- Ralf Spatzier, University of Michigan
- TBA
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Abstract: TBA
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- Thursday April 28 at 3pm in E308
- David Fisher, Indiana University
- TBA
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Abstract: TBA
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- Thursday May 5 at 3pm in E308
- Charles Pugh, UC Berkeley
- TBA
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Abstract: TBA
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- Thursday May 12 at 3pm in E308
- Romain Tessera, CNRS at Ecole Normale Superieure Lyon
- TBA
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Abstract: TBA
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- Thursday June 2 at 3pm in E308
- Melody Chan, UC Berkeley
- TBA
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Abstract: TBA
For questions, contact