Sponsored by
Boston College
HMI
NSF
SFI
|
The 7th William
Rowan
Hamilton Geometry and
Topology Workshop
on
The
Geometry and Dynamics of Teichmüller Spaces
August 30-September 3, 2011
The Hamilton Mathematics
Institute,
Trinity College Dublin
Mini-Course
on
Teichmüller
Theory
by Jeff
Brock and Howard Masur
August
30-31, 2011
The mini-course will consist of a series
of lectures and problem sessions directed by Jeff Brock and Howard
Masur.
Schedule of Mini-Course
All minicourse lectures will be held in the Salmon Lecture Hall, in the
Hamilton Building (see campus map).
Time
|
Speaker
|
Talk
|
Tuesday
|
|
9:00
-
10:30
|
Howard Masur (Chicago)
|
Quasiconformal maps,
Teichmüller's theorem and the Teichmüller geodesic flow I |
10:00 - 11:00
|
COFFEE & DISCUSSION
|
11:00
-
12:30
|
Jeff Brock (Brown) |
Coarse and synthetic
geometry of Teichmüller space I
|
12:30
-
2:00
|
LUNCH |
1:30
-
2:30
|
Will
Cavendish
(Princeton)
|
The Weil-Petersson
Completion of Moduli Space |
2:30
-
3:30
|
|
3:30
-
4:30
|
Johanna
Mangahas
(Brown)
|
TBA
|
Wednesday
|
|
| 9:00
-
10:30
|
Jeff Brock (Brown) |
Coarse and synthetic
geometry of Teichmüller space II
|
10.30 - 11:00
|
COFFEE &
DISCUSSION |
| 11:00
-
12:30 |
Howard Masur
(Chicago) |
Quasiconformal maps,
Teichmüller's theorem and the Teichmüller geodesic flow II |
| 12:30
-
2:00 |
LUNCH |
| 2:00
-
3:00 |
Vaibhav
Gadre
(Harvard)
|
Random walks on the
mapping class group and hitting measures on the Thurston boundary
|
3:00 - 4:00
|
DISCUSSION |
Lecture
Series
on
The
Geometry and Dynamics of Teichmüller Spaces
September
1-3,
2011
Schedule of Talks
All talks will be held in the East End Lecture Theatre 3 (LTEE3) of the
Hamilton
Building, TCD (see campus map).
Time
|
Speaker
|
Talk
|
Thursday
|
|
9.30
-
10.30
|
Howard Masur (Chicago)
|
Schmidt games and
bounded geodesics in moduli space |
10.30-10.45
|
COFFEE
|
10:45
-
11:45
|
Steve
Kerckhoff
(Stanford) |
Hyperbolic and AdS
geometry in dimension 3 |
11:45
-
1:30
|
LUNCH |
1:30
-
2:30
|
Corinna Ulcigrai
(Bristol)
|
Ergodic properties
of extensions of area-preserving flows |
2:45
-
3:45
|
William
Goldman
(Maryland)
|
Complete flat
Lorentz 3-manifolds and
two-generator Fuchsian groups |
4:00
-
5:00
|
Francois Labourie (Orsey)
|
Cubic holomorphic
differentials and real projective structures
|
Friday
|
|
| 9:30
-
10:30
|
Alex Eskin (Chicago)
|
Rational billiards
and the SL(2,R) action on modui space |
10.30 - 11:00
|
POSTER SESSION
|
| 11:00
-
12:00 |
Scott Wolpert (Maryland) |
Products of twists,
geodesic-lengths and Thurston shears |
| 12:00
-
1:30 |
LUNCH |
| 1:30
-
2:30 |
Ken Bromberg (Utah)
|
The projection
complex and its applications |
| 2:45
-
3:45 |
Mladen
Bestvina
(Utah) |
Teichmüller
theory in Outer space |
| 4:00
-
5:00 |
Jeff Brock (Brown)
|
Weil-Petersson
billiards on moduli space and the geometry of 3-dimensional manifolds |
Saturday
|
|
| 10:00
-
11:00 |
David
Dumas
(UIC)
|
Floyd's theorem and
Lambda-trees |
11.00 - 11.15
|
COFFEE
|
| 11.15-
12:15 |
Chris
Leininger
(UIUC)
|
Hyperbolic spaces in
Teichmuller spaces |
12.15 - 2.00
|
DISCUSSIONS
AND
LUNCH
|
2.00 - 3.00
|
Kasra Rafi (Oklahoma, Norman) |
Coarse
Differentiation and the Rank of Teichmüller Space |
3.00 - 4.00
|
Alexandra Pettet
(British Columbia)
|
Geometry and
dynamics of fully irreducible outer automorphisms of a free group |
| END OF
WORKSHOP |
Friday Poster Session
There will be a poster session Friday. The posters will be
presentations from students who attended the minicourse and
will be up from 10:30 am to 5:00 pm Friday. The presenters will be
available during the morning break from 10:30-11:00 am to discuss the
work.
Abstracts for Lecture Series
Mladen Bestvina(University
of Utah, Salt Lake)
Title: Teichmuller theory in Outer space
Abstract: I will explain the basic dictionary between Teichmuller space
and Outer space, including the notions corresponding to Teichmuller
distance and Teichmuller geodesics. To focus the discussion, I will
outline the proof (joint with Mark Feighn) of the hyperbolicity of the
complex of free factors.
Jeff Brock (Brown
University)
Title: Weil-Petersson billiards on moduli space and the geometry of
3-dimensional manifolds
Kenneth Bromberg
(University of Utah)
Title: The projection complex and its applications
Abstract: We will describe a new combinatorial complex called a
projection complex. The construction is axiomatic and applies to a wide
variety of situations where there are some features of negative
curvature. In particular a wide variety of groups act on a projection
complex. The key property of this complex is that it is quasi-isometric
to a tree. We will describe the construction and give some
applications. This is joint work with with M. Bestvina and K. Fujiwara.
David Dumas (University of
Illinois, Chicago)
Title: Floyd's theorem and Lambda-trees
Abstract: Floyd proved that for a 3-manifold M with nonempty
incompressible boundary S, the set of boundary curves of incompressible
and boundary-incompressible surfaces is contained in a "thin" subset of
the measured lamination space of S; specifically, they lie in a
piecewise linear cone that is isotropic with respect to the Thurston
symplectic form. (This generalizes an earlier result of Hatcher
for manifolds with torus boundary.) We discuss a generalization
of this theorem involving actions of surface groups and 3-manifold
groups on R-trees and Lambda-trees. As time permits we will also
discuss how this result is used to study Thurston's skinning maps.
Alex Eskin (University of
Chicago)
Title: Rational billiards and the SL(2,R) action on modui space
Abstract: I will discuss ergodic theory over the moduli space of
compact Riemann surfaces and its applications to the study of polygonal
billiard
tables. There is an analogy between this subject and the theory of
flows on homogeneous spaces; I will talk about some successes and
limitations of this viewpoint. This is joint work with Maryam
Mirzakhani.
William Goldman
(University of Maryland)
Title: Complete flat Lorentz 3-manifolds and
two-generator Fuchsian groups
Abstract: Consider a hyperbolic surface S with fundamental group F_2
the rank two free group. S is homeomorphic to either a 3-holed sphere,
a 2-holed cross-surface (projective plane), a 1-holed torus or a
1-holed Klein bottle. An affine deformation of S is a complete affine
3-manifold whose associated hyperbolic surface is S. Complete affine
3-manifolds with fundamental group F_2 fall into four types
corresponding to these four topological types of S. The space of affine
deformations of S is a convex cone, whose boundary is determined by the
measured geodesic laminations on S. The 3-manifold itself is a
handlebody of genus two. In his talk, I will describe the geometry and
the topology of these 3-manifolds and how they relate to deformations
of the hyperbolic structure on S.
Steve Kerckhoff (Stanford
University)
Title: Hyperbolic and AdS geometry in dimension 3
Abstract: In the early 90's Geoff Mess discovered a beautiful analogue
in AdS geometry to the Ahlfors-Bers theory of quasi-Fuchsian groups.
This analogy has been reinforced by the connection between the work of
Series and of Bonahon on almost Fuchsian quasi-Fuchsian groups and
parallel work of Bonsante and Schlenker in the AdS setting. We
will take a new look at this phenomenon from the point of view of
Transitional Geometry, as recently developed by Jeff Danciger.
Francois Labourie (Orsey University)
Title: Cubic holomorphic differentials and
real projective structures
Abstract: I will present not so recent results, proved independently
by John Loftin and myself about the paramatisation of the moduli space
of convex real projective structures by cubic holomorphic
differentials. I wil then explain the interpretation of this result as
the existence of a minimal surface in a locally symmetric space, as
well as existence of holomorphic curves in SL(3,R)/SL(2,R).
Chris Leininger
(university of Illinois, Urbana-Champaign)
Title: Hyperbolic spaces in Teichmuller spaces
Abstract: I'll describe a construction of quasi-isometric
embeddings of hyperbolic space into Teichmuller spaces with nice
geometric properties---specifically, the image is quasi-convex and lies
in the thick part. As a corollary, this produces quasi-isometric
embeddings of hyperbolic space into curve complexes. I'll also
explain the motivation for such a construction relating to
Gromov-hyperbolic surface bundles over hyperbolic manifolds. This
is joint work with Saul Schleimer.
Howard Masur (Chicago
University)
Title: Schmidt games and bounded geodesics in moduli space.
Abstract: In 1966 W.Schmidt introduced a game, now called a
Schmidt game, that can be played in any metric space and introduced the
notion of winning sets for this game. Winning sets have various nice
properties. The first main example are real numbers with bounded
coefficients in their continued fraction expansions. Equivalently,
these correspond to hyperbolic geodesics in the upper half place that
stay in a bounded subset of the quotient modular surface; the
Teichmuller space of a torus. Here we consider higher genus
generalizations. We prove that in each Teichmuller disc the geodesics
that are bounded in the moduli space form a winning set in the Schmidt
game. In fact they are winning in an improved game devised by
McMullen. I will talk about all of these notions. This is joint
work with Jonathan Chaika and Yitwah Cheung.
Alexandra Pettet
(University of British Columbia)
Title: Geometry and dynamics of fully irreducible outer automorphisms
of a free group
Abstract: The outer automorphism group of a free group of finite rank
shares many properties with the mapping class group of a surface,
however its geometry is not nearly as well understood. Motivated by
analogy, I will present some results concerning the geometry and
dynamics of fully irreducible elements which were previously well-known
for their mapping class group counterpart, the pseudo Anosov elements.
This is joint work Matt Clay.
Kasra Rafi (University of
Oklahoma, Norman)
Title: Coarse Differentiation and the Rank of Teichmüller Space
Abstract: Let S be a possibly disconnected surface of finite hyperbolic
type and let T (S) be the Teichmüller space of S. We study
quasi-Lipschitz maps from a large subset of R^n into T (S). We give a
local description of the image of such a map. That is, locally, the
image lies in a neighborhood of a standard flat (product of lines) up
to a small linear error. As a corollary, we compute the maximum
dimension n where a large box in R^n can be mapped quasi-isometrically
into the Teichmüller space.
Corinna Ulcigrai
(University of Bristol)
Title: Ergodic properties of extensions of area-preserving flows
Abstract: Consider smooth area preserving flows on a surface S of genus
g (locally Hamiltonian flows). In a series of papers, we studied
the ergodic properties (in particular mixing) of typical locally
Hamiltonian flows using Rauzy-Veech induction, which is a discrete
version of the Teichmueller flow on Abelian differentials.
In a joint work with K. Fraczek, we consider extensions of locally
Hamiltonian flows given by a real valued function f on S, that are
infinite 3-dimensional volume preserving flows on SxR. Extensions were
studied in genus one, but very little is known about the ergodic
properties of extensions for higher genus. For any genus g>1, we
construct extensions which are ergodic with respect to the infinite
invariant measure. We use the Teichmuller flow (in the version of
Rauzy-Veech induction) to renormalize locally Hamiltonian flows.
Scott Wolpert (University
of Maryland)
Title: Products of twists, geodesic-lengths and Thurston shears
Abstract: We consider ideal geodesics on hyperbolic surfaces with
cusps. Thurston defined the shear deformation for weighted ideal
geodesics as a generalized weighted twist deformation. Weights
satisfy a completeness condition at each cusp. The weighted sum
of ideal geodesic length functions is defined similarly. We
consider the Weil-Petersson Riemannian and symplectic geometry of
shears and weighted ideal length sums. The basic analytic
formulation of shear deformations and weighted ideal length sums is
divergent.
Results include:
. An analytic formulation for shear
deformations and weighted ideal length sums without divergences.
. The symplectic geometry of shears
and weighted ideal length sums - the symplectic dual of a shear is the
corresponding weighted ideal length sum.
. Defining an elementary 2-form for
shear weights - the symplectic pairing of shears is given by the 2-form
and the Poisson bracket for weighted ideal length sums is given by the
2-form.
. A distances-sum formula for the
Riemannian pairing of shears.
. An example - an exact relation
for distances between lines in the classical modular tessellation.
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