Program - June 18-20, 2009
|
Thursday |
Friday |
Saturday |
10:00 - 11:00am
|
Alex
Counting problems on flat surfaces |
Saul
Hyperbolicity, I |
Yitwah
Characterisations of non-ergodic parameters |
| 11:30 - 12:30am |
Saul
The complex of curves |
Yitwah
Recurrence implies Unique Ergodicity |
Saul
Hyperbolicity II |
| 12:30am - 2:00pm |
LUNCH at CROUS |
LUNCH at CROUS |
Picnic |
2:00 - 3:00am
|
Yitwah
Non-ergodic Measured Foliations |
Chris
Boundary I |
|
| 3:30 - 4:30pm |
Chris
Flat structures |
Alex
Counting lattice points on
Teichmuller space |
Alex
Counting closed geodesics
in moduli space |
| 5:00 - ...pm |
talks
Title |
talks
Title |
Chris
Boundary II |
Saul Schleimer and Chris Leininger
Lecture 1 [Saul]: The complex of curves.
In the first lecture we will introduce the complex of curves, C(S), of a
surface S. The objective will be to state the two main theorems that will
be proved during the course of these six lectures. The first result is
due to Masur and Minsky, and states that C(S) is Gromov hyperbolic. The
second result is due to Klarreich, and gives a geometric description of
the Gromov boundary as the space of ending laminations.
Lecture 2 [Chris]: Flat structures.
In this lecture we will discuss the role of Teichmuller theory in the
proofs of the two main theorems. The main idea in both is to study the
``systole map'' from the Teichmuller space T(S) to the curve complex C(S)
sending a hyperbolic structure to one of its systoles. This talk will
introduce some of the tools used throughout, in particular the flat
metrics used to describe Teichmuller geodesics and some of their basic
geometric properties.
Lecture 3 [Saul]: Hyperbolicity, I.
We begin the proof of hyperbolicity, following Bowditch.
Lecture 4 [Chris]: Boundary I.
We begin the proof of Klarreich's theorem.
Lecture 5 [Saul]: Hyperbolicity II.
Lecture 6 [Chris]: Boundary II.
Yitwah Cheung: Non-ergodic Measured Foliations.
Lecture 1: Non-ergodic Measured Foliations.
In the first lecture I will begin by defining what is
a measured foliation and explaining how they arise
from studying directional flows on flat surfaces.
Intuitively, a measured foliation is uniquely ergodic
if all its leaves are uniformly distributed on the surface;
in particular, every leaf is dense, in which case the
measured foliation is said to be minimal. In this talk,
I will describe a construction of Masur and Smillie
that produces (lots of) examples of non-ergodic
measured foliations by realising them as a limits
of a sequence of partitions of a flat surface.
Lecture 2: Recurrence implies Unique Ergodicity.
In the second lecture I will discuss a theorem of Masur's
that provides a sufficient condition, expressed in terms
of the Teichmuller flow, for a measured foliation to be
uniquely ergodic. More precisely, the condition says a
measured foliation is uniquely ergodic if the associated
Teichmuller geodesic is recurrent when projected to the
moduli space of Riemann surfaces. As an application
of Masur's theorem, we shall prove Veech dichotomy.
Lecture 3: Characterisations of Non-ergodic parameters.
In the last lecture, I will describe some extensions and
generalisations of Masur's theorem that allow us to show
that for some flat surfaces, every non-ergodic direction
arises from a Masur-Smillie construction. Historically,
Masur and Smillie were motivated by a paper of Veech
in 1969 exhibiting examples of minimal but not uniquely
ergodic Z/2-skew products over the rotations of the circle.
As an application, we shall give a complete characterisation
of the set of parameters in Veech's examples that give rise
to non-ergodic Z/2-skew products.
Alex Eskin
Lecture 1: Counting problems on flat surfaces.
I will discuss how to count cylinders of closed geodesics and
other related quantities on a flat surface. The main tool is
the dynamics of the SL(2,R) action.
Lecture 2: Counting lattice points on Teichmuller space, and
volume growth of balls in the Teichmuller metric.
I will discuss briefly the dynamics of the Teichmuller geodesic
flow, which is the main tool for this problem.
Lecture 3: Counting closed geodesics in moduli space.
I will go over the Margulis argument using mixing of the geodesic
flow, and discuss how to modify it for Teichmuller space.
