# Schedule of the Conference

## Monday, June 14

09:00-09:30 |
Registration |

09:30-10:30 |
Howard Masur: Ergodicity of the Weil-Petersson geodesic flow on moduli space |

10:30-11:00 |
Coffee break |

11:00-12:00 |
Gabriele Mondello: On the horocyclic flow |

12:00-14:00 |
Lunch break |

14:00-15:00 |
Thomas Koberda: Mapping class groups and virtual homology of surfaces |

15:00-15:30 |
Coffee break |

15:30-16:30 |
Erwan Lanneau: Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction |

## Tuesday, June 15

09:30-10:30 |
Eriko Hironaka: The "shape" of small dilatation pseudo-Anosov mapping classes |

10:30-11:00 |
Coffee break |

11:00-12:00 |
W. Patrick Hooper: The invariant measures for some infinite interval exchange maps |

12:00-14:00 |
Lunch break |

14:00-15:00 |
Dmitry Korotkin: Tau function and moduli of differentials |

15:00-15:30 |
Coffee break |

15:30-16:30 |
Samuel Grushevsky: Meromorphic differentials with real periods and the geometry of the moduli space of Riemann surfaces |

## Wednesday, June 16

09:30-10:30 |
Alex Eskin: Invariant and stationary measures for the SL(2,R)-action on moduli space |

10:30-10:45 |
Coffee break |

10:45-11:45 |
Barak Weiss: Geometry of REL leaves |

11:45-14:00 |
Lunch break |

12:00-13:00 |
Jing Tao: Diameter of the thick part of moduli space |

afternoon: Hike |

## Thursday, June 17

09:30-10:30 |
Curtis McMullen: Entropy on Riemann surfaces |

10:30-11:00 |
Coffee break |

11:00-12:00 |
Pascal Hubert: Rosen continued fractions and trancendence |

12:00-14:00 |
Lunch break |

14:00-15:00 |
Jon Chaika: Diophantine properties of IETs |

15:00-15:30 |
Coffee break |

15:30-16:30 |
Luca Marchese: Khinchin-type conditions and asymptotic law for the Teichmuller flow |

## Friday, June 18

09:30-10:30 |
Gaven John Martin: Mappings of finite distortion and Teichmüller spaces |

10:30-11:00 |
Coffee break |

11:00-12:00 |
Matthew Bainbridge: The classification problem for Teichmueller curves |

12:00-14:00 |
Lunch break |

14:00-15:00 |
Ronen Mukamel: Teichmueller curves in genus two: orbifold points and jacobians with complex multiplication |

15:00-15:30 |
Coffee break |

15:30-16:30 |
Giovanni Forni: The Kontsevich-Zorich exponents beyond the canonical measures |

## Abstracts:

Jon Chaika: Diophantine properties of IETs

This talk is on shrinking target and diophantine properties for IETs. The results are motivated by results of Khinchin, Kurzweil and Cassels for rotations. I will give descriptions of generic behavior, some discussion of behavior that is always present and present some strange exceptional behavior.

I will show:

1) A result on homogenous approximation for generic IETs.

2) A discussion of what shrinking targets properties almost every IET satisfies.

3) A shrinking target property that all IETs satisfy.

4) Some exotic behavior that can occur in minimal but not uniquely ergodic IETs.

Some of this is joint work with Michael Boshernitzan.

Giovanni Forni: The Kontsevich-Zorich exponents beyond the canonical measures

I will present a criterion for the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle which improves upon my proof of non-uniform hyperbolicity for the canonical absolutely continuous invariant measures on the strata of moduli space of abelian differentials. Besides that case, the criterion covers other previously known cases (algebraically primitive Veech surfaces, due to Bouw-Moeller, "stairs" square-tiled surfaces and "stairs" square-tiled cyclic covers, due to Eskin-Kontsevich-Zorich) and yields some new results (McMullens Prym surfaces). Presumably, the criterion can be used to prove the non-uniform hyperbolicity of the cocycle for the measures coming from strata of quadratic differentials via double cover construction.

Samuel Grushevsky: Meromorphic differentials with real periods and the geometry of the moduli space of Riemann surfaces

We use meromorphic differentials on Riemann surfaces with prescribed singular parts and all periods being real to define a foliation of the moduli space of Riemann surfaces. We then use this foliation to give a quick direct proof of the Diaz bound on the dimension of compact subvarieties of the moduli space of Riemann surfaces, and indicate an approach to proving the vanishing results for the cohomology of the moduli space. Based on joint work with Igor Krichever.

Eriko Hironaka: The "shape" of small dilatation pseudo-Anosov mapping classes

We will talk about the "shape" of small dilatation pA mapping classes, both in terms of quasi-symmetries of the train track and in terms of the pattern of associated characteristic polynomials.

W. Patrick Hooper: The invariant measures for some infinite interval exchange maps

I will give a construction of some infinite interval exchange maps for which it is possible to classify the locally finite ergodic invariant measures. These interval exchange maps arise from infinite translation surfaces admitting a non-elementary affine automorphism group, and we can interpret these interval exchange maps as deterministic "random" walks on graphs. In many cases, these ergodic measures can be interpreted arising from invariant potentials for the corresponding random walk. The interval exchange maps we study arise from a modification of a construction of Thurston which produced examples of surfaces admitting pseudo-Anosov automorphisms using a pair of multi-twists. (For finite interval exchange maps, the types of questions we ask are answered by work of Thurston, Masur and Veech.)

Thomas Koberda: Mapping class groups and virtual homology of surfaces

I will discuss some of the methods and difficulties in analyzing mapping class groups by looking at their action on the virtual homology of a surface and on various tractable quotients of its fundamental group.

Dmitry Korotkin: Tau function and moduli of differentials

The tau function on the moduli space of generic holomorphic 1-differentials arises in the holomorphic factorization formula of determinant of Laplacian on Riemann surfaces in flat metrics with trivial holonomy. We interpret the tau-function as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. The talk is based on joint works with A. Kokotov and P. Zograf.

Erwan Lanneau: Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction

In this talk I will explain the link between pseudo-Anosov homeomorphisms and Rauzy-Veech induction. We will see how to derive properties on the dilatations of these homeomorphisms (I will recall the definitions) and as an application, we will use the Rauzy-Veech-Yoccoz induction to give lower bound on dilatations.

Luca Marchese: Khinchin-type conditions and asymptotic law for the Teichmuller flow

We study a diophantine condition for interval exchange transformations and translation surfaces, which is inspired by the classical Khinchin theorem. We prove that we have the same dichotomy as in Khinchin result. Then we deduce a sharp estimation on how fast the typical Teichmüller geodesic wanders towards in finity in the moduli space of translation surfaces.

Gaven John Martin: Mappings of finite distortion and Teichmüller spaces

Mappings of finite distortion are generalisations of quasiconformal mappings. They allow much greater flexibility in studying degenerating structures. Thus there is the possibility of having a mapping in ones hand on the boundary of Teichmüller spaces (of various dynamical systems) and the dream of an analytic proof of the ELC. Typically the defining equations for these mappings are degenerate elliptic (quasiconformal <=> unifomrly elliptic) and there are very interesting analytic problems involved in their study. Here we discuss the extremal theory of mappings whose distortion lies in an L^{p} space (or nearby Zygmund space). There are interesting connections with the theory of harmonic mappings and particularly the Nitsche conjecture (now Theorem) and the Schoen conjecture. We may talk about interesting applications in nonlinear materials science too, if time allows.

Howard Masur: Ergodicity of the Weil-Petersson geodesic flow on moduli space

This represents joint work with Keith Burns and Amie Wilkinson. Let S be a surface of genus g with n punctures. We assume 3g-3+n is at least 1. Associated is the Teichmuller space T(S) and the Weil-Petersson metric WP on T(S). Among the properties of the metric are that is not complete and that it has negative curvature which is not bounded below, or above, away from 0. The metric is invariant under the action of the mapping class group and so descends to a finite volume metric on the quotient moduli space. We show that the geodesic flow is ergodic on the quotient. The proof uses techniques from Teichmuller theory, smooth ergodic theory, and differential geometry. I will give some indications of the main ideas of the proof.

Gabriele Mondello: On the horocyclic flow

In this talk we will examine the relation between two Hamiltonian flows on the Teichmüller space: the earthquake and the Teichmüller horocyclic flow.

Ronen Mukamel: Teichmueller curves in genus two: orbifold points and jacobians with complex multiplication

For each D>4 the discriminant of an order in a real quadratic field, the Weierstrass curve W is the union of one or two Teichmuller curves in genus two. Let E(D) be the orbifold points on W other than the pentagon, and let E = \cup_D E(D) be the collection of all such points. We will show that E coincides with the complex multiplication points on a single Shimura curve in M_{2}. As a consequence, we obtain a formula for the number of points in E(D), show the genus of WD tends to infinity with D, and give equations defining algebraic curves on W_{D} for most D.

Jing Tao: Diameter of the thick part of moduli space

Let S be a surface of finite type. We study the shape of moduli space of S. We show that, in either the Teichmuller, Lipschitz, or bi-Lipschitz metric, the diameter of the thick part of moduli space grows like the logarithm of the Euler characteristic of S. A similar result is true for moduli space of metric graphs (the quotient of Outer space by Out(F_{n})). This is joint work with Kasra Rafi.

Barak Weiss: Geometry of REL leaves

I will present joint work with Erwan Lanneau and John Smillie regarding the geometry of infinite translation surfaces with a large Veech group (dense in SL(2,R)) which arise as leaves of the REL foliation in the moduli space of translation surfaces with two singularities. I will focus on the stratum H(1,1).