# Schedule, Titles & Abstracts

Schedule

Monday | Tuesday | Wednesday | Thursday | Friday | ||

09:30-10:30 | Wright | Masur | Marchese | Lenzhen | 09:30-10:30 | Hamenstaedt |

11:00-11:30 | Ruhr | Apisa | Taha | Dozier | 11:00-11:30 | Fougeron |

11:35-12:35 | Moeller | Skripchenko | Matheus | Delecroix | 11:35-12:05 | Ghazouani |

LUNCH BREAK | ||||||

14:30-15:00 | Gutierrez | Pardo | Free Afternoon | 13:30-14:30 | Weiss | |

15:30-16:30 | Zvonkine | Lanneau | 14:35-15:35 | Chaika |

**Paul APISA**

Title: Shouting across the void-Low dimensional orbit closures in hyperelliptic components.

Abstract: A simple technique for studying rank one orbit closures will be introduced and I will explain why it is strong enough to show that the rank one orbit closures in hyperelliptic components of strata are either Teichmuller curves or branched covering constructions.

**Jon CHAIKA**

Title: Horocycle orbits in strata of translation surfaces I

Abstract: By work of Ratner, Margulis, Dani and many others, the horocycle flow on homogeneous spaces has strong measure theoretic and topological rigidity properties. By work of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohommadi, the action of SL(2,R) and the upper triangular subgroup of SL(2,R) on strata of translation surfaces have similar rigidity properties. We will describe how

some of these results fail for the horocycle flow on strata of translation surfaces. In particular,

1) There exist horocycle orbit closures with fractional Hausdorff dimension.

2) There exist points which do not equidistribute under the horocycle flow with respect to any measure.

3) There exist points which equidistribute distribute under the horocycle flow to a measure, but they are not in the topological

support of that measure.

This is joint work with John Smillie and Barak Weiss.

**Vincent DELECROIX**

Title: Counting meanders with combinatorial constraints

Abstract: Meanders are pairs of curves on the sphere (considered up to homotopy). They were considered in theoretical physics as well as combinatorics and their enumeration (as the number of intersection tends to infinity) is known to be delicate. A meander can equivalently be considered as a square tiled surface of genus 0 whose horizontal and vertical cylinder decompositions are made of a unique cylinder of height one. With E. Goujard, P. Zograf and A. Zorich we provide explicit asymptotics of meanders with further combinatorial constraints (corresponding to fixing the stratum). The two principal ingredients are an equidistribution theorem (close in spirit to Mirzakhani simple curves counting result on hyperbolic surfaces) and the explicit volume computation of J. Athreya, A. Eskin and A. Zorich in genus 0.

**Ben DOZIER**

Title: Convergence of Siegel-Veech constants

Abstract: I will prove a continuity property of Siegel-Veech constants, namely that given a converging sequence of ergodic SL(2,R)-invariant probability measures on a stratum, the associated Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. The proof uses a certain recurrence-type result for the SL(2,R) action and works for any of the commonly considered types of Siegel-Veech constants (saddle connection, cylinder, area, etc). I'll use this, together with tools developed by Eskin-Mirzakhani-Mohammadi and the classification of measures in the stratum H(2) due to McMullen, to prove that Siegel-Veech constants of Teichmuller curves in H(2) converge to the Siegel-Veech constant for the whole stratum. Evidence for this phenomenon was found by Lelievre and Bainbridge.

****

**Charles FOUGERON**

Title: Rauzy induction for affine interval exchanges.

Abstract: Rauzy induction has been widely explored for interval exchanges in the last decades and has appeared to be a key tools in studying translation surfaces.

The same induction can be applied to affine interval exchanges and happen to have very different properties.

We will explore some of these particular properties, and explain new construction of interesting dynamical behaviours they yield.

**Selim GHAZOUANI**

Title: Teichmueller dynamics and dilation tori

Abstract: In this talk I will endeavour to explain how ideas from Teichmüller dynamics can be implemented in the more general context of dilation structures on surfaces. These structures are structures modelled on the complex plane via the group of dilations and are the natural objects that arise when one forms the suspension of an affine interval exchange transformation. Like translations surfaces, they come with a 1-parameter family of directional foliations which are transversally affine.

**Rodolfo GUTIERREZ**

Title: Rauzy–Veech groups of flat surfaces

Abstract: The Rauzy–Veech induction is a powerful renormalization procedure for (half-)translation flows. By tracking the changes it induces in homology, we define the Rauzy–Veech monoid (or group) of a connected component of a stratum of Abelian or quadratic differentials. This monoid was proven to be pinching and twisting by Avila and Viana, which implies the Kontsevich–Zorich conjecture stating the simplicity of the Lyapunov spectrum of almost every translation flow with respect to the Masur–Veech measures.

In this talk, I will present a full classification of the Rauzy–Veech groups of Abelian strata: they are explicit finite-index subgroups of their ambient symplectic groups. This is strictly stronger than pinching and twisting and solves a conjecture of Zorich about the Zariski-density of such groups. Moreover, some techniques can be extended to the quadratic case to prove that the indices of the “plus” and/or “minus” Rauzy–Veech groups of certain connected components of quadratic strata are also finite. This proves that the Lyapunov spectra of such strata are simple, which was not previously known.

**Ursula HAMENSTAEDT**

Title: Ergodic properties of strata with a simple zero

Abstract: Given a stratum of abelian differentials with a simple zero, we show that the absolute period foliation as well as the real rel folations are ergodic.

We will also discuss some applications to the Kontsevich Zorich cocycle.

**Erwan LANNEAU**

Title: Diffusion rate in non generic directions in the wind-tree model

Abstract: I will discuss a recent work on the wind-tree model (this is a joint work with Sylvain Crovisier and Pascal Hubert). We show that any real number in [0,1) is a diffusion rate for the periodic wind-tree model.

**Anja LENZHEN**

Title: Limit sets of Teichmuller geodesic rays in the Thurston boundary of Teichmuller space

Abstract: H. Masur showed in the early 80s that almost every Teichmuller ray converges to a unique point in PMF. It is also known since a while that there are rays that have more than one accumulation point in the boundary.

I will give an overview of what is understood so far about the limit sets of Teichmuller rays, mentioning some recent progress. For example, I will mention recent joint work with K. Rafi and B. Modami where we give a construction of a ray whose limit set in PMF is a d-dimensional simplex.

**Luca MARCHESE**

Title: Hitting time on some translation surfaces.

Abstract: We consider the flow in a given direction on a translation surface and we study the asymptotic behaviour for r that tends to 0 of the time needed by generic orbits to hit the r-neighbourhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction. In higher genus, we consider a generalized geometric notion of Diophantine type and we seek for relations with hitting time.

For surfaces in the stratum H(2) we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any origami in the same stratum the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. This is a joint work with D. H. Kim and S. Marmi.

**Howard MASUR**

Title: Hausdorff dimension of the set of non uniquely ergodic interval exchange transformations

Abstract: Let π a permutation in the Rauzy class of the symmetric permutation in 4 or more letters. The set of IET with permutation π is parametrized by the standard d-1 dimensional simplex Δ.

I will discuss the following result and some consequences for translation surfaces. Theorem: The subset of Δ consisting of non uniquely ergodic IET has Hausdorff dimension d-1-1/2. This is joint work with Jonathan Chaika.

**Carlos MATHEUS**

Title: Local conjugacy classes of Diophantine interval exchange transformations

Abstract: In this talk, we discuss an extension of the results of Marmi-Moussa-Yoccoz on local conjugacy classes of interval exchange transformations of restricted Roh type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.

This is a joint work with G. Forni and S. Marmi.

**Martin MOELLER**

Title: A Gothic modular form

Abstract: The gothic locus is an exceptional SL(2,R)-invariant submanifold of the moduli space of genus four flat surfaces that contains an infinite collection of primitive “gothic” Teichmüller curves.

We construct a modular form vanishing along these Teichmüller curves and show how this can be used for the computation of various topological invariants of gothic Teichmüller curves.

**Angel PARDO**

Title: A non-varying phenomenon: Hidden symmetries in hyperelliptic surfaces

Abstract: In this short talk I will exhibit a non-varying phenomenon for a refinement of the counting problem on translation surfaces in a special case. Namely, the (area) Siegel-Veech constant associated to the counting problem of cylinders passing through two marked regular Weierstrass points of a translation surface in a hyperelliptic connected component.

In the generic situation, the Weierstrass points of a translation surface in an hyperelliptic component are symmetric and their numbering is irrelevant to specialized counting. However, as we will see, even in non-generic situations there is some "hidden symmetry" that makes the counting symmetric.

An application to the wind-tree model will be given and, if time allows, I will present a family of counter-examples showing that the non-varying phenomenon for this refinement does not hold in general for other hyperelliptic loci.

**Rene RUHR**

Title: Effective Counting of Saddle Connections

Abstract: In 1998, Veech introduced the concept of Siegel measures,which can be utilised to obtain counting statements of holonomy vectors of saddle connection on a generic translation surface -under assumption of a spectral gap of the SL2 action and L^2-integrabity of what is now called Siegel-Veech transform. Eskin-Masur showed pointwise asymptotic without the full knowledge of these properties. Combining both works, one may also read of a polynomial error term (joint with Amos Nevo and Barak Weiss).

**Sasha SKRIPCHENKO**

Title: Rauzy gasket and its adventures in dynamics, topology and physics

Abstract: In 1981 P.Arnoux and J.-C.Yoccoz constructed their famous foliation on a surface of genus three with zero flux. Later it was shown that this example can be generalized, and in particular that there is an interesting fractal set of parameters that give rise to the foliations with similar properties. This fractal was named in honour of Gerard Rauzy.

In my talk I will briefly discuss how such a family of foliations appeared in different branches of mathematics (symbolic dynamics, Teichmuller dynamics, low-dimensional topology, geometric group theory) and even in theoretical physics (conductivity theory in monocrystals) and explain what do we know about this family from ergodic point of view.

The talk is mainly based on work in progress with Ivan Dynnikov and Pascal Hubert.

**Diaaheldin TAHA**

Title: A Gap Theorem (Steinhaus Conjecture) for General Interval Exchange Transformations via Zippered Rectangles.

Abstract: The Three Gap Theorem, also known as Steinhaus Conjecture, is a historic result asserting that for any $\alpha \in (0, 1)$ and any integer $N \geq 1$, the fractional parts of the sequence $0, \alpha, 2\alpha, \cdots, (N - 1)\alpha$ partition the unit interval into $N$ subintervals having at most three distinct lengths. We are presenting:

* a geometric interpretition in terms of sections to flows on surfaces that provides yet another proof of TGT

* a similar (and independent) approach by Alan Haynes, Jens Marklof, and Andreas Strombergsson to TGT that proves TGT and higher dimensional analogues (linear forms)

* a gap theorem for orbits of general interval exchanges maps with optimal bound

If there is time, we will touch on some applications of TGT that are of contemporary interest.

**Barak WEISS**

Title: Horocycle orbits in strata of translation surfaces II

Abstract: By work of Ratner, Margulis, Dani and many others, the horocycle flow on homogeneous spaces has strong measure theoretic and topological rigidity properties. By work of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohommadi, the action of SL(2,R) and the upper triangular subgroup of SL(2,R) on strata of translation surfaces have similar rigidity properties. We will describe how some of these results fail for the horocycle flow on strata of translation surfaces. In particular,

1) There exist horocycle orbit closures with fractional Hausdorff dimension.

2) There exist points which do not equidistribute under the horocycle flow with respect to any measure.

3) There exist points which equidistribute distribute under the horocycle flow to a measure, but they are not in the

topological support of that measure.

This is joint work with Jon Chaika and John Smillie.

**Alex WRIGHT**

Title: Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

Abstract: We will present joint work with Jeremy Kahn proving that any complete cusped hyperbolic three manifold contains many "nearly isometrically immersed" closed hyperbolic surfaces.

**Dimitri ZVONKINE**

Title: Plugging r=0 into the space of r-th roots

Abstract: Consider a complex curve C endowed with line bundle L whose r-th tensor power is the trivial or the canonical line bundle. The moduli space of pairs (C,L) is a ramified covering of the moduli space Mbar_{g,n} of algebraic curves. It carries several natural cohomology classes whose projections to Mbar_{g,n} turn out to be polynomial in r. We will state several theorems and conjectures that relate the constant term of this polynomial (obtained by plugging r=0) to the Poincaré dual cohomology classes of important geometric loci in Mbar_{g,n}. This is joint work with F. Janda, R. Pandharipande, and A. Pixton.