Moscow, June 5-9, 2017

Dynamics in moduli spaces is an intersection point for several different branches of mathematics. Dynamical systems, algebraic geometry, topology and geometric group theory offer various tools and approaches that can be useful to study this subject. Several important advances were recently obtained in this field.

We plan to bring together small group of experts and young scientists working in this area to discuss the last achievements, develop existing collaborations and start possible new ones.

Organizer:

Skoltech

Organizing Committee:

Alexandra Skripchenko *(HSE & Skoltech)*

Anton Zorich *(Inst. of Mathematics of Jussieu, Paris)*

Invited participants:

Mauro Artigiani *(Scuola Normale Superiore, Pisa) *

Alexander Bufetov *(Aix-Marseille Univ., Steklov Inst., Moscow, HSE, Moscow, IITP, Moscow & S.-Petersburg Univ.)*

Vincent Delecroix *(LaBRI, Bordeaux) *

Ivan Dynnikov *(Steklov Inst., Moscow) *

Charles Fougeron *(Ecole Normale Superieure, Paris)*

Elise Goujard *(Univ. of Paris-Sud)*

Pascal Hubert *(Aix-Marseille Univ.) *

Igor Krichever *(Skoltech, HSE & Columbia Univ.)*

Erwan Lanneau *(Grenoble Alpes Univ.) *

Barak Weiss *(Tel-Aviv Univ.) *

**Poster** [ download as pdf ]

June 1st // Thursday Math Faculty of HSE (Usacheva, 6) |
17:00 (room 212) | Vincent Delecroix |
Experimental mathematics with SageMath |

June 5th // Monday SkolTech |
11:00 – 12:00 | Anton Zorich |
Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes |

12:00 – 12:30 | coffee-break |
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12:30 – 13:30 | Igor Krichever |
Real-normalized differentials and their applications | |

13:30 – 14:30 | lunch |
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14:30 – 15:30 | Charles Fougeron |
Lyapunov exponents: from windtree models to hypergeometric equations | |

June 6th // Tuesday Math Faculty of HSE (Usacheva, 6) |
14:00 – 15:00 (room 110) | Alexander Bufetov |
Limit theorems for parabolic systems |

15:20 – 16:20 (room 108) | Pascal Hubert |
Windtree model | |

16:40 – 17:40 (room 108) | Barak Weiss |
New phenomena for the horocycle flow on the moduli space of surfaces | |

June 7th // Wednesday Math Faculty of HSE (Usacheva, 6) |
13:30 – 14:30 (room 306) | Erwan Lanneau |
Diffusion rate in non generic directions in the wind-tree model |

14:30 – 16:00 | lunch |
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16:00 – 17:00 (room 306) | Ivan Dynnikov |
Integrabiligy and chaos in Novikov’s problem on plane sections of 3-periodic surfaces | |

17:15 – 18:15 (room 211) | Vincent Delecroix |
Experimental mathematics with SageMath | |

19:00 | conference dinner |
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June 8th // Thursday SkolTech |
11:00 – 12:00 | Vincent Delecroix |
Checking minimality of interval exchange transformations |

12:00 – 12:30 | coffee-break |
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12:30 – 13:30 | Mauro Artigiani |
Hall ray for Fuchsian groups | |

13:30 – 14:30 | lunch |
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14:30 – 15:30 | Elise Goujard |
Counting square-tiled surfaces : quasimodularity of generating functions |

**Mauro Artigiani** // Hall ray for Fuchsian groups

*Abstract: The Lagrange spectrum is a classical object in Diophantine approximation on the real line. It can be also seen as the spectrum of asymptotic penetration of hyperbolic geodesics into the cusp of the modular surface. This interpretation yielded many generalizations of the Spectrum to non-compact, finite volume, negatively curved surfaces and higher dimensional manifolds.*

A remarkable property of the classical Spectrum is that it contains an infinite interval, called Hall ray. The presence of the Hall ray is a common feature of the generalizations of the Lagrange spectrum to higher dimensions. Together with L. Marchese and C. Ulcigrai, we show that the Lagrange spectrum of hyperbolic surfaces contains a Hall ray. Moreover, we show that the same result holds if we measure the excursion into the cusps with a proper function that is close in the Lipschitz norm to the hyperbolic height.

**Alexander Bufetov** // Limit theorems for parabolic systems

*While hyperbolic dynamical systems obey the Sinai Central Limit Theorem, parabolic systems do not: limit distributions have compact support, exist only along subsequences of times and essentially depend on the subsequence: for instance, for translation flows, the delta mass at zero typically arises as a limit distribution along a subsequence. The survey talk, aimed at undergraduate students, will aim to give a gentle introduction to limit theorems for parabolic dynamical systems. The main step is a construction of a special family of finitely-additive measures on the arcs of trajectories of our flows.*

Specific examples include translation flows, horocycle flows (joint work with Giovanni Forni) and tiling flows (joint work with Boris Solomyak)

**Vincent Delecroix** // Checking minimality of interval exchange transformations

*Abstract: In 1975, M. Keane provided a condition that implies minimality of interval exchange transformations (iet) either called “i.d.o.c” or “Keane condition”. However, checking this condition involves countably many linear equalities and inequalities. The decidability (in the algorithmic sense) of Keane condition is a challenging problem.*

In 1988, M. Boshernitzan showed that in the case of “rank 2″ iet that minimality is decidable. Following M. Boshernitzan, we will explain how one can generalize his algorithm to nonzero SAF invariant iet. Finally, we will present experimental data made with B. Weiss around the Dynnikov-Skripshenko conjecture and minimality of SAF zero iet.

**Vincent Delecroix** // Experimental mathematics with SageMath

*Abstract: SageMath is a general purpose open source math software started in 2005. It is free to download, install and can even be used online. In this mini-course I will give*

* a short overview of the SageMath project

* a tutorial on how to use it

* some concrete problems that can efficiently be solved with it .

**Ivan Dynnikov** // Integrabiligy and chaos in Novikov’s problem on plane sections of 3-periodic surfaces

*Abstract: Novikov’s problem provides an application of the theory of singular mesured foliations on surfaces to physics, namely, the conductivity theory. The question is to describe possible asymptotic behavior of unbounded trajectories that arise as intersections of a 3-periodic surface (which is the Fermi surface of a metal) with a plane (which is defined by the magnetic field). This problem gives rise to a very specific family of foliations defined by a closed 1-form on a closed surface. In particular, minimality and unique ergodicity are not typical properties for this family. I will review old and more recent results on the subject.*

**Charles Fougeron** // Lyapunov exponents: from windtree models to hypergeometric equations

*Abstract: Lyapunov exponents and their Oseledets flag decomposition are a very useful tool for describing dynamical systems. I will explain some results which relate the diffusion rate of windtree models with Lyapunov exponents. These results rely on understanding closed SL(2,R) invariant loci in strata of moduli space of translation surfaces using recent work of Eskin-Mirzakhani and Chaika-Eskin.*

Nonetheless there are almost no explicit formula for these exponents. The only known result we can use in this setting was discovered in the 90’s by M. Kontsevich, and relates the sum of Lyapunov exponents to the degree of some holomorphic subbundle.

Recently, a similar result was observed on higher weight variation of Hodge structure (which decomposition has more flags), A. Eskin, M. Kontsevich, M. M”oller and A. Zorich showed indeed a lower bound of their associated Lyapunov exponents given by the parabolic degrees of their variation of Hodge structure.

I will present this result on the example of variation of Hodge structure yielded by hypergeometric equations of arbitrary order. Starting with the computation of their degrees, and presenting some computer experiments. This will motivate questions and conjectures about the equality case

**Elise Goujard** // Counting square-tiled surfaces : quasimodularity of generating functions

*Abstract: In this talk I will focus on arithmetic questions arising in the study of dynamics in polygonal billiards. These questions appear naturally in the evaluation of volumes of moduli spaces of flat surfaces and in the computation of Siegel-Veech constants. In particular I will present a joint work with M. M”oller concerning the quasimodularity of generating functions for counting square-tiled surfaces.*

**Pascal Hubert** // Windtree model

*Abstract: The windtree model is a model of Z*^{2}periodic billiard, identical rectangular obstacles are periodically located along the Z^{2}lattice. It is a concrete example on which the great results obtained in Teichmueller dynamics during the last 15 years can be applied to solve very natural questions from dynamical origin (recurrence, ergodicity, periodic orbits, diffusion). I will give general results about the windtree model, Z^{d}periodic translation surfaces and I will quote results by Angel Pardo about the counting of periodic orbits.

This talk will be elementary and I will address many open questions. It will present the background for the talk by Lanneau, nevertheless both talks will be essentially self contained.

**Igor Krichever** // Real-normalized differentials and their applications

*Abstract: A general notion of real normalized differentials is central to the algebro-geometric spectral theory of quasi-periodic linear operators and to the Whitham perturbation theory of soliton equations. Relatively recently, real normalized differentials have become instrumental to the study of geometry of moduli spaces of algebraic curves with punctures. In the talk basic constructions of this theory will be presented and certain connections to the theory of Teichmuller dynamic will be established.*

**Erwan Lanneau** // 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 wind-tree model.

**Barak Weiss** // New phenomena for the horocycle flow on the moduli space of surfaces

*Abstract: The recent breakthrough of Eskin, Mirzakhani and Mohammadi, which revolutionized the study of translation surfaces, concerns the orbit-closures and invariant measures for the action of SL(2,R) on the moduli space of translation surfaces. This result is an analogue of results of Ratner in the setting of homogeneous dynamics. If the analogy to homogeneous dynamics were a perfect one, there would be a corresponding result for the horocycle flow. However, in joint work with Jon Chaika and John Smillie, we show that the horocycle flow on the moduli space violates some of the conclusions of Ratner’s theorem. For example, in genus 2, we construct an orbit which is equidistributed for some measure, whose support is strictly contained in the closure of the orbit.*

**Anton Zorich** // Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes (joint work with V. Delecroix, E. Goujard, P. Zograf)

*Abstract: We show how recent equidistribution results allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.*

We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero.

We present a very recently discovered bridge between flat and hyperbolic worlds giving a formula for the Masur-Veech volume of the moduli space of quadratic differentials in the spirit of Mirzakhani-Weil-Peterson volume of the moduli space of curves.

Finally we present several ambitious conjectures.