# Past meetings

## slides

Abstract: We will discuss recent work showing that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic growth like $c R^2$ where the constant $c$ depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel-Veech transform is in $L^2$. This is joint work with Jayadev Athreya and Howard Masur.

## Erwan Lanneau (Institut Fourier)Title: Pseudo-Anosov stretch factors

Abstract: Pseudo-Anosov mapping classes first appeared in Thurston's work in connection to classification of surface homeomorphisms. Nowadays, their study is a theory by itself combining Teichmüller theory, dynamics, flat geometry and number theory. An important aspect of this theory emerged with Fried's work and concerns the study of the stretch factors. They reflect the geometry of the moduli spaces (e.g. by the lengths spectrum for the Teichmüller metric) and the fine properties of the dynamics of the map (e.g. by the Ruelle spectrum). I will review several old and new results on pseudo-Anosov stretch factors.

## Jenya Sapir (Binghamton University)Title: A projection from geodesic currents to Teichmuller space

Abstract: Given a genus g surface S, we consider the space of projective geodesic currents on S. This space contains many objects of interest in low dimensional topology, such as the set of all closed curves on S up to homotopy, the set of all marked, negatively curved metrics on S, as well as some higher Teichmuller spaces. We show that there is a mapping class group invariant, length minimizing projection from the space of filling projective currents onto Teichmuller space, and that this projection is continuous and proper. This is joint work with Sebastian Hensel.

## Alexandra Skripchenko (HSE University)Title: Real-normalized differentials with a single order 2 pole: the first steps

Abstract: A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. This notion was introduced by I. Krichever in connection with the study of geometry of moduli spaces.

Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. In our joint work with Sergei Lando and Igor Krichever we propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze the corresponding absolute period foliation.

## Adrien Sauvaget (CNRS)Title: Moduli of large pluricanonical divisors

Abstract: We will study moduli spaces of k-canonical divisors. A standard invariant of these spaces is their (Masur-Veech) volume which can be computed by means of intersection theory. Considering the large k behavior of these volumes one may compute volumes of moduli spaces of flat surfaces (by "approximation" of these spaces). I’ll also explain how different choices of limit should allow to compute the Weil-Petersson volumes of moduli spaces of hyperbolic surfaces.

## Pedram Safaee (Universität Zürich)Title: Quantitative Weak Mixing For Interval Exchange Transformations

Abstract: An interval exchange transformation (IET) is an orientation preserving piecewise isometry of the unit interval. These transformations are low complexity systems that exhibit interesting spectral properties; They are never mixing, typically uniquely ergodic, typically rigid, and typically weakly mixing. Weak mixing is equivalent to having the Cesaro averages of correlations tend to zero. In this talk, we will focus on the rate of decay of the Cesaro averages of correlations for sufficiently regular observables for typical IETs. We show that a (rather unexpected) dichotomy holds for this decay depending on whether the IET is of rotation class or not. In the former case, we provide logarithmic lower and upper bounds for the decay of Cesaro averages whereas we provide polynomial upper bounds in the latter case. This is joint work with Artur Avila and Giovanni Forni.

# Maxim Kontsevich (IHÉS)

slides

### Title: Wall-crossing for abelian differentials

Abstract: For an abelian differential on a complex curve one can count saddle connections in all possible relative homology classes. These numbers jump when one crosses a wall in the moduli space of abelian differentials. I will show that the jumping formula is a particular case of the general wall-crossing formalism of Y.Soibelman and myself. The corresponding graded Lie algebra is the algebra of matrices over the ring of Laurent polynomials in several variables. The wall-crossing structure is explicitly calculable, and is determined by a finite collection of invertible matrices over the field of rational functions. The whole story generalizes from curves to higher-dimensional complex algebraic varieties.

# Nalini Anantharaman (IRMA)

### Title: The bottom of the spectrum of a random hyperbolic surface

Abstract: In an ongoing project with Laura Monk, we are trying a strategy to prove that with high probability (and in the limit when the volume goes to infinity) there are no small eigenvalues. A large part of the talk will be dedicated to giving motivations, and describing the strategy to solve a similar question for random regular graphs (due to Friedman and Bordenave). We will then describe what we have done so far for surfaces.

# Samuel Grushevsky (Stony Brook University)

slides

### Title: Equations for affine invariant manifolds, via degeneration

Abstract: Studying the closures of the orbits of the $SL(2,\RR)$ action on the strata of holomorphic differentials is a central question in Teichmueller dynamics. By the results of Eskin-Mirzakhani-Mohammadi, locally in period coordinates these orbit closures are given by linear equations. We use the compactification of the strata given by the moduli space of multi-scale differentials to restrict the kinds of linear equations that can thus appear, by using a mix of algebraic and dynamic techniques, and in particular obtaining a new proof of Wright's cylinder deformation theorem as a byproduct of our study. Based on joint work with F. Benirschke and B. Dozier.

# Sam Payne (University of Texas)

### Title:The moduli space of tropical curves and top weight cohomology of M_g

Abstract: I will discuss a natural proper and surjective map from the moduli space of Riemann surfaces of genus g to the moduli space of tropical curves of genus g and its applications. In joint work with Chan and Galatius, we show that the pullback on compactly supported cohomology is an injection and that the compactly supported cohomology of the tropical moduli space is isomorphic to the cohomology of Kontsevich’s commutative graph complex. Combining this with deep results of Brown and Willwacher from Grothendieck-Teichmüller theory, we deduce that the dimension of H^{4g-6}(M_g, Q) grows exponentially with g. This growth was unexpected and disproves conjectures of Church-Farb-Putman and Kontsevich.

# Bertrand Deroin (CNRS)

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### Title: Irreducible lattices in semi-simple Lie groups of rank at least 2 are not left-orderable

Abstract: I'll report on the problem of the left orderability of lattices in semi-simple Lie groups, and give some insight of our joint proof with Sebastian Hurtado that in rank at least two an irreducible lattice is not left-orderable.

# Umberto Zannier (Scuola Normale Superiore di Pisa)

slides

### Title: Torsion values of sections, elliptical billiards and diophantine problems in dynamics.

Abstract: We shall consider sections of (products of) elliptic schemes, and their "torsion values". For instance, what can be said of the complex numbers b for which (2, \sqrt{2(2-b)}) is torsion on y^2=x(x-1)(x-b)? In particular, we shall recall results of "Manin-Mumford type" and illustrate some applications to elliptical billiards. Finally, we shall frame these issues as special cases of a general question in arithmetic dynamics, which can be treated with different methods, depending on the context. (Most results refer to work with Pietro Corvaja and David Masser.)

# Corinna Ulcigrai (Universität Zürich)

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### Title: Rigidity of foliations in genus two and renormalization of generalized IETs

Abstract: It follows from a celebrated result by Michel Herman on circle diffeomorphisms (later improved by Yoccoz) that minimal smooth orientable foliations on surfaces of genus one, under a full measure arithmetic condition on, are geometrically rigid, namely: if they are topologically conjugated to a linear flow, then they are actually differentiably conjugated to it.

In very recent joint work with Selim Ghazouani, we prove a generalization of this result to genus two, in particular by showing that smooth, orientable foliations with non-degenerate (Morse) singularities on surfaces of genus two, under a full measure arithmetic condition, are geometrically rigid.

At the level of Poincare maps, this can be translated in a statement about generalized interval exchange transformations (or GIETs, for short) and answers a conjecture by Marmi, Moussa and Yoccoz in genus two.

The result is based on the study of the dynamics of a renormalization operator on the space of GIETs (which is a suitable acceleration of Rauzy-Veech induction). We prove in particular a dynamical dichotomy for orbits under renormalization which is valid in any genus.

In the talk we will motivate and explain the result, by giving a brief survey of some of the key results in the theory of circle diffeos and in the study of GIETs and then an brief overview the main steps of the proof.

# Gabriele Mondello (Università di Roma)

slides

### Title: On spherical surfaces of genus 1 with 1 conical point

Abstract: A spherical metric on a surface is a metric of constant curvature 1, which thus makes the surface locally isometric to S^2. Such a metric has a conical point x of angle 2\pi\theta if its area element vanishes of order 2(\theta-1) at x. If the conformal class is prescribed, a spherical metric can be viewed as a solution of a suitable singular Liouville equation. If the conformal class is not prescribed, isotopy classes of spherical metrics can be considered as flat (SO(3,R),S^2)-structure, and so their deformation space has a natural finite-dimensional real-analytic structure. Additionally, the moduli space of spherical surfaces of genus g with n conical points comes endowed with a natural forgetful map to the moduli space of Riemann surfaces of genus g with n marked points.

I will begin by giving an overview of what is known about the topology of the moduli space of spherical surfaces and the above mentioned forgetful map.

I will then focus on the case of genus 1 with 1 conical point (joint work with Eremenko-Panov).

slides

# Charles Fougeron (Université de Paris)

### Title: A cyclotomic family of thin groups

Abstract: Thin matrix groups are a delicate object: they are by definition a sparse subgroup of a lattice but Zariski-dense in the ambient Lie group. Despite much recent work, a lot remains to be understood about these groups and explicit examples are still rare.

In this talk, we will focus on matrix monodromy groups associated to hypergeometric differential equations. It was noticed a few years ago by Eskin-Kontsevich-Möller-Zorich that in a family of 14 of these matrix groups (associated to moduli spaces of Calabi-Yau varieties) the 7 cases that were known to be thin coincide with cases that numerically satisfied an equality between their Lyapunov exponents and some algebraic invariant.

By exploring numerically the Lyapunov exponents of these differential equations, we found candidates for an infinite family of thin groups in Sp4(R). After explaining the path to these numerical observations, I will explain how we proved their thinness. (j.w. Simion Filip)

# Pedram Safaee (Universität Zürich)

### Title: Quantitative Weak Mixing For Interval Exchange Transformations

Abstract: An interval exchange transformation (IET) is an orientation preserving piecewise isometry of the interval $[0,1]$. These transformations are low complexity systems that exhibit interesting spectral properties; They are never mixing, typically uniquely ergodic, typically rigid, and typically weakly mixing. Weak mixing is equivalent to having the Cesaro averages of correlations tend to zero. In this talk, we will focus on the decay rate of the Cesaro averages of correlations for sufficiently regular observables for typical IETs. We show that a (rather unexpected) dichotomy holds for this decay depending on whether the IET is of rotation class or not. In the former case, we provide logarithmic lower and upper bounds for the decay of Cesaro averages whereas we provide polynomial upper bounds in the latter case. This is joint work with Artur Avila and Giovanni Forni.

# Osama Khalil (University of Utah)

### Title: On the Mozes-Shah phenomenon for horocycle flows on moduli spaces

Abstract: The Mozes-Shah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) SL(2,R). We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.

# Dawei Chen (Boston College)

### Title: Connected components of the strata of k-differentials

Abstract: k-differentials on Riemann surfaces correspond to (1/k)-translation structures. The moduli space of k-differentials can be stratified according to the multiplicities of zeros and poles of k-differentials. While these strata are smooth, some of them can be disconnected. In this talk I will review known results and open problems regarding the classification of their connected components, with a focus on geometric structures that can help distinguish different components. This is joint work with Quentin Gendron.

# Yunhui Wu (Tsinghua University, Beijing)

slides

### Title: Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$

Abstract: Let M_g be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus g goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$. This is a joint work with Yuhao Xue.

# Viveka Erlandsson (University of Bristol)

### Title: Mirzakhani’s curve counting theorem

Abstract: In her thesis, Mirzakhani established the asymptotic behavior of the number of simple closed geodesics of a given type in a hyperbolic surface. Here we say that two geodesics are of the same type if they differ by a homeomorphism. In this talk I will discuss this theorem, the extension to geodesics which are not simple, and some applications.

# Howard Masur (UChicago)

### Title: Counting finite order elements in the mapping class group

Abstract: Let S be a closed surface of genus g at least 2 and Mod(S) the mapping class group. Mod(S) acts by isometries on the Teichmuller space of S with respect to the Teichmuller metric. The lattice counting problem was considered in a paper by Athreya, Bufetov, Eskin, Mirzakhani. They showed that for any pair of points x and y, the number of orbit points of y under the action of Mod(S) that lie in a ball of radius R about x has an asymptotic growth rate of the form C exp((6g-6)R), as R goes to infinity, for a constant C. In this talk I will discuss estimates for the number of finite order elements in this lattice counting problem. This is joint work with Spencer Dowdall.

# Olga Paris-Romaskevich CNRS, I2M (Institut de Mathématiques de Marseille)

slides

### Title: Tiling billiards and Novikov’s problem: from dynamics to topology

Abstract: Imagine a ray of light moving through a tiling of a plane (by polygons). Each time it crosses an edge, it refracts with a refraction coefficient equal to $-1$.

Such a dynamical system is called a tiling billiard.

I will give an elementary introduction to tiling billiards in two families of tilings :

a. periodic tilings by triangles

b. periodic tilings by cyclic quadrilaterals.

The study of the dynamics of these two families happens to be equivalent to the topology of plane sections of symmetric genus 3 subsurfaces of the 3-torus (the so-called Novikov’s problem).

I will explain this and other elementary geometric constructions related to tiling billiards.

Homework: Watch a following 5-MIN movie (here is a link to Youtube) by a wonderful mathematician and animator Ofir David.

slides

# Serge Cantat (IRMAR)

### Title: Stationary measures on real projective surface

Abstract: Consider a real projective surface $X(\R)$, and a group $\Gamma$ acting by algebraic diffeomorphisms on $X(\R)$. If $\nu$ is a probability measure on $\Gamma$, one can randomly and independently choose elements $f_j$ in $\Gamma$ and look at the random orbits $x$, $f_1(x)$, f_2(f_1(x))$, … How do these orbits distribute on the surface ? This is directly related to the classification of stationary measures on$X(\R)\$. I will describe recent results on this problem, all obtained in collaboration with Romain Dujardin. The main ingredients will be ergodic theory, notably the work of Brown and Rodriguez-Hertz, algebraic geometry, and complex analysis. Concrete geometric examples will be given.

# Chaya Norton (UMich)

### Title: Obtuse Veech Triangles

Abstract: The question of which obtuse triangles ufold to Veech surfaces has been open since Kenyon and Smillie's results on acute and right triangles. There are two known infinite families of obtuse Veech triangles due to Veech and Ward. More recently Hooper showed that the unfolding of the sporadic example (pi/12, pi/3, 7*pi/12) generates a Teichmuller curve, and he conjectures that these are all the obtuse Veech triangles. We prove this conjecture when the largest angle is at least 135 degrees. Our method relies on a criterion of Mirzakhani and Wright which builds on work of Moeller and McMullen studying the variation of the period matrix along the GL(2,R) action. This is joint work with Anne Larsen and Bradley Zykoski completed during the 2020 University of Michigan REU.

## 23 November 2020Time: 9am PST / 12pm EST / 6pm CET

slides

### Title: Quantitative weakly mixing of flows over Salem type substitutions

Abstract: Suspension flows over Vershik automorphisms provide a powerful symbolic frame for study linear flows over translation surfaces. The simplest case is the periodic one, which leads us to substitutions. Spectral properties depend strongly on the algebraic nature of the Perron eigenvalue of the adjacency matrix of the substitution, as shown in the work of Bufetov and Solomyak. In this talk I will consider the "border case" in which this eigenvalue is a Salem number and I will show a modulus of continuity for spectral measures in a family of algebraic points.

slides

# Scott Mullane (Frankfurt)

### Title: Strata of exact differentials and the birational geometry of Hurwitz spaces

Abstract: The strata of exact differentials are obtained from Hurwitz spaces of covers of the rational line with specified branching profiles and form linear manifolds inside the strata of meromorphic differentials. Despite the utility of Hurwitz spaces in the study of a number of the birational aspects of the moduli space of curves, many open questions on Hurwitz spaces persist. I'll show how the perspective of the strata of exact differentials can be used to prove, that as conjectured, the rational Picard group of the moduli space of simply branched degree d covers of the rational line by smooth genus g curves is trivial for d>g-1.

Further, this perspective yields results on open questions on the irreducibility of non-simple Hurwitz spaces and has applications to the birational geometry of moduli spaces of pointed rational curves.

# Frederik Benirschke (Stony Brook)

slides

### Title: The boundary of orbit closures

Abstract: Moduli spaces of translation surfaces carry a natural GL(2,R)-action by acting linearly on the periods of the translation surface.

Recent breakthroughs by Eskin, Mirazakhani, Mohammadi and Filip, which extend results of McMullen in genus 2, show that orbit closures for the GL(2,R)-action are surprisingly well behaved: Orbit closures are algebraic varieties that are locally defined by linear equations among periods. Orbit closures are never compact and it is natural to search for "nice" compactifications. One simple way of compactifying orbit closures is by taking the closure inside the moduli space of multi-scale differentials, which was constructed recently by Bainbridge-Chen-Gendron-Grushevsky-Möller. Our main result is a description of the boundary of an orbit closure inside the moduli space of multi-scale differentials. In particular the boundary is again given by linear equations among periods. Time permitting, we explain how our description of the boundary can be used to extend Wrights cylinder deformation theorem to the case of meromorphic strata, which is partially joint work with Benjamin Dozier and Samuel Grushevsky.

# Yilin Wang (MIT)

### Title: SLE, energy duality, and foliations by Weil-Petersson quasicircles

Abstract: Schramm-Loewner evolution (SLE) is a one-parameter family of random simple planar curve. It first arises as interfaces in scaling limits of 2D statistical mechanics lattice models which exhibit conformal invariance. The small-parameter asymptotic behaviors give rise to the Loewner energy for Jordan curves, which is finite if and only if the curve is a Weil-Petersson quasicircle, and is moreover a Kahler potential on the Weil-Petersson Teichmuller space. I will survey the link between SLE and Weil-Petersson quasicircles, then show the large-parameter asymptotic behaviors of SLE giving rise to Loewner-Kufarev energy, provides a further duality via foliations of the Riemann sphere by Weil-Petersson quasicircles.

# Laura Monk (IRMA)

slides

### Title: Geometry and spectrum of random hyperbolic surfaces

Abstract: The aim of this talk is to describe the geometry and spectrum of most random hyperbolic surfaces, picked with the Weil-Petersson probability measure.

In this model, one can get a good understanding of the geometry of a typical surface: Cheeger constant, diameter (Mirzakhani), injectivity radius, number of short closed geodesics (Mirzakhani-Petri), length of the shortest non-simple closed geodesic, improved collar theorem (joint work with Joe Thomas), Benjamini-Schramm convergence.

I will explain how these geometric properties, together with the Selberg trace formula, lead to precise estimates on the distribution of the eigenvalues of the Laplacian on a typical surface.

# Brian Chung (UChicago)

slides

### Title: Stationary measure and orbit closure classification for random walks on surfaces

Abstract: We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces.

We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the Out(F2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

# Karl Winsor (Harvard)

slides

### Title: Navigating absolute period leaves and the Arnoux-Yoccoz surface in genus 3

Abstract: The moduli space of holomorphic 1-forms on genus g Riemann surfaces has a foliation whose leaves consist of 1-forms with locally constant absolute periods. Individual leaves have a natural flat structure, recording changes in relative periods along paths between the zeros. In genus 2, a typical leaf is topologically a disk, after being completed. One can also restrict this foliation to strata of 1-forms with given zero orders, and we will mainly focus on strata in genus greater than 2. We will describe closed geodesics on these leaves, give an example of a leaf with infinite genus, and show how to upgrade this to a statement about a typical leaf in the ambient stratum component.

# Junho Peter Whang (MIT)

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### Title: Integral points on moduli of local systems

Abstract: Moduli spaces for special linear rank two local systems on topological surfaces are basic objects in geometry. The study of integer points on these algebraic varieties can be traced back to 1880 work of Markoff, in the case where the surface is the once-punctured torus. In the first part of the talk, we describe a structure theorem for the integral points on these moduli spaces for general surfaces, proved using mapping class group dynamics and differential geometric tools. In the second part (based on joint work with Fan), we discuss exceptional isomorphisms between these varieties and moduli spaces of points on (algebraic) 3-spheres. Using this connection and the previous structure theorem for the twice-punctured torus, we can deduce a Diophantine finiteness result for integral Stokes matrices of rank 4.

# Gaëtan Borot (Humboldt-Universität zu Berlin)

slides

### Title: Geometry of combinatorial moduli spaces and multicurve counts

Abstract: The Teichmuller space of bordered surfaces can be described via metric ribbon graphs, leading to a natural geometry (the symplectic form introduced by Kontsevich in his proof of Witten's conjecture). I will show that many tools of hyperbolic geometry can be adapted to this combinatorial geometry: there are Fenchel-Nielsen coordinates that are Darboux, Mirzakhani-McShane type identity, integration formulas, recursions for volume and statistics of multicurves, etc. Besides, combinatorial geometry is hyperbolic geometry for large boundary lengths converges to combinatorial geometry: we extend some results of Mondello in this direction, but also stress some non-uniformity than manifests itself in a different integrability behavior of the Thurston measure of unit balls wrt combinatorial length in the space of measured foliations than the one found in the hyperbolic setting by Arana-Herrera and Athreya.

# Bram Petri (IMJ-PRG)

### Title: The minimal diameter of a hyperbolic surface.

Abstract: The main question in this talk is what the "most connected" closed hyperbolic surface of a given genus is. There are multiple measures of the connectivity of a hyperbolic surface, but as the title suggests, we will focus on their diameter. I will explain how random constructions of hyperbolic surfaces help with this question. This is joint work with Thomas Budzinski and Nicolas Curien.

# Paul Apisa (Yale)

### Title: Reconstructing an orbit closure from its boundary, holomorphic retracts of Teichmuller space, and new Eierlegende-Wollmilchsau-like orbit closures!

Abstract: Work of McMullen in genus two and Eskin, Mirzakhani, Mohammadi, and Filip in general established that the GL(2, R) orbit closure of any translation surface is an affine invariant subvariety (AIS). Myriad questions abound about AIS. We focus on the following - how does the boundary of an AIS constrain the AIS?

We will begin by explaining how different boundary components of an AIS can be accessed by cylinder degenerations. While considering one degeneration is often insufficient to completely determine an AIS, we will show that one can often identify the AIS from two degenerations that form what will be called a diamond. These results are key to work in progress showing that any sufficiently large orbit closure of a genus g translation surface is a locus of covers (sufficiently large means that the rank is greater than g/2). To explain the connection, we take a seeming detour.

The Eierlegende-Wollmilchsau square-tiled surface has the property that every cylinder is parallel to exactly one other cylinder, which is isometric to it. In this talk, we will generalize this property to AIS beyond those generated by square-tiled surfaces, saying, roughly, that an AIS on which every cylinder on every surface has an isometric “twin” is called geminal. Loci of double covers are examples of geminal AIS. Less trivially, every sufficiently large AIS (with rel zero) is geminal. Moreover, work of Markovic and Gekhtman showed that if M is the collection of points, in a stratum of quadratic differentials, whose corresponding Teichmuller disk is a holomorphic retract of Teichmuller space, then the locus of holonomy double covers of elements of M is geminal.

Using the “reconstructing an AIS from its boundary” technique described above, we will show that geminal AIS are loci of covers. This result has implications for the complex geometry of Teichmuller space and is a key step in the aforementioned work showing that sufficiently large AIS are loci of covers. Finally, we will sketch the construction of new geminal AIS. These examples negatively resolve two questions of Mirzakhani and Wright and illustrate new behavior in the finite blocking problem. This work is joint with Alex Wright.

## Title: Integral PL actions from birational geometry

Abstract: Theory of flat surfaces provides a series of interesting actions of SL2(Z) on finite sets (isomorphism classes of square-tiled surfaces with a given integer area). I will talk on a different construction, with the origin in mirror symmetry/tropical geometry, producing somewhat similar actions. For example, in the case of K3-surfaces, an arithmetic subgroup of SO(1,18) acts on S2 by Z-piecewise-linear transformations, inducing a tower of non-trivial finite actions. I will describe a general construction, and give numerous examples which could be interesting from the dynamical point of view.

## Title: The number of components of a multicurve in large genus

Abstract: A multicurve on a closed surface S of genus g >= 2 is a homotopy class of a disjoint collection of simple closed curves on S. A hyperbolic metric on S allows to measure the length of a multicurve. We study the number of components of a multicurve taken at random among all multicurves of length at most L on S. We then let L tend to infinity and talk about a random multicurve on S. M. Mirzakhani proved that the number of components of a random multicurve on S only depends on the topology of S and not on the specific hyperbolic metric. It hence makes sense to talk about the number of components of a random multicurve of genus g. Furthermore M. Mirzakhani provided explicit formulas for this distribution involving the Kontsevich-Witten correlators. Thanks to the recent work of A. Aggarwal on the asymptotics of these correlators we describe its behavior as the genus g tend to infinity. We show that it asymptotically behaves as the number of cycles of a random permutation in Sym_{3g-3} taken with respect to a very explicit probability distribution.

The number of components of a random multicurve of genus g coincide with the number of cylinders of a random square-tiled surface in genus g. Hence our work equivalently provides results on the geometry of random square-tiled surfaces.

## Title: On weak mixing for translation flows and billiards in polygons

Abstract: How chaotic can a polygonal billiard be? We will present a recent joint result with Jon Chaika that the set of weak mixing (non-rational) polygons is dense (hence a dense G_delta). Along the way we will discuss results and open questions on weak mixing and effective weak mixing of translation flows and interval exchange transformations.

## Title: Framed mapping class groups and strata of abelian differentials

Abstract: Strata of abelian differentials have long been of interest for their dynamical and algebro-geometric properties, but relatively little is understood about their topology. I will describe a project aimed at understanding the (orbifold) fundamental groups of non-hyperelliptic stratum components. The centerpiece of this is the monodromy representation valued in the mapping class group of the surface relative to the zeroes of the differential. For g \ge 5, we give a complete description of this as the stabilizer of the framing of the (punctured) surface arising from the flat structure associated to the differential. This is joint work with Aaron Calderon.

## Amol Aggarwal (Harvard University)Title: Large Genus Asymptotics for Intersection Numbers and Strata Volumes

Abstract: Correlators, or intersection numbers between psi-classes on the moduli space of stable curves, are fundamental invariants ubiquitous in mathematical physics, algebraic geometry, geometric topology, and dynamical systems. In this talk, we analyze the large genus asymptotics for these correlators using a comparison between the recursive relations (Virasoro constraints) that uniquely determine them with the jump probabilities of a certain asymmetric simple random walk. By combining this result with a combinatorial analysis of recently proven formulas of Delecroix-Goujard-Zograf-Zorich, we further provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal strata in the moduli space of quadratic differentials.

## Ivan Smith (Cambridge University)Title: Symplectic mapping class groups and flat surfaces

Abstract: I will try to explain why one particular approach to studying the mapping class groups of higher-dimensional symplectic manifolds leads to thinking about flat surfaces and their cousins, and some of the open questions that arise in that context. The talk will try to be reasonably self-contained, but will therefore necessarily be somewhat impressionistic.

## 14 May 2020

### 9:00 am (PDT) / 12:00 pm (EDT) / 6:00 pm (MESZ)

Slides of the talk can be found here.

## 7 May 2020

### 9:00 am (PDT) / 12:00 pm (EDT) / 6:00 pm (MESZ)

Slides of the talk can be found here.

## Aaron Calderon (Yale University)Shear-shape coordinates for Teichmüller space and applications to flat and hyperbolic geometry

Abstract: There is a deep yet mysterious connection between the hyperbolic and singular flat geometry of Riemann surfaces. Using Bonahon and Thurston’s “shear coordinates” for maximal laminations, Mirzakhani related the earthquake and horocycle flows on Teichmüller space, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. In this talk, I will describe joint work (in progress) with James Farre in which we construct new “shear-shape coordinates” for Teichmüller space adapted to any lamination. Using these coordinates, we extend Mirzakhani’s conjugacy to strata of quadratic differentials as well as produce new examples of geodesics for the Lipschitz (asymmetric) metric with given stretch locus. These coordinates also yield information about the global structure of certain subloci in both Teichmüller space and its cotangent bundle of quadratic differentials.

## Matteo Costantini (Universität Bonn)

The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials

Abstract: Recently, Bainbridge-Chen-Gendron-Grushevsky-Möller defined the moduli space of multi scaled differentials, which is a compactification of the moduli spaces of abelian differentials with very similar properties as the Deligne-Mumford compactification of the moduli space of curves. During the talk I will explain how it is possible to develop intersection theory on this moduli space and how to use it, together with a twisted Euler sequence, in order to compute its Chern classes. As a special case, via Gauss-Bonnet, we compute a formula for the Euler characteristic of the moduli spaces of abelian homolorphic and meromorphic differentials and obtain values in small genera. This is based on a joint work with Martin Möller and Jonathan Zachhuber.

Slides of the talk can be found here.

## Kasra Rafi (University of Toronto)

### Absolutely continuous stationary measures for the mapping class group.

Abstract: We prove a version of a Theorem of Furstenberg in the setting of Mapping class groups. Thurston measure defines a smooth measure class on PML. For every measure \nu in this measure class, we produce a measure \mu with finite first moment on the mapping class group such that \nu is the unique \mu-stationary measure. In particular, this gives an coding-free proof of the already known result that the Lyapunov spectrum of Kontsevich-Zorich cocycle on the principal stratum of quadratic differentials is simple.