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# Jayadev Athreya (Seattle) and Elise Goujard (Orsay)

## Schedule of mini-courses:

All mini-course talks will be held in the Maxwell Lecture Hall in the Hamilton Building, TCD (see campus map).

 Time Speaker Talk Tuesday 9.00-10.30 Jayadev Athreya (Seattle) Volumes of moduli spaces of meromorphic quadratic differentials I 10.30-11.00 COFFEE 11.00-12.30 Elise Goujard (Orsay) Volumes of moduli spaces of meromorphic quadratic differentials II 12.30-2.00 LUNCH 2.00-3.00 Benjamin Dozier (Stanford) Equidistribution of saddle connections on translation surfaces 3.00-3.30 DISCUSSION 3.30-4.30 David Aulicino (CUNY) Computing the Lyapunov Exponents of a Particular Orbit Closure 5:00 BEERS AT PAVILION BAR Wednesday 9.00-10.30 Elise Goujard (Orsay) Volumes of moduli spaces of meromorphic quadratic differentials III 10.30-11.00 COFFEE 11.00-12.30 Jayadev Athreya (Seattle) Volumes of moduli spaces of meromorphic quadratic differentials IV 12.30-2.00 LUNCH 2.00-3.00 Nicholas Vlamis (Michigan) Identities on hyperbolic manifolds 3.00-3.30 DISCUSSION 3.30-4.30 Rene Ruhr (Tel Aviv) Counting saddle connections on translation surfaces

# Geometry and Dynamics of Moduli Spaces

## Schedule of lectures:

All lectures will be held in the Maxwell Lecture Hall in the Hamilton Building, TCD (see campus map).

 Time Speaker Talk Thursday 9.30-10.20 Anton  Zorich (Jussieu) Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes 10.30-11.00 COFFEE 11:00-11:50 Matt Bainbridge (Indiana) Smooth compactifications of strata 12.00-2:00 LUNCH 2.00-2.50 Sarah Koch (Michigan) On the connectivity of Milnor curves in moduli space 3:00-4.00 DISCUSSION 4.00-4.50 Viveka Erlandsson (Aalto) Counting curves on surfaces Friday 9.30-10.20 Ronen Mukamel (Rice) Teichmuller curves in positive characteristic 10.30-11.00 COFFEE 11:00-11:50 Paul Apisa (Chicago) Holomorphic Sections, Teichmuller Dynamics, and Billiards - From Finiteness to Classification 12.00-2:00 LUNCH 2.00-2.50 Kasra Rafi (Toronto) Updates on the limit set of a Teichmüller geodesic in the Thurston boundary 3.00-4.00 DISCUSSION 4.00-4.50 Maxime Bourque (Toronto) Non-convex balls in the Teichmüller metric Saturday 9.30-10:20 Vaibhav Gadre (Glasgow) Random geodesics in moduli spaces 10.30-11.00 COFFEE 11.00-11.50 Kathryn Lindsey (Boston College) Entropies of P.C.F. generalized $\beta$ transformations 12:10-1:00 John Smillie (Warwick) Renormalisation dynamics for polygonal billiards END OF WORKSHOP

## Abstracts for Mini-Courses:

 Jayadev Athreya (Seattle) and Elise Goujard (Orsay) Title: Volumes of moduli spaces of meromorphic quadratic differentials Abstract: We survey recent progress in explicit computations of volumes of moduli spaces of meromorphic quadratic differentials, and connections to problems in dynamics, combinatorics, number theory, and algebra. We will work out several concrete examples and will attempt to keep the lectures as self-contained as possible. Speaker: David Aulicino (CUNY) Title: Computing the Lyapunov Exponents of a Particular Orbit Closure Abstract: We consider a particular degree 6 covering of the pillowcase with a free marked point.  The resulting surface has genus 4 with Kontsevich-Zorich spectrum 1 > 5/12 = 5/12 > 0.  We will explain how to prove this by considering the area Siegel-Veech constant and the action of the automorphism group on homology.  A general observation concerning computing area Siegel-Veech constants for rank one affine submanifolds will be presented.  All necessary background will be given accompanied by many figures. Speaker: Benjamin Dozier (Stanford) Title:  Equidistribution of saddle connections on translation surfaces Abstract: I will show that on any translation surface the collection of saddle connections of length at most R becomes equidistributed on the surface, as R tends to infinity.  This is an analog of classical results on equidistribution of closed geodesics on hyperbolic surfaces.  The proof in our setting uses the powerful machinery pioneered by Eskin-Masur, in particular, a new result about angles of saddle connection, together with the theorem of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.  Speaker: Rene Ruhr  (Tel Aviv) Title: Counting saddle connections on translation surfaces Abstract: It is known that for most translation surfaces the number of saddle connections whose length is less than T grows asymptotically like T^2 by works of Eskin and Masur.One main idea in their proof is to use ergodicity of the SL_2(R) action on the space of translation surfaces. We will review the dynamically part of their proof. As it is now known that this action also exhibits a spectral gap, one can give an additional error term, which we shall also outline. This effectivization is joint work with Amos Nevo and Barak Weiss. Speaker: Nicholas Vlamis (Michigan) Title: Identities on hyperbolic manifolds Abstract:  We will give an overview of four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn, and Luo-Tan.  Each of these identities calculates a geometric quantity, such as the volume of the boundary or the volume of the unit tangent bundle, as a summation over spectral data, such as the lengths of simple closed geodesics or the lengths of orthogeodesics.  We will give a sketch of how these identities are proved with a focus on their relation to cross ratios.  The cross ratio formulation of identities lends itself to generalizations to higher Teichmüller theory; we will end with discussing some recent generalizations, which includes joint work with Andrew Yarmola.

## Abstracts for Lecture Series:

 Paul Apisa (Chicago) Title: Holomorphic Sections, Teichmuller Dynamics, and Billiards - From Finiteness to Classification Abstract: Given a family of Riemann surfaces when are there point markings that vary holomorphically with the surfaces? Given a polygonal billiard table and two points when can all billiard shots from one point to the other be blocked by a finite collection of points? These questions connect via Teichmuller dynamics. We will use work of Eskin-Mirzkhani and Eskin-Mirzakhani-Mohammadi to show finiteness results for the holomorphic sections and finite blocking problems. These results exhibit an elliptic-parabolic-hyperbolic trichotomy. We will also explain how to use these results as a tool to prove that certain subvarieties of moduli space are loci of branched covers. (This is joint with Alex Wright). In the second half of the talk, we will discuss how to promote the finiteness results to classifications using - (1) Filip’s characterization of the equations defining affine invariant submanifolds, (2) McMullen’s classification of genus two complex geodesics, and (3) the geometry of hyperelliptic curves. These three ingredients will solve finite blocking problems for prime triangles, genus two translation surfaces, and isosceles triangles respectively. Bizarrely, in some cases, no knowledge of a translation surface’s orbit closure is necessary to solve the finite blocking problem. Matt Bainbridge (Indiana) Title: Smooth compactifications of strata Abstract: In recent work with Chen, Gendron, Grushevsky, and Moeller,we studied the Incidence Variety Compactification of strata ofholomorphic one-forms.  It turns out that this compactification hasmany defects, for example it is highly singular.  In this talk, wepropose a better commpactification. Maxime Fortier Bourque (Toronto) Title: Non-convex balls in the Teichmüller metric Abstract: In joint work with Kasra Rafi, we discovered the existence of non-convex balls in the Teichmüller metric. This non-convexity is equivalent to the fact that the extremal length of a measured foliation can have a local maximum along a Teichmüller geodesic. In turn, the existence of such local maxima can be shown with numerical calculations. This gives another proof that the Teichmüller metric is not non-positively curved, contrary to an infamous claim of Kravetz (first disproved by Masur).Since balls are in general not convex, are there other natural subsets which are? In particular, is the convex hull of a finite set in Teichmüller space always compact? The answer is positive for some 2-dimensional Teichmüller spaces of surfaces with boundary (joint with Yudong Chen, Roman Chernov, Marco Flores, Seewoo Lee, and Bowen Yang). The next simplest case which remains open is the 3-dimensional Teichmüller space of hexagons, or equivalently, of pairs of pants. Viveka Erlandsson (Aalto) Title: Counting curves on surfaces Abstract: I will discuss the asymptotic growth of (non-simple) closed curves on a surface S with negative Euler characteristic . Mirzakhani showed that when S is equipped with a hyperbolic metric the number of curves in each mapping class group orbit with length bounded by L grows asymptotically as a constant times L^{6g-6+2r} (where g and r is the genus and number of cusps, respectively). In this talk I will show that the same asymptotics hold for other metrics, such as any Riemannian metric, Euclidean cone metric, or the word metric with respect to any generating set of the fundamental group. Vaibhav Gadre (Glasgow) Title: Random geodesics in moduli spaces. Abstract: Typical geodesics in moduli spaces show analogies with typical geodesics on a finite area hyperbolic surface with cusps. The Masur-Sullivan logarithm laws are a vivid example of this analogy. This talk will survey recent progress in theory of random geodesics in moduli space. Part of this talk is joint work with Maher, Tiozzo and with Maher. Sarah Koch (Michigan) Title: On the connectivity of Milnor curves in moduli space Abstract: Living inside the moduli space of quadratic rational maps is the Milnor curve' S_n, which consists of all f:P^1 -> P^1 with a periodic critical point of period n. It is unknown if S_n is connected for all n. Recently, Arfeux and Kiwi proved that the analogous curves in the moduli space of cubic polynomials are always connected. In this talk, we discuss some unexpected challenges that arise in the quadratic rational map setting which are absent in the cubic polynomial setting. Based on joint work with E. Hironaka. Kathryn Lindsey (Boston College) Title:   Entropies of P.C.F. generalized $\beta$ transformations Abstract: Ronen Mukamel (Rice) Title:  Teichmuller curves in positive characteristic Abstract:  We will investigate the arithmetic nature of Teichmuller curves in the spirit of the celebrated theory developed for the classical modular curves.  For arithmetic Teichmuller curves in genus two, we introduce the Weierstrass polynomial, an analogue of the classical modular polynomial associated to modular curves.  We find that Weierstrass polynomials often have integer coefficients and surprising congruences with modular polynomials in positive characteristic. Kasra Rafi (Toronto) Title: Updates on the limit set of a Teichmüller geodesic in the Thurston boundary Abstract: In her thesis, Lenzhen showed that a Teichmüller geodesic may have more than one limit point in the Thurston boundary. This opened the question of what behaviors for a Teichmüller geodesic are possible near the boundary. We give an overview of some old and several recent results including methods for constructing examples and a necessary condition about the shape of the limit set. Various results in this talk are joint work with Brock-Leininger-Modami, Leininger-Lenzhen and Lenzhen-Modami John Smillie (Warwick) Title: Renormalisation dynamics for polygonal billiards Abstract: Billiard tables are studied in dynamics because they often give aproachable examples of deeper and more technical dynamical phenomena. The study of rational polygonal billiards is an instance of this and it leads to the deep dynamical idea of renormalisation. This means that many of the interesting features of polygonal billiards are actually approached by studying a related renormalisation dynamics on a higher dimensional  `parameter space” of dynamical systems.  The renormalisation dynamics that arise in connection with polygonal billiards have many features in common with homogeneous dynamics. Homogeneous dynamical systems are elegant examples with some very nice properties. The study of homogeneous dynamics goes back to work of Hopf, Hedlund, Hopf, Morse and continued with striking results of Ratner, Margulis and many others. Renormalisation dynamics has properties similar to homogeneous dynamics but it also has some surprising differences. I will describe some recent positive  results joint with Barak Weiss and Matt Bainbridge and some recent negative results joint with Barak Weiss and Jon Chaika. Anton  Zorich (Jussieu) Title: Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes 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  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  conjectures  around  large  genus asymptotics of Masur-Veech volumes. (joint work with V. Delecroix, E. Goujard, P. Zograf).

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