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.

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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.

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.

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



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).