
Time 
Speaker 
Talk 
Thursday 

9.3010.20  Anton Zorich (Jussieu)  Equidistribution of squaretiled surfaces, meanders, and MasurVeech volumes 
10.3011.00 
COFFEE 

11:0011:50 
Matt Bainbridge (Indiana)  Smooth compactifications of strata 
12.002:00 
LUNCH  
2.002.50 
Sarah Koch (Michigan)  On the connectivity of Milnor
curves in moduli space 
3:004.00 
DISCUSSION 

4.004.50 
Viveka Erlandsson (Aalto)  Counting curves on surfaces 
Friday


9.3010.20  Ronen Mukamel (Rice) 
Teichmuller
curves in positive characteristic 
10.3011.00  COFFEE  
11:0011:50  Paul Apisa
(Chicago) 
Holomorphic
Sections, Teichmuller Dynamics, and
Billiards  From Finiteness to
Classification 
12.002:00  LUNCH  
2.002.50  Kasra Rafi (Toronto)  Updates on the limit set of a Teichmüller geodesic in the Thurston boundary 
3.004.00  DISCUSSION  
4.004.50  Maxime Bourque (Toronto)  Nonconvex balls in the Teichmüller metric 
Saturday 

9.3010:20  Vaibhav Gadre (Glasgow)  Random
geodesics in moduli spaces 
10.3011.00 
COFFEE 

11.0011.50  Kathryn Lindsey (Boston College)  Entropies
of P.C.F. generalized $\beta$
transformations 
12:101:00 
John Smillie (Warwick)  Renormalisation dynamics for polygonal billiards 
END OF WORKSHOP 
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
selfcontained 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 KontsevichZorich spectrum 1 > 5/12 = 5/12 > 0. We will explain how to prove this by considering the area SiegelVeech constant and the action of the automorphism group on homology. A general observation concerning computing area SiegelVeech 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 EskinMasur, in particular, a new result about angles of saddle connection, together with the theorem of KerckhoffMasurSmillie 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, BridgemanKahn, and LuoTan. 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) 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 EskinMirzkhani and
EskinMirzakhaniMohammadi to show
finiteness results for the holomorphic
sections and finite blocking problems.
These results exhibit an
ellipticparabolichyperbolic
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) Abstract: In recent work with Chen,
Gendron, Grushevsky, and Moeller,we
studied the Incidence Variety
Compactification of strata ofholomorphic
oneforms. It turns out that this
compactification hasmany defects, for
example it is highly singular. In
this talk, wepropose a better
commpactification.
Abstract: In joint work with Kasra
Rafi, we discovered the existence of
nonconvex balls in the Teichmüller
metric. This nonconvexity 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
nonpositively 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
2dimensional 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 3dimensional Teichmüller space of
hexagons, or equivalently, of pairs of Viveka
Erlandsson (Aalto)
Title: Counting curves on surfaces Abstract: I will discuss the asymptotic growth of (nonsimple) 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^{6g6+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 MasurSullivan
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) 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 BrockLeiningerModami, LeiningerLenzhen and LenzhenModami 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 squaretiled surfaces, meanders, and MasurVeech volumes Abstract: We show how recent equidistribution results allow to compute approximate values of MasurVeech 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 MasurVeech volume of the moduli space of quadratic differentials in the spirit of MirzakhaniWeilPeterson volume of the moduli space of curves.Finally we present several conjectures around large genus asymptotics of MasurVeech volumes. (joint work with V. Delecroix, E. Goujard, P. Zograf). 
Trinity
College Dublin, College Green, Dublin 2. Tel:
+35316081000.
