Summer School Around Moduli Spaces

The summer school of the SFB-TRR 195 in 2019 takes place from 2nd to 6th September 2019 at Saarland University in Saarbrücken.

General Info

This summer school is founded by the SFB-TRR 195.

The goal of the summer school is to introduce young researchers to different areas in mathematics that are
related to moduli spaces.
There will be mini courses on moduli and Hurwitz spaces, Schottky spaces, tropical 
geometry, translation surfaces, and topological
recursion.

The mini courses will be given by the following speakers

 

Schedule

The summer school starts on Monday at 2pm and ends on Friday at noon. The detailed schedule will be announced soon.

Abstracts

Paul Apisa: Moduli Space of Translation Surfaces

A holomorphic one-form on a Riemann surface induces a flat metric on the surface with metric singularities 
at the zeros of the one-form. This
flat surface – a translation surface – can be described quite simply. It is a 
disjoint union of polygons with sides identified by
translation. Moreover, any translation surface defines a 
holomorphic
one-form on a Riemann surface. We will use our intuition for polygons in the plane to explore 
moduli space!


In this series, we will study the moduli space of translation surfaces (equivalently of holomorphic one-forms 
over Riemann surfaces).


In the first lecture, we will begin by fleshing out the aforementioned equivalence. Then we will construct a 
stratification of the moduli
space, equip the strata with coordinates (period coordinates), and introduce the 
GL(2, R) action on the moduli space. The first lecture
will conclude with a sketch of the proof of ergodicity of 
the GL(2, R)
action and a statement of the “magic wand” theorem of Eskin, Mirzakhani, and Mohammadi. 
Time permitting we will also sketch a result
of Kontsevich and Zorich that shows that the components of 
strata are
distinguished by their spin structure and hyperellipticity.

In the second lecture we will use cylinders on flat surfaces to navigate around moduli space. In this lecture 
we will – show the existence of
cylinders on all translation surfaces, prove the cylinder deformation theorem, 
and study the isoperiodic foliation of moduli space. We will
also relate cylinders to the dynamics of the 
horocycle flow by sketching
a proof of the Smillie-Weiss theorem.

In the third and final lecture we will discuss different partial compactifications of moduli space and illustrate 
their uses. We will
then apply the results of all the sections to prove results in rational billiards and sketch 
results on the structure of linear submanifolds in
moduli space!

 

Vincent Delecroix:

The Masur-Veech volumes are SL(2,R)-invariant volume forms on the moduli spaces of translation surfaces 
(see Paul Apisa lecture).
The total mass of these measures are intriguing numbers related to the geometry of 
translation surfaces (eg area Siegel-Veech constant)
and to dynamics of translation flow (e.g. Eskin-Kontsevich-
Zorich
formula). More recently, the asymptotics of these volumes have been used to give some geometric features 
of surfaces of high genera
(e.g. Masur-Rafi-Randecker, Delecroix-Goujard-Zograf-Zorich).

Topological recursion is a combinatorial framework that is relevant to study enumerative geometry of surfaces 
(e.g.
counting combinatorial maps on surfaces, Gromov-Witten invariants, ...). It was invented in the context of 
random matrices and its
use in a more geometric context was pioneered by Eynard-Orantin. We will explain 
how this framework is relevant for computing the
Masur-Veech volumes.

 

Frank Herrlich: Schottky groups and moduli spaces

This mini-course will be devoted to the uniformization of curves by Schottky groups both in the complex and the \(p\)-adic setting. Here, “curve” will mostly mean smooth projective curve, but we shall also consider stable curves.

We shall begin with the classical construction of a compact Riemann surface out of a Schottky group.
The uniformization theorem tells us that all compact Riemann surfaces can be obtained that way. Then we shall introduce coordinates for Schottky groups and define the (marked) Schottky space. It admits a natural action of Out(\(F_g\)) and a surjective map to the moduli space of curves.

The definition of a Schottky group literally carries over to a \(p\)-adic field, and the quotient again is a nonsingular projective curve. But unlike the complex situation, not all projective curves admit such a uniformization, but only the so called Mumford curves. We shall see how they can be characterised by their reduction mod \(p\). The analogously defined \(p\)-adic Schottky space is sometimes also called \(p\)-adic Teichmüller space, since it shares several properties with the classical Teichmüller space. In addition, it has an intersting relation to Outer space.

The ideas used to establish this relation can be applied to construct “Schottky coverings” of Riemann surfaces with nodes. Instead of on the Riemann sphere \(\hat{\mathbb{C}}\), the Schottky group now acts on a tree of projective lines. The construction leads to a larger complex manifold called “extended Schottky space” which admits a surjective map to the Deligne-Mumford compactification of the moduli space of curves.

 

Martin Ulirsch: Geometry of tropical curves and their moduli spaces

One of the archetypal starting points of tropical geometry is the basic analogy between Riemann surfaces
and metric graphs. Many of the geometric constructions on Riemann surfaces have natural counterparts on
metric graphs. Surprisingly, in many cases this relationship goes beyond a mere analogy and the geometry
of tropical curves has interesting consequences in the classical realm of Riemann surfaces. In this minicourse
I will give an introduction to the intrinsic geometry of tropical curves, motivated by the theory of Riemann
surfaces, and to some of its applications to the proof of classical geometric theorems. Main topics: Covering
theory of tropical curves and the Hurwitz correspondence principle, divisors and rational functions on tropical
curves and the Riemann-Roch theorem, Brill-Noether theory, and the moduli space of tropical curves

 

Registration and Funding

Please register for the summer school by sending an email to This email address is being protected from spambots. You need JavaScript enabled to view it. until Friday, 9 August, 2019. There will be limited funding available. If you are interested in receiving funding, please apply by including to your mail a short description of your current work, the name of your advisor, and your CV.  In the case of support we expect contribution by a 5 minutes long snap shot talk. Please let us know by Friday, 9 August, 2019.

Contact

If you have any questions, please feel free to contact us via This email address is being protected from spambots. You need JavaScript enabled to view it..

Organizers

 Poster

poster-summer-school