Summer School Around Moduli Spaces

The summer school of the SFB-TRR 195 in 2019 takes place from 2^nd to 6^th 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

• Paul Apisa (Yale University)
• Vincent Delecroix (Université Bordeaux 1, MPI Bonn)
• Frank Herrlich (Karlsruhe Institute of Technology)
• Navid Nabijou (University of Glasgow)
• Martin Ulirsch (Goethe-Universität Frankfurt)

Schedule

The print version can be found here.

The talks take place in room Hörsaal III (lecture hall III) in building E 2.5. A map of the campus can be found  at www.math.uni-sb.de/CMS/index.php/de/kontakt-de. The closest bus stop is called Universität Mensa.

On Wednesday we plan to visit Saarbrücken Castle. We booked a guided tour starting at 4:30pm. The tour including the entry costs 3€. We will take a bus together to get there.

On Thursday evening we plan a joint dinner in the restaurant "Die Kartoffel" in Saarbrücken. The address is St. Johanner Markt 32, 66111 Saarbrücken.

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!

References:

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.

References:
Kontsevich, Maxim: Intersection theory on the moduli space of curves and the matrix Airy function
Commun. Math. Phys. 147, No. 1, 1-23 (1992). Zbl 0756.35081
Norbury, Paul: Counting lattice points in the moduli space of curves
Math. Res. Lett. 17, No. 3, 467-481 (2010). Zbl 1225.32023
Chapman, Kevin M.; Mulase, Motohico; Safnuk, Brad: The Kontsevich constants for the volume of the moduli
of curves and topological recursion
Commun. Number Theory Phys. 5, No. 3, 643-698 (2011). Zbl 1259.14028
Athreya, Jayadev S.; Eskin, Alex; Zorich, Anton: Right-angled billiards and volumes of moduli spaces of quadratic
differentials on CP1
Ann. Sci. Éc. Norm. Supér. (4) 49, No. 6, 1311-1386 (2016). Zbl 1372.32020
Jorgen Ellegaard Andersen, Gaëtan Borot, Leonid O. Chekhov, Nicolas Orantin: The ABCD of topological recursion
https://arxiv.org/abs/1703.03307
Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich: Masur-Veech volumes, frequencies of simple closed
geodesics and intersection numbers of moduli spaces of curves
https://arxiv.org/abs/1908.08611
Jørgen Ellegaard Andersen, Gaëtan Borot, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto,
Danilo Lewanski, Campbell Wheeler: Topological recursion for Masur-Veech volumes
https://arxiv.org/abs/1905.10352
Maxim Kontsevich, Yan Soibelman: Airy structures and symplectic geometry of topological recursion
https://arxiv.org/abs/1701.09137

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.

References:
For the complex part of the mini course see sections 1 to 5 of:
F. Herrlich: Schottky Space and Teichmüller Disks. Handbook of Group Actions, vol. II
(eds. L. Ji, A. Papadopoulos, S.-T. Yau), Higher Education Press 2015, pp. 289-308.
For mor detail see also:
B. Maskit: Kleinian groups. Grundlehren der math. Wiss. 287, Springer 1988.

For the second part of the mini course see sections 1 to 7 of:
F. Herrlich: p-adic origamis. Contemporary Mathematics 629 (2014), 225-243.
For mor detail see also:
L. Gerritzen and M. van der Put, Schottky groups and Mumford curves. Lecture Notes in
Mathematics 817, Springer 1980.

Navid Nabijou: Moduli and Hurwitz spaces

In the first half, we will give an introduction to how algebraic geometers think about moduli spaces. The key is Grothendieck's functorial point of view, which allows us to give a rigorous definition of what we really mean when we say the words "moduli space." Geometric properties of moduli spaces, such as smoothness and compactness, are translated via this point of view into questions in deformation theory. We will also see examples in which automorphisms preclude the existence of a moduli space. The exposition will lean heavily on examples, in particular moduli spaces of curves (of various flavours).

In the second half, we will turn our attention to Hurwitz theory. This is a classical subject, concerned with counting ramified covers of Riemann surfaces - in retrospect, it can be viewed as one of the earliest branches of enumerative geometry. Although the objects considered are simple, the theory is incredibly rich, with deep connections to Gromov-Witten theory, representation theory and tropical geometry (for more on the latter, see Martin Ulirsch's lectures). We will explore some of these connections, with a focus on concrete computations, and see what all this has to do with moduli spaces and their cohomology.

References:
Arbarello-Cornalba-Griffiths: "Geometry of algebraic curves, volume II."
Cavalieri: "On Hurwitz theory and applications."
Cavalieri-Miles: "Riemann surfaces and algebraic curves: a first course in Hurwitz theory."
Eisenbud-Harris: "3264 and all that: a second course in algebraic geometry."
Harris-Morrison: "Moduli of curves."
Kock: "Notes on psi classes."
Vakil: "The rising sea", Chapter 19.

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.

References:
Abramovich, Caporaso, Payne: Tropicalization of the moduli space of curves
Baker, Jensen: Degeneration of Linear Series From the Tropical Point of View and Applications
Baker: Specialization of linear series from curves to graphs
Baker, Faber: Metric properties of the Abel-Jacobi map
Baker, Payne, Rabinoff: Nonarchimedean geometry, tropicalization and metrics on curves
Baker, Rabinoff: The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic
functions from tropical to algebraic curves
Caporaso: Algebraic and tropical curves: comparing their moduli spaces
Caporaso: Recursive combinatorial aspects of compactified moduli spaces
Chan: Lectures on tropical curves and their moduli spaces
Cavalieri, Markwig, Ranganathan: Tropicalizing the moduli space of admissible covers
Cavalieri, Markwig, Johnson: Tropical Hurwitz numbers
Cools, Draisma, Payne, Robeva: A tropical proof of the Brill-Noether-Theorem
Mikhalkin, Zharkov: Tropical curves, their Jacobians, and theta functions
Möller, Ulirsch, Werner: Realizability of tropical canonical divisors

Group Picture

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

• Gabriela Weitze-Schmithüsen
• Andrea Thevis
• Christian Steinhart

Participants

Lev Blechman (Tel Aviv University)
Sven Caspart (Karlsruhe Institute of Technology)
Alejandro Giangreco (Aix-Marseille University)
Daniel Gromada (Saarland University)
Michael Hoff (Saarland University)
Cedric Holle (Saarland University)
Birte Johansson (TU Kaiserslautern)
Manuel Kany (Saarland University)
Christoph Karg (Heidelberg University)
Pascal Kattler (Saarland University)
Felix Leid (Saarland University)
Mingkun Liu (Université Paris Diderot)
Biao Ma (Université de Nice)
Juan Marshall (Aix-Marseille University)
Jesús Martín Ovejero (Universidad de Salamanca)
Felix Röhrle (Goethe University Frankfurt)
Emil Rotilio (TU Kaiserslautern)
Fabian Ruoff (Karlsruhe Institute of Technology)
Sogo Pierre Sanon (TU Kaiserslautern)
Johannes Schwab (Goethe University Frankfurt)
Carlo Sircana (TU Kaiserslautern)
Roland Speicher (Saarland University)
Mirko Stappert (Saarland University)
Christian Steinhart (Saarland University)
Isabel Stenger (Saarland University)
Andrea Thevis (Saarland University)
Nikolaos Tsakanikas (Saarland University)
Ruihua Wang (Leiden University)
Yvonne Weber (TU Kaiserslautern)
Gabriela Weitze-Schmithüsen (Saarland University)

Poster