KMPB Workshop: Holomorphic Differentials & RTG 2965 "From geometry to numbers" : Scientific opening

20 Nov 2024, 00:00 → 22 Nov 2024, 18:00 Europe/Berlin

Raum 1.107 (IRIS), Raum 1.013 (RUD 25) (Humboldt-Universität zu Berlin)

Gaëtan Borot (Humboldt-Universität zu Berlin), Gavril Farkas (Humboldt-Universität zu Berlin), Bruno Klingler (Humboldt-Universität zu Berlin)

Holomorphic differentials on Riemann surfaces give rise to flat metrics with singularities. They have been studied (as well as their moduli space) from many different perspectives and with diverse motivations: algebraic and arithmetic geometry of strata of differentials with fixed singularity type, enumerative questions (computation of volumes, statistics of saddle connections, intersection theory on the moduli space), dynamics (on flat surfaces, on Teichmüller space, etc.), relation with BPS structures in the physics of supersymmetric gauge theory, stability conditions and wall-crossing phenomena in homological algebra, etc.

The workshop (Wednesday 20 and Thursday 21) opens the second phase of the Kolleg Mathematik Physik Berlin. It will bring together experts whose research has been related in some way to holomorphic differentials.

This workshop will be followed on Friday 22 by the kick-off event for the new DFG-funded Research Training Group 2965 From geometry to numbers: moduli, Hodge theory and rational points established between the Humboldt-Universität zu Berlin and the Leibniz Universität Hannover.

The lecturers are:

• Gregorio Baldi (Institut de Mathématiques de Jussieu)
• Matteo Costantini (Universität Duisburg-Essen)
• Samuel Grushevsky (SCGP Stony Brook University)
• Fabian Haiden (SDU Odense)
• Carlos Matheus (École Polytechnique)
• Martin Möller (Goethe-Universität Frankfurt)
• Anja Randecker (tbc, Universität des Saarlandes)
• David Urbanik (IHES)
• Ivan Yakovlev (MPIM Bonn)
  
  

The lecturers for the scientific opening of the RTG are:

• Jarod Alper (University of Washington)
• Tim Browning (IST Austria)
• Soheyla Feyzbakhsh (Imperial College London)
• Rahul Pandharipande (ETH Zürich)
  
  

Registration for attendance is free but mandatory.

Venue for Wednesday & Thursday:
1.107, IRIS Adlershof, Zum Großen Windkanal 2, 12489 Berlin
Coffee breaks in Room 1.202

Venue for Friday:
1.013, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin

Humboldt-Universität_Logo.jpg

Poster_Holomorphic_Differentials.pdf

RTG 2965 Webseite

Slides_Carlos_Matheus.pdf

Slides_Fabian_Haiden.pdf

Slides_Ivan_Yakovlev.pdf

Wednesday 20 November

09:00 Welcome

1 Fabian Haiden

Spaces of quadratic differentials versus spaces of stability conditions

Spaces of quadratic differentials are special cases of spaces of Bridgeland stability conditions. I will recall this surprising correspondence and discuss the following questions: 1) Does this tell us something new about quadratic differentials? 2) In what way do more general spaces of stability conditions behave like spaces of quadratic differentials? Based on arxiv:1409.8611, 1808.06364, 2104.06018, 2410.08028.

10:30 Coffee break

2 Martin Möller

Siegel-Veech constants and intersection theory

We give an overview over the methods to compute characteristic quantities of strata, Siegel-Veech constants, Masur-Veech volumes and sum of Lyapunov exponents, by flat geometry and by intersection theory, with a view towards answering the same questions for linear submanifolds

12:00 Lunch break

3 Carlos Matheus

Siegel-Veech constants of certain cover constructions

Siegel-Veech constants are quantities describing counting problems (of saddle-connections, cylinders, etc.) on flat surfaces and, after the works of many mathematicians (including Aggarwal, Chen, Eskin, Kontsevich, Moller, Sauvaget, Zagier, Zorich just to mention a few), we know that these constants are related to several interesting mathematical objects such as quasi-modular forms, Lyapunov exponents, slopes of holomorphic bundles, intersection numbers, ...

In this talk, we shall explain how a precise control of the monodromy actions of the orbifold fundamental group of connected components of strata of Abelian differentials on relative cohomology groups allow to compute (and get some surprising phenomena about) the Siegel-Veech constants of loci of cyclic covers of translation surfaces. This is based on a joint work with D. Aulicino, A. Calderon, N. Salter and M. Schmoll.

14:30 Coffee break

4 Ivan Yakovlev

Counting differentials combinatorially

I will present a couple of results about asymptotic enumeration of holomorphic differentials with periods in Z+iZ (square-tiled surfaces) and meromorphic differentials with periods in Z (integral metric ribbon graphs). These results were obtained by pure combinatorics, and it might be interesting to understand their algebraic meaning.

5 Gregorio Baldi

Hodge Locus and differential geometry (Ho-Lo-Diff)

I will survey various applications of algebraic differential geometry and Galois theory of foliations to finiteness results in differential geometry. Specifically, I will discuss the study of totally geodesic submanifolds of ball quotients (joint work with Ullmo) and affine invariant submanifolds of strata of abelian differentials (with Urbanik). I will aim to highlight a unifying theme throughout: atypical intersections.

Thursday 21 November

6 Anja Randecker

Lengths of saddle connections for large genus

For a holomorphic differential on a given surface, we can consider saddle connections, that is, geodesic segments between the zeros of the differential, and we can measure their lengths. We consider the number of saddle connections in a given length range as a random variable on a stratum and show that for genus going to infinity, this converges in distribution to a Poisson distributed random variable. In the talk, I will introduce the geometric aspects of the topic and connect it to Siegel-Veech constants. This is based on joint work with Howard Masur and Kasra Rafi.

10:30 Coffee break

7 David Urbanik

Orbit Closures as Atypical Intersections

We review the Hodge-theoretic characterization of orbit closures due to Filip. Using this characterization, we then explain how the theory of atypical intersections from Hodge theory naturally lets us classify such orbit closures as "typical" and "atypical". This classification recovers a finiteness theorem of Eskin, Filip and Wright for atypical orbit closures. If we have time, we explain how this finiteness can be made effective. Joint work with Greg Baldi.

12:00 Lunch break

8 Matteo Costantini

Geometry of strata of differentials

The geometry of spaces of algebraic curves together with meromorphic k-forms of a fixed type is still quite mysterious. The multi-scale compactification of these strata of differentials allowed to compute some of their topological and algebraic invariants. In this talk we describe the ideas behind such computations and possible applications.

14:30 Coffee break

9 Samuel Grushevsky

Ends of strata of differentials

Using the multi-scale compactification, we determine the number of ends of strata of meromorphic differentials. It turns that in almost all cases all connected components of the strata of differentials have only one end. This is joint work with Ben Dozier.

19:00 Social dinner

Friday 22 November

10 Jarod Alper

11:00 Coffee break

11 Tim Browning

12:30 Lunch break

12 Soheyla Feyzbakhsh

15:30 Coffee break

13 Rahul Pandharipande