Geometric, Algebraic, and Physical structures around the moduli of Meromorphic Quadratic Differentials: May 13, 2024 – June 21, 2024

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Organizing by:

  • Tom Bridgeland (University of Sheffield, UK)
  • Samuel Grushevsky (Stony Brook University, USA)
  • Andrew Neitzke (Yale University, USA)
  • Martin Moeller (Universitaet Frankfurt, Germany)

This program in Mathematics is inspired by a circle of ideas originating in quantum field theory and string theory, particularly the study of the quantum field theories “of class S”. These field theories have been the subject of intense study in the high energy theory community over the last thirteen years. Many of the constructions from physics have now found their natural place in vibrant areas of research in pure mathematics. Physical insights have helped to catalyze the realization that the mathematics of flat surfaces, triangulated categories, cluster algebras, symplectic geometry, and ordinary differential equations have a common Rosetta stone, in the theory of meromorphic quadratic differentials on Riemann surfaces.

This program, partly motivated by physics,  is structured around meromorphic quadratic differentials on Riemann surfaces, and their relations with stability conditions and enumerative invariants in geometry. We aim to bring these diverse mathematical communities together to focus on the core constructions of mutual interest, to see if techniques from one field can shed light on questions from another, and to make progress in each of the directions separately.

WEEK 1: May 13 – 17, 2024


  • Tuesday 5/14/24: 10:30AM – Seminar Room – 313
  • Wednesday 5/15/24: 10:30AM – Seminar Room – 313
  • Thursday 5/16/24: 10:30AM – Seminar Room – 313

Speaker: Anton Zorich (University of Paris 7)
Title: Some basic facts about geometry and dynamics of moduli spaces of quadratic differentials

  • Friday 5/17/24: 10:30AM – Seminar Room – 313

Speaker: Fabian Haiden (Centre for Quantum Mathematics, Denmark)
Title: Introduction to stability conditions

WEEK 2: May 20 – 24, 2024


  • Monday 5/20/24: 10:30AM – Seminar Room – 313
  • Tuesday 5/21/24: 10:30AM – Seminar Room – 313

Speaker: Fabian Haiden (Centre for Quantum Mathematics, Denmark)
Title: Introduction to stability conditions

  • Wednesday 5/22/24: 10:30AM – Seminar Room – 313
  • Thursday 5/23/24: 10:30AM – Seminar Room – 313
  • Friday 5/24/24: 10:30AM – Seminar Room – 313

Speaker: Tom Bridgeland (University of Sheffield)
Title: Geometric structures on spaces of stability conditions induced by Donaldson-Thomas invariants


  • Wednesday 5/22/24: 1:30PM – Seminar Room – 313

Speaker: Alexander Goncharov (Yale)
Title: Exponential volumes of moduli spaces of hyperbolic surfaces

Abstract: Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of the spaces M(S,L) are finite. Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics. However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that the exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani’s recursions. This talk is based on the joint work with Zhe Sun.

  • Friday 5/24/24: 1:30PM – Seminar Room – 313

Speaker: Hulya Arguz (University of Georgia)
Title: The KSBA moduli space of log Calabi-Yau surfaces

WEEK 3: May 27 – 31, 2024


Tuesday 5/28/24: 10:30AM – Seminar Room – 313
Wednesday 5/29/24: 10:30AM – Seminar Room – 313
Thursday 5/30/24: 10:30AM – Seminar Room – 313

Speaker: Pierrick Bousseau (University of Georgia)
Title: Scattering diagrams with a view toward spectral networks

Program Talk:

Thursday 5/30/24: 1:30PM – Seminar Room – 313

Speaker: Simion Filip (University of Chicago)
Title: Finiteness of totally geodesic hypersurfaces in variable negative curvature

Friday 5/31/24: 10:30AM – Seminar Room – 313

Speaker: Andy Neitzke (Yale University)
Title: Introduction to spectral networks

WEEK 5: June 10 – 14, 2024

Program Talk:
Monday 6/10/24: 10:30AM – Seminar Room – 313

Speaker: Michael Wolf
Title: Rays of Holomorphic Differentials
Abstract: We study the geometry of rays of holomorphic Hitchin differentials in low rank. Mostly, we restrict ourselves to the classical case of PSL(2,R), where we can depict the rays (and a dual version) as interpolating between Teichmüller and particular Thurston geodesics; there are new consequences for Thurston metric geometry. This is joint with Huiping Pan.

Tuesday 6/11/24: 10:30AM – Seminar Room – 313
​Thursday 6/1​3/24: 10:30 AM – Seminar Room – 313

Speaker: Andy Neitzke (Yale University)
Title: Introduction to spectral networks

Program Talk:
Wednesday 6/1​2/24: 10:30 AM – Seminar Room – 313

Speaker: Francois Labourie
Title: Poisson algebra, combinatorics and representations of surface groups
Abstract: In this talk, I will start by recalling what a Poisson algebra is by first concentrating on the basic example of the algebra of polynomials in 2 variables. I will then briefly recall the relationship of Poisson algebras with Hamiltonian and Quantum Dynamics. Then I will explain a beautiful combinatorial construction due to Bill Goldman giving the structure of a Lie algebra on the vector space formally generated by loops on a surface $S$. The motivation underlying Goldman’s construction is motivated by the Poisson algebra of regular functions on the character variety of $\pi_1(S)$ in a Lie group $G$. From that I will move to the deformation space of Anosov representations of $\pi_1(S)$ in a non-compact Lie group $G$ and show that there are more natural functions than regular ones (think of length or cross-ratio functions on Teichmüller space) and show that their Poisson Structure can also be described by a combinatorial object called Ghost Bracket. This is a joint work with Martin Bridgeman.

​Friday 6/1​4/24: 10:30 AM – Seminar Room – 313
Speaker: Lotte Hollands
Title: Spectral networks and spectral determinants
Abstract: Spectral networks encode an awful lot of information about​ four-dimensional N=2 theories of class S, but are also a very useful​ tool to study spectral problems that come up in this context. In this​ talk I will explain a better understanding of their spectral​ determinants in terms of spectral coordinates, and as an explicit​ instance of the “non-perturbative” partition function. This is based on​ 1906.04271 with Andy Neitzke and work in progress with Alba Grassi and Qianyu Hao.

WEEK 6: June 17 – 21, 2024
Program Talks:
Monday 6/17/24: 10:30AM – Seminar Room – 313

Speaker: Dmitry Korotkin
Title: WKB expansion of Yang-Yang generating function for second order equation and monodromy dependence of the Jimbo-Miwa tau-function.
Abstract: We study the symplectic properties of the monodromy map of the second order equation on a Riemann surface whose potential is meromorphic with double poles. The Poisson bracket defined in terms of periods of meromorphic quadratic differential in the potential implies
the Goldman Poisson structure on the monodromy manifold. We show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the “Yang-Yang function” of Nekrasov, Rosly and Shatashvili) is determined by the Bergman tau-function on the moduli space of meromorphic quadratic differentials.
We discuss the relationship of this result to the problem of dependence of the isomonodromic tau-function by Jimbo-Miwa on the monodromy data and appearance of dilogarithms in the connection formulas for the tau-function. The talk is based on joint works with Marco Bertola.

Tuesday 6/18/24: 10:30AM – Seminar Room – 313

Speaker:Nicholas Williams
Title: Donaldson–Thomas invariants for the Bridgeland–Smith correspondence
Abstract: Christ, Haiden, and Qiu recently refined the correspondence of Bridgeland and Smith between quadratic differentials and stability conditions. In this setting, we compute the Donaldson–Thomas invariants associated to the finite-length trajectories of the quadratic differential, showing that they match predictions from topological recursion. This is joint work with Omar Kidwai.

Thursday 6/20/24: 10:30AM – Seminar Room – 313

Focus: Informal Discussions, research and future outlook talks.