Workshop in Geometry and Mathematical Physics

May 25, 2026, 9:00 AM → May 26, 2026, 3:00 PM Europe/Rome

7th floow (SISSA)

This is a 2-day workshop that includes the award of the Dubrovin medal.

The winners are 

Elba Garcia-Failde (Universitat Politècnica de Catalunya and Institut de Mathematiques de Jussieu, Sorbonne Universite)

Yang Li (University of Cambridge)

 

Speakers

 

Gaetan Borot       (Humboldt University,  Berlin)

Marco Bertola     (Concordia University, Montreal)

Guido Carlet        (IMB,  Dijon)

Marta Mazzocco  (ICREA Barcelona)

Paolo Rossi   (University of Padua)

Giulio Ruzza  (University of Lisbon)

 

 

Organizers: Tamara Grava, Davide Guzzetti, Sara Perletti, Zechuan Zhang,  Mahmoud Abdelrazek, Leilei Shi and Danilo Lewanki

 

 

 

Mon, May 25

1 MARCO BERTOLA, Hitchin systems and r-matrix structure on (framed) Higgs bundles

Abstract.
On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson structure, the ``r-matrix structure’’.
It is an essential structure underlying the Hamiltonian dynamics of the vast majority of integrable systems, isospectral and isomonodromic evolution equations.
The known r-matrices depend on parameter in rational way (trig/elliptic, respectively) and hence we think of them on the Riemann sphere (cylinder/torus).

In a relatively abstract Hamiltonian framework the isospectral evolution equations are generalized to higher genus Riemann surfaces as the “Hitchin systems”, an evolutionary integrable system on the moduli space of vector bundles. On the isomonodromic side main progress is attributable to Krichever who used a quite explicit coordinatization of vector bundles on Riemann surfaces that we can call “Tyurin parametrization”.

In this talk I report on the fully explicit generalization of the r-matrix structure to an arbitrary genus Riemann surface merging the Tyurin-Krichever approach with the general framework of Hitchin’s. The key tool is a (fully explicit) matrix-valued kernel that plays crucial role also in setting up integral equations in related area of the "non-abelian steepest descent” method.

2 GUIDO CARLET, Spectral sequences and Poisson brackets

The methods of homological algebra offer a remarkably efficient tool to study the structure and the deformations of Poisson and bi-Hamiltonian structures in the variational setting. We review some recent applications.

10:45 AM Coffe break

3 PAOLO ROSSI, The extended r-spin F-cohomological field theories and relations in the kappa ring of compact type

Abstract: In a recent joint paper with G. Borot and S. Ragni we extend the Givental-Teleman theorem on reconstruction a cohomological field theories from the underlying Frobenius manifold to the context of F-cohomological field theories on the moduli space of curves of compact type and flat F-manifolds. In the same way as Pixton, Pandharipande and Zvonkine use Givental-Teleman theory on the r-spin (for r=3, in particular) Witten CohFT to produce all known relations in the tautological reign of the moduli space of stable curves, we apply our generalization to the extended r-spin F-CohFT, introduced by Buryak and myself) to produce relations in the kappa ring of the moduli spaces of curves of compact type.

4 GAETAN BOROT, Gaussian fluctuations in random lozenge tilings and the Kenyon-Okounkov conjecture

I will describe results for the macroscopic asymptotics in random lozenge tilings in a large class of two-dimensional domains, in particular identifying Gaussian free field fluctuations for the height field. The results rely on an analysis on a discrete Coulomb gas extending the one known for invariant ensembles of random matrices.
This is based on the joint work https://arxiv.org/abs/2601.16377 with Vadim Gorin and Alice Guionnet

5 GIULIO RUZZA Multiplicative statistics of Poissonized Plancherel random partitions

After reviewing the Plancherel measure on partitions and its relevance in combinatorics and (asymptotic) representation theory, I will introduce a class of multiplicative statistics of Poissonized Plancherel random partitions. Their study is motivated by connections to integrable systems (Toda equations) and to important stochastic growth models (polynuclear growth models).
In particular, with Mattia Cafasso and Matteo Mucciconi we tackled the asymptotic study of these statistics. Building on the log-gas structure of the Poissonized Plancherel measure we derived optimal shapes for Poissonized Plancherel random partitions (which generalize the celebrated Vershik-Kerov-Logan-Shepp density and exhibit new behaviors naturally described in terms of elliptic functions) as well as refined asymptotic expansions for the statistics themselves.

3:30 PM Coffe break

6 MARTA MAZZOCCO, Decorated Betti moduli space

Abstract: The focus of this talk le is the study of moduli spaces of representations of fundamental groups of surfaces S with boundaries with values in GL(,n C). In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on S, as the moduli space of representations of the fundamental groupoid, as the moduli space of monodromy data and as character variety. By adding marked points to the boundary of S in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. In this talk, I will develop a categorical framework that allows for a clear and systematic definition of the decorated Betti moduli space designed to specialize to the different points of view encountered in the literature.

7:30 PM Conference dinner Chimera di Bacco

Tue, May 26

7 DIRECTOR PRESENTATION

8 MEDIALAB and LMP presentation

9 Mathematics Area Coordinator presentation

10 Laudation by Gaetan Borot

11 Elba Garcia-Failde, Whispers of Topological Recursion: The negative Witten conjecture and a universal duality

opological recursion (TR) is a universal procedure that helps connect diverse areas of mathematics and physics. Starting from a spectral curve—a Riemann surface equipped with additional data—it produces a family of differentials that often encode enumerative invariants, such as volumes of moduli spaces, matrix model correlators, and intersection numbers. After a brief introduction, we explore two stories, in principle unrelated to, yet guided by, TR—almost in a whisper—towards their resolution:
Witten conjectured that the generating series of psi-class intersection numbers is a tau function of the KdV hierarchy, a result first proved by Kontsevich. Norbury later conjectured an analogous statement for intersection numbers of psi-classes times a negative square root of the canonical bundle. We prove Norbury’s conjecture and derive polynomial relations among kappa-classes. We further introduce a new cohomological field theory (CohFT): the negative analogue of Witten’s r-spin CohFT. We show that its intersection numbers are recursively computable via W-constraints and obtain new tautological relations on the moduli space of curves.
The generating series of maps—graphs embedded on surfaces—obeys TR. Exchanging the roles of x and y in the initial data of TR transforms these maps into fully simple maps. Ordinary and fully simple maps are related via monotone Hurwitz numbers, unveiling a universal duality. This duality translates into functional relations between moments and free cumulants, resolving an open problem in free probability that generalises Voiculescu’s R-transform. In this way, we uncover a general theory of freeness which takes into account higher genus corrections and captures the full, all-order asymptotic behaviour of random matrices.

12 Laudation by Cristiano Spotti

13 Yang Li, Metric SYZ conjecture

The Strominger-Yau-Zaslow conjecture asks for a special Lagrangian fibration on the generic region of a Calabi-Yau manifold close to the large complex structure limit. I will give a survey talk for the recent progress on this question.