20/03, 10:00h 

Around non-commutative analogs of the Painlevé monodromy manifolds 

Abstract: The famous differential Painlevé equations define the most general non-linear special functions. They are connected to various branches of mathematics as well as mathematical physics. In recent years, there has been growing interest in generalizing these equations to the non-commutative case, motivated by problematics and needs of modern physics. Furthermore, it is natural to study the properties of these non-commutative analogs, which extend those in the commutative setting. In particular, one may inquire about the non-commutative analogs of the so-called monodromy manifolds associated with the isomonodromic problems of the Painlevé equations.

In this talk, we will discuss a method for deriving these analogs and defining the associated Poisson structure. The talk is based on the paper arXiv:2302.10694 and an ongoing project in collaboration with S. Arthamonov.

18/03, 16:15h 

An example of Fock--Goncharov theory and Stokes data 

This talk will focus on the study of a specific example of a meromorphic connection on the projective line. More precisely, it will be shown how the Stokes data arising from the higher order pole can be computed exploiting a triangulation of the decorated real oriented blowup of the surface involved. This way, Stokes data will arise as representations of specific automorphisms arising from Fock--Goncharov theory and the generalized monodromy matrices will be parametrized by Fock--Goncharov variables. 

Part One: 18/03/25, 10:00h 

Part of a minicourse entitled: From Classical to Quantum Vistas: The Art of Geometric Quantization

 From Newton to Schrödinger: Through the Quantum Looking Glass

Is our reality classical or quantum? The wave-particle duality hints at a world where both coexist, challenging our fundamental understanding of nature. Geometric quantization offers a path to bridge these realms, but there’s no universal formula—quantization is an art, blending intuition, structure, and deep mathematics.

This talk takes a dynamic journey through geometric quantization, from Darboux’s theorem to the Heisenberg uncertainty principle, exploring how symplectic geometry, polarization, and Bohr-Sommerfeld conditions shape the way we transition from classical observables to quantum operators. Along the way, we uncover the beauty, the paradoxes, and the ongoing quest to make sense of the quantum world.

Part Two: 19/03/25, 15:00h

Part of a minicourse entitled: From Classical to Quantum Vistas: The Art of Geometric Quantization

When Symmetry Meets Polytopes – A Path to Quantization

Symmetry is not just a fundamental principle in physics—it is a powerful tool that shapes how we transition from classical to quantum mechanics. In the process of quantization, symmetry and reduction principles become deeply intertwined. A particularly elegant approach arises from polytopes, which serve as blueprints for quantization in toric manifolds. In this setting, Bohr–Sommerfeld leaves align with integer lattice points, unveiling a rich interplay between symplectic geometry, combinatorial structures, and formal geometric quantization. This perspective not only provides a framework for quantizing symplectic manifolds but also extends naturally to certain classes of Poisson manifolds which are not symplectic.

18/03/2025, 15:00h

Resurgence of the Deformed Painlevé I Equation

ABSTRACT: The first Painlevé equation with a perturbation parameter ħ is easy to solve in formal ħ-power series, but this solution is necessarily divergent. We prove that this divergent series is resurgent, which means it is Borel summable in almost every direction and the associated Stokes phenomenon is describable in terms of the Borel resummation of singularities of the Borel transform. In particular, the Borel transform of this formal solution admits endless analytic continuation with the following remarkable geometric structure: it naturally defines a global multivalued singular function on an algebraic surface isomorphic to the Fermat quintic surface x^5 + y^5 + z^5 = 0 modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with the generic fibre a smooth quintic curve. The locus of singularities of the Borel transform intersects each fibre in exactly five points. Each fibre has a fivefold branched covering map to a complex plane (the Borel plane) with five ramification points of order five which are precisely the five singular points of the Borel transform. Based on joint work in preparation with M. Alameddine, O. Marchal, and N. Orantin.

 20/03/2025, 15:00h

Deformations and quantizations of moduli spaces of meromorphic connections on curves 

The de Rham/Betti spaces of wild nonabelian Hodge theory parameterize isomorphism classes of meromorphic connections on principal bundles (or their monodromy/Stokes data). When the base of the bundle is a Riemann surface, they assemble into local systems of (complex) Poisson/symplectic algebraic varieties over any `admissible' deformation of the base: this goes beyond the deformations of pointed Riemann surfaces, due to the additional local moduli of irregular-singular connections at each pole, which lead in particular to cabled braid groups. Moreover, it possible to quantize the moduli spaces of meromorphic connections to construct (projectively) flat vector bundles over the same base, which instead lead to conformal blocks in the WZW model.

In this talk we will aim at a review of (a small) part of this story, and then describe extensions in the direction of:

(i) deformation quantization (based on past work with D. Calaque, G. Felder, R. Wentworth);
and (ii) the twisted/ramified case (based on work in progress with J. Douçot and past work with P. Boalch, J. Douçot, M. Tamiozzo).