Speakers

Title: Painlevé monodromy manifolds and related structures: towards non-commutative setting 

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.

Title: Contact topology and symplectic invariants in hydrodynamics 

Abstract: More than twenty years ago, in their "Contact Topology and Hydrodynamics" saga, Etnyre and Ghrist introduced a fruitful connection between contact topology and the study of stationary solutions of the 3D Euler equations for ideal fluids. After a historical review of this connection, I will present a completely new framework that allows assigning contact/symplectic invariants to large sets of time-dependent solutions to the Euler equations on any closed Riemannian three-manifold. Applications include a general non-mixing result for the infinite-dimensional dynamical system defined by the equation and the existence of new conserved quantities of the PDE obtained from Floer theories in contact topology. This is based on joint work with Francisco Torres de Lizaur. 

Title: Stokes data as representations of real decorated blowups 

Abstract: This talk will focus on the study of a specific example of a meromorphic connection on a vector bundle of rank 3 on the projective line. More precisely, it will be shown how the Stokes data arising from the higher order pole can be computed exploiting the decorated real oriented blowup of the surface involved and Fock-Goncharov theory. 

Title:  Gravitating vortices and symplectic reduction by stages 

Abstract: The self-dual Einstein-Maxwell-Higgs equations describe a special instance of Einstein’s field equations of gravity in four dimensions, coupled to a electromagnetic field and an abelian Higgs field, which saturate a Bogomol’nyi energy bound. Via a natural ansatz, due to Comtet and Gibbons, their solutions can be recast as vortices on a Riemann surface with back-reaction of the metric, known as `gravitating vortices'.

We undertake a novel approach to the existence problem for gravitating vortices based on symplectic reduction by stages. The main technical tool for our study is the reduced α-K-energy, for which we establish convexity properties by means of finite-energy pluripotential theory. Using these methods, we prove that the existence of solutions to the gravitating vortex equations on the sphere implies the polystability of the effective divisor defined by the zeroes of the Higgs field. This approach also enables us to establish the uniqueness of gravitating vortices in any admissible Kähler class, in the absence of automorphisms. Joint work with L. Álvarez-Cónsul, O. García-Prada, V. 

Pingali, and C. Yao, in arXiv:2406.03639. 

Title: Dynamics of computable functions 

Abstract: The interplay between computability and dynamics has fascinated mathematicians and computer scientists since the inception of the first models of computation. Recently, renewed interest has been sparked in this relationship due to T. Tao's programme to find blowing-up solutions of the Navier-Stokes equations via an embedded computer and the relation between static solutions and contact structures. In this context, several fascinating works have emerged in recent years, demonstrating that Turing machines can be simulated using the flow lines of various interesting dynamical systems.

In this talk, we will present a novel approach to studying computability in dynamics, inspired by Topological Quantum Field Theory. We will prove that any computable function can be represented through the flow of a volume-preserving vector field on a smooth bordism. This result introduces a new model of computation and opens up exciting perspectives on computability, revealing deep connections between the topological properties of the flow, the existence of compatible contact-like geometric structures on the bordism, and the computational complexity of the function.

Joint work with E. Miranda and D. Peralta-Salas 

Title: Chasing Symmetries: special functions and quantum algebra

Abstract:

Title: Actions of finite abelian groups on weak Lefschetz cohomologically symplectic manifolds 

Abstract:  I will discuss some results on actions of finite abelian groups on weakly Lefschetz cohomologically symplectic (WLS) manifolds, a collection of topological manifolds that includes all compact connected Kaehler manifolds. Let X be a WLS manifold. 

The main results are: (1) for a big enough prime p, if G=(Z/p)^m acts on X, then there is a splitting G=G_0 x G_1 such that G_0 acts freely on X and the action of G_1 on X has a fixed point, (2) for a big enough prime p, if (Z/p)^k acts freely on X, then the sum of the Betti numbers of X is at least 2^k. The combination of (1) and (2) gives an analogue for finite group actions of an old theorem of V. Ginzburg on symplectic actions of tori, which was extended by Lupton and Oprea to continuous actions of tori. 

Title: New local invariants in generalized complex geometry 

Abstract: After recalling some classical geometric structures, I will review generalized complex geometry for even-dimensional manifolds and introduce its extension to manifolds of any dimension, known as Bn-generalized complex geometry. Then, I will focus on the case of 3-manifolds and mention the novel appearance of local invariants, which is joint work with Joan Porti. 

Title: Double symplectic groupoids in a braided world 

Abstract: In Poisson geometry there are two basic "integration" constructions: any Poisson manifold integrates to a local symplectic groupoid and any Lie bialgebra to a Poisson-Lie group. These two facts were united by A. Weinstein and by K. Mackenzie and P. Xu to a picture, where at the top we have a double symplectic groupoid, at the bottom a Lie bialgebroid, and on the sides two Poisson groupoids. I will talk about a cute generalization of this to the world of quasi-Hamiltonian spaces, which form a braided monoidal category and thus somewhat entangle the picture. A part of the motivation is coming from braided Hopf algebras, and examples from moduli spaces of flat connections. 

Title: Lagrangian Rabinowitz Floer homology in celestial mechanics 

Abstract: The origins of the field of symplectic geometry and Hamiltonian dynamics go back to early nineteenth century when Lagrange introduced the notion of a symplectic manifold to study the three body problem. The three body problem considers the motions of three heavenly bodies under the influence of their gravitational field. Our project is motivated by the following practical question: Can we send a rocket between any two points in the gravitational field of the Moon and the Earth, using the engines only at the beginning and at the end of the journey? In my talk I will explain how to answer this question using the powerful and robust techniques of Floer theory, which combines tools from algebraic topology (Morse homology) and variational analysis (action functional) to investigate the evolution of Hamiltonian systems in time.

Joint work with: Kai Cieliebak, Urs Frauenfelder and Eva Miranda.