Title: WKB Asymptotics of Stokes matrices and rhombus inequalities
Abstract: Stokes matrices of first order ODEs are conjectured to have WKB type asymptotics with respect to the small parameter in front of the derivative term. For the simple case of a rank n system on P^1 with a simple pole at zero and a double pole at infinity, the Boalch Theorem in Poisson Geometry implies an interesting system of rhombus inequalities on leading WKB exponents. For very large values of the coefficient in front of the double pole (the caterpillar line), these inequalities reduce to Cauchy interlacing inequalities. We show that near the caterpillar line one can relate these inequalities to positivity of certain periods on the spectral curve of the system. Our analysis relies on the Gaiotto-Moore-Neitzke spectral network theory which makes predictions for WKB asymptotics in terms of the data of the spectral curve.
The talk is based on a joint work with A. Neitzke, X. Xu, and Y. Zhou, see https://arxiv.org/abs/2403.17906.
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: Enhancements of symplectic groupoids and semiclassical limit of star products
Abstract: In this talk, we will introduce a notion of "enhanced" symplectic groupoid which is motivated by the semiclassical limit of Fourier Integral operators. The idea is that the graph of groupoid multiplication, which is a Lagrangian submanifold, is now enhanced with a half-density which must satisfy a non-linear associativity condition. Our main result yields the existence and classification of such enhancements. Finally, we apply this formalism to understand the "1-loop diagrams" contribution of Kontsevich's quantization formula on R^n. This is joint work with Gabriel Ledesma.
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: Batalin-Vilkovisky and hypercommutative algebras in complex geometry
Abstract: I will review some constructions of BV and hypercommutative algebras for manifolds with additional geometric structures, ranging from Poisson to Hermitian and generalized complex manifolds. Such algebra structures are related to the extended deformation theory introduced by Barannikov and Kontsevich for Calabi-Yau manifolds. I will explain how, using mixed Hodge theory at the homotopical level, one can prove hypercommutative formality of compact Kähler manifolds. This talk includes joint results with Geoffroy Horel and with Scott Wilson.
Title: Shadowing of non-transversal heteroclinic chains in lattices
Abstract: In this paper we deal with dynamical systems on complex lattices possessing chains of non-transversal heteroclinic connections between several periodic orbits. The systems we consider are inspired by the so-called toy model systems (TMS) introduced in the paper `Transfer of energy to high frequencies in the cubic defocusing NLS equation', J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Invent. Math. (2010) 181:39-113, used to prove the existence of energy transfer from low to high frequencies in the nonlinear cubic Schrödinger equation (NLS) or other generalizations. Using the geometric properties of the complex projective space as a base space, we generate in a natural way collections of such systems containing this type of chains, both in the Hamiltonian and in the non-Hamiltonian setting. On the other hand, we characterize the property of block diagonal dynamics along the heteroclinic connections that allows these chains to be shadowed, a property which in general only holds for transversal heteroclinic connections. Due to the lack of transversality, only finite chains are shadowed, since there is a dropping dimensions mechanism in the evolution of any disk close to them. The main shadowing technical tool used in our work is the notion of covering relations as introduced by one of the authors.
Title: The cosymplectic Chern--Hamilton conjecture
Abstract: We study a functional on compatible metrics on compact, 3-dimensional, cosymplectic manifolds. It generalizes the functional studied by Chern and Hamilton in 1984 for contact manifolds.
We classify which manifolds admit critical metrics: they are either co-Kähler or homogeneous spaces of AO(1, 1). In the first case, the Reeb vector field is Killing, in the second it is algebraic Anosov and the suspension flow of a hyperbolic toral automorphisms of the 2-torus. We also show that any critical metric minimizes the energy functional.
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: An overview of b-contact dynamics
Abstract: Singularities in contact forms of b- type or of other similar tractable types appear when one compactifies (some) energy levels of many important Hamiltonian systems. This motivates the study of b-contact dynamics. After considering a few examples from physical systems, we formulate an analog of the Weinstein conjecture for closed b-contact manifolds: the singular Weinstein conjecture. This conjecture turns out to be false, and we give a counterexample at the end of the talk. To get to this counterexample, we give a detailed description of the dynamics on the singular hypersurface and we give a sort of plug construction by inserting a "singular bubble" into a contact manifold via contact surgery. This talk is based on joint work with Eva Miranda, Cédric Oms, and Daniel Peralta-Salas.
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: E-symplectic and almost regular Poisson manifolds
Abstract: This talk investigates interactions among three notions; almost regular foliations, and two generalizations of b-symplectic manifolds: E-symplectic manifolds and almost regular Poisson structures. E-manifolds are almost regular foliations with a symplectic structure. They give rise to a Poisson manifold. Almost regular Poisson manifolds are Poisson manifolds whose symplectic foliation is almost regular. These two notions differ but they are deeply connected. The holonomy groupoid of the almost regular foliation have a natural Poisson structure that one can write explicitly. In some cases this Poisson groupoid gives enough information to get the Symplectic groupoid integrating the underlying Poisson manifold.
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: Approaching the continuum: convergent discretizations of a Riemannian manifold.
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Title: Chasing Symmetries: special functions and quantum algebra
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Title: Singular Symplectic Geometry and the Poisson Frontier
Abstract: Drawing on examples from pseudo-Riemannian geometry, celestial mechanics, Painlevé equations, and deformation quantization, I will explore and compare several classes of singular symplectic manifolds, such as b/log-symplectic and folded symplectic manifolds. This framework will set the stage to tackle exciting open challenges, including defining the Symplectic Topology program counterpart for Poisson manifolds.
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: Isomorphisms and characteristic classes of singular tangent bundles
Abstract: Singular tangent bundles have arisen as one framework to develop a calculus for manifolds with singularities. Notorious instances of this construction are the study of escape orbits in celestial mechanics and the blow-up of certain almost-complex structures in twistor spaces.
In this talk we discuss the characteristic classes and obstructions to isomorphism for these bundles. We show any $b^m$-tangent bundle can be described in terms of the standard or $b$-tangent bundle. This result is intimately related to the desingularization procedure of Guillemin, Miranda and Weitsman. We prove that the existence of an isomorphism between standard and $b$-tangent bundles is obstructed by a combinatorial gadget called the associated graph. In general, there are additional topological obstructions. We give a criterion for isomorphism when the base manifold is a sphere and discuss the relationship of certain algebraic invariants, such as characteristic classes, between both bundles. We conclude the talk with some remarks on the topology of edge-tangent bundles.
This talk is based on joint work with Eva Miranda.
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: Higher Bessel, N-valued groups and symmetric polynomials
Abstract: I shall discuss several results about multiplication kernels, which include the famous Sonine –Gegenbauer formulas, examples of polynomials for symmetric Buchstaber–Kontsevich polynomials given as addition laws for special 2-valued formal groups (Buchstaber–Novikov–Veselov) as well as a connection of discriminant-type singularity loci for the N-Bessel kernel with N-valued multiplication laws given by Buchstaber-Rees polynomials.
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: Knots-quivers correspondence and beyond
Abstract: I will review the status of the knots-quivers correspondence, which – as the name indicates –is a relation between quiver representation theory and knot theory. While it follows from physical considerations and properties of brane systems in string theory, it also leads to quite non-trivial statements that connect various areas of mathematics. To start with, it was shown that this correspondence enables to express various invariants of knots in terms of invariants of quivers. Subsequently the relation to quivers has been generalized to other systems, in particular 3-manifolds and toric Calabi-Yau manifolds. These developments also turn out to be related to (and also enable to solve) problems in combinatorics, number theory, conformal field theory, and other fields. In this talk I will review various results and open problems in this research area.
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.
Title: Generalizes Kahler geometry and sigma models
Abstract: I will try to give an overview of the relation between the generalized Kahler geometry and two dimensional supersymmetric sigma models stressing the interplay between local and global issues. I will discuss some latest developments related to the understanding of generalized Kahler potential.