Seminário Simplético do Rio - 2020

Abstract: There are three important settings for studying actions of reductive Lie groups: modules, algebraic group actions, and Hamiltonian group actions. In the study of modules one encounters various constructions of nice bases which are in some sense canonical (e.g. Gelfand-Zeitlin, Lusztig). In the study of algebraic group actions canonical bases give rise to toric degenerations; deformations of the G-variety to a toric variety (cf. Caldero, Alexeev-Brion). In this talk I will discuss the symplectic analogue of these constructions: integrable systems. We show how toric degenerations give rise to integrable systems on arbitrary symplectic manifolds equipped with Hamiltonian group actions. This generalizes a family of well-known examples called Gelfand-Zeitlin integrable systems due to Guillemin and Sternberg. As a by-product, we generalize results of Harada and Kaveh on construction integrable systems from toric degenerations. This talk is based on joint work with Benjamin Hoffman. arXiv:2008.13656

Abstract: Transverse foliations for 3 dimensional flows naturally generalize the concepts of global surfaces of section and open book decompositions. These are singular foliations whose singular set consists of finitely many periodic orbits, called binding orbits, and the regular leaves are transverse to the flow. In this talk we discuss the existence of transverse foliations for Reeb flows on the tight 3-sphere. These flows include Hamiltonian flows on $\mathbb{R}^4$ restricted to star-shaped regular energy levels. We discuss the existence of a particular type of transverse foliation, called 3-2-1 foliation, which has exactly three binding orbits with Conley-Zehnder indices respectively 3, 2 and 1.

Abstract: Let K be a compact Lie group and V a unitary K-module. Considering V as a real symplectic manifold, the action of K admits a homogeneous quadratic moment map J : V → k* where k denotes the Lie algebra of K. The zero fiber Z := J−1(0) of J is K-invariant, and the real symplectic quotient M0 is defined to be the quotient of Z by the action of K. Though usually singular, the real symplectic quotient X admits several structures including that of a semialgebraic set, a symplectic stratified space, and a differentiable space. The algebra of smooth functions on M0 admits a graded Poisson subalgebra R[M0] of real regular functions which can be described using invariant theory. We will discuss recent progress on a program to understand the algebra R[M0] focusing on the case that K is the 2-dimensional torus. We will discuss a computation of the Hilbert series of R[M0] as well as the first few coefficients of the Laurent expansion in this case.

Joint work with Hans-Christian Herbig and Daniel Herden.

Abstract: A leaf-wise intersection point is a certain type of fixed point that can occur when one perturbs a dynamical system. Existence results for leaf-wise intersection points have been proved for a variety of dynamical systems on exact symplectic manifolds. In this talk, we explore the question: what can be said about leaf-wise intersection points in monotone symplectic manifolds? Adapting Floer techniques discovered by Albers, we prove an existence result for Hamiltonian flows on some negative line bundles. We finish with a discussion about the relationship between closed Lagrangian submanifolds and leaf-wise intersection points.

Abstract: I will discuss the problem of obtaining a "Holonomy Groupoid" for a singular Riemannian foliation (SRF). Throughout the talk I will try to explain why we want to obtain such a Lie groupoid by stating results which are valid for regular foliations and how they can be obtained from the Holonomy groupoid of the foliation. Although we do not yet know how to associate a holonomy groupoid to any SRF, we can obtain the holonomy groupoid of the linearization of the SRF in a tubular neighbourhood of (the closure of) a leaf. I will explain this construction.

I will not assume that the audience has prior knowledge of Singular Riemannian Foliations or of Lie Groupoids.

Abstract: In this talk I will present a systematic study of complex Dirac structures (i.e., Dirac structures in the complexification (TM ⊕ T*M) ⊗ C of the generalized tangent bundle of a manifold M) with constant real index. I will begin by introducing its invariants: the order and the type. Then I will show that every complex Dirac structure has associated a real Dirac structure and finally I will prove a splitting theorem for complex Dirac structure which extends the splitting theorem for generalized complex structures of Abouzaid and Boyarsjenko.

Abstract: O termo Dinâmica Simplética foi criado por Hofer tendo em vista as recentes aplicações de métodos em Geometria/Topologia Simplética a problemas em dinâmica Hamiltoniana. A idéia deste encontro é discutir tais problemas e métodos de maneira informal.

Abstract: We will show how to construct volume filling ellipsoid embeddings in some 4-dimensional toric domain using mutation of almost toric compactification of those. In particular we recover the results of McDuff-Schlenk for the ball, Fenkel-Müller for product of symplectic disks and Cristofaro-Gardiner for E(2,3), giving a more explicit geometric perspective for these results. To be able to represent certain divisors, we develop the idea of symplectic tropical curves in almost toric fibrations, inspired by Mikhalkin's work for tropical curves. This is joint work with Roger Casals.

Obs: The same result appears in "On infinite staircases in toric symplectic four-manifolds", by Cristofaro-Gardiner -- Holm -- Mandini -- Pires. Both papers were posted simultaneously on arXiv.

Abstract: Given a Riemannian manifold M one may attempt at exploring its geometry by sounding it with hard spheres. Regarding these spheres as the atomic constituents of a gas filling M, one can compute an equation of state relating its several macroscopic quantities such as pressure, volume, and temperature. This talk is an exercise in statistical mechanics where I shall explain how the resulting equation depends on the underlying Riemannian geometry.

Abstract: I will give a survey of results on a relatively new system of equations in $G_2$-geometry that comes from heterotic string theory. The system is interesting in that it combines ideas of special Riemannian metrics, gauge theory and generalized geometry. In particular, we will see a new means of constructing solutions, and if there remains space and time, how some of these solutions are dual to one another. This is joint work with Mario Garcia-Fernandez and Carl Tipler.

Abstract: A Poisson manifold is calibrated if the symplectic structures on leaves come from an ambient closed 2-form. This is an interesting and useful notion, but only applies to regular Poisson manifolds. We propose here a weaker notion of subcalibration, which applies to non-regular Poisson manifolds as well, and discuss how this notion relates to pencils, and its effect in Poisson homology.

Abstract: We present a description of the low-degree cohomology of Poisson structures in neighborhoods of symplectic leaves. Our approach is based on the coupling method, and on some semilocal caracterization of the Lichnerowicz-Poisson complex in terms of a bigraded complex. In particular, we apply our results to the description of the modular class, providing some unimodularity criteria for neighborhoods of symplectic leaves.

Abstract: The first part of the talk will discuss work in https://arxiv.org/abs/1908.04227 on constructing a Donaldson-Fukaya-Seidel type category for the generalized SYZ mirror of a genus 2 curve. We will explain the categorical mirror correspondence on the cohomological level. The key idea uses that a 4-torus is SYZ mirror to a 4-torus. So if we view the complex genus 2 curve as a hypersurface of a 4-torus V, a mirror can be constructed as a symplectic fibration with fiber given by the dual 4-torus V^. Hence on categories, line bundles on V are restricted to the genus 2 curve while fiber Lagrangians of V^ are parallel transported over U-shapes in the base of the mirror. Next we describe ongoing work with H. Azam, H. Lee, and C-C. M. Liu on extending the result to a global statement, namely allowing the complex and symplectic structures to vary in their real six-dimensional families. The mirror statement for this more general result relies on work of A. Kanazawa and S-C. Lau.

Abstract: We study homology classes in the mirror quintic Calabi-Yau threefold which can be realized by special Lagrangian submanifolds. We have used Picard-Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p$ we can compute the orbit modulo $p$. We conjecture that the orbit in homology with coefficients in $\mathbb{Z}$ can be determined by these orbits with coefficients in $\mathbb{Z}_p$.

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Abstract: Let G be a simply connected Lie group. We denote by C(G) the differential graded Hopf algebra of smooth singular chains on G. We will explain how the category of modules over C(G) can be described infinitesimally in terms of representations of the differential graded Lie algebra Tg, which is universal for the Cartan relations. In case G is compact, this correspondence can be promoted to an A-infinity equivalence of dg-categories. We will also explain how this equivalence is related to Chern-Weil theory and higher local systems on classifying spaces. This talk is based on joint work with Alexander Quintero Vélez.

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Abstract: We study Lie–Hamilton systems on the plane, i.e. systems of first-order (nonlinear) ODEs describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. Lie–Hamilton systems enjoy a plethora of properties, e.g., they admit their general solution expressed as a (nonlinear) function, the so-called superposition rule, of a finite set of particular solutions and some constants. Lie–Hamilton systems are important because of their appearance in the physics, mathematics and biology literature. For example, they can be used to study Milne–Pinney, second-order Kummer–Schwarz, complex Riccati and Buchdahl equations, which occur in cosmology, relativity and classical mechanics; or in the investigation of Lotka–Volterra, predator-prey or growth of a viral infection models.

In this talk, I present the geometrical properties of Lie–Hamilton systems and their application to SIS-pandemic models. Furthermore, I will present a complete classification of Lie–Hamilton systems on the plane and we will devise methods for the derivation of their solutions in terms superposition rules with the aid of the theory of coalgebras. With it, we propose a solution to the SIS-pandemic system. To conclude this talk, we will review some other geometric structures that are compatible with Lie systems, their generalization to the quantum framework, among other things.

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Abstract: An LA-groupoid is a special type of Lie algebroid that also has the structure of a Lie groupoid. In this talk, after reviewing some general aspects of these objects and motivating their study, we focus on the class of LA-groupoids whose base is a Lie group and explore their structure.

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Abstract: Hyperpolygons spaces are a family of hyperkähler manifolds, that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these space are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied.In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic -Higgs bundles over on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them.Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski -space.This is based in joint works with Leonor Godinho.

Abstract: In this talk we explain how billiard dynamics can be used to relate a symplectic isoperimetric-type conjecture by Viterbo with an 80-years old open conjecture by Mahler regarding the volume product of convex bodies.

The talk is based on a joint work with Shiri Artstein-Avidan and Roman Karasev.

Abstract: Since the introduction of Nijenhuis recursion operators as a mechanism to generate integrals in bihamiltonian systems, the study of the so-called Poisson-Nijenhuis structures has become an important area of research. In this talk, we shall focus on how the relevant compatibility equations between Nijenhuis and Poisson structures can be generalized to Lie (bi)algebroids and Dirac structures by means of new connection-like objects. An important case which will permeate the talk is the Nijenhuis operator coming from a holomorphic structure. Part of this talk will be based in joint work with H. Bursztyn and C. Netto.

Abstract: In this talk I will introduce a higher geometrical version of a Hamiltonian action. More precisely, I will speak about Lie 2-groups acting in a Hamiltonian manner on a symplectic groupoid. We will start by recalling what a Lie 2-group is and what are the main examples of these objects. Then we will explain the infinitesimal picture, namely Lie 2-algebras. We will explain the notion of a Lie 2-group action and then we will restrict our attention to symplectic and Hamiltonian actions of Lie 2-groups. Finally, a reduction procedure will be presented.

Abstract: A symplectic realization is a way to present a given Poisson manifold as a quotient of a symplectic one. Such a desingularization, which always exists, may not reflect the original dynamics, it may fail to be complete. The existence of complete symplectic realizations is equivalent to the integrability of the associated Lie algebroids. This problem is ruled by topological obstructions and forces us to deal with non-Hausdorff manifolds. In this talk I will review symplectic realizations, the interplay between Poisson manifolds and Lie algebroids, and a recent collaboration with D. Lopez where we pay special attention to Hausdorff integrations.

Abstract: In this talk, I will show some results that use pseudo-holomorphic curves in simplectizations to find global surfaces of sections. In our main theorem, we proved that a dynamically convex Reeb flow on the standard tight lens space (L(p, 1), ξ std ), p > 1, admits a p-unknotted closed Reeb orbit P which is the binding of a rational open book decomposition with disk-like pages.

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Abstract: In this talk, we will review the notion of a generating function for a local symplectic groupoid. The idea is that this is a function that generates all the groupoid structure maps. We will show that any Poisson manifold M can be integrated by a local symplectic groupoid admitting such a generating function S. When M is a coordinate space, we provide explicit formulas for S, which generalize the Baker-Campbell-Haussdorff series to non-linear brackets. We also show that a formal Taylor expansion of S yields the tree-level part of Kontsevich's star product formula. Finally, we make contact with functional calculus and comment on the relation between S and the so-called Poisson Sigma Model functional.