VI Mini-Workshop em Geometria Simplética 

Mais fotos

Local: 

Universidade Federal do Espírito Santo (UFES)

Auditório do PPGMat - Edifício da Pós-graduação em Matemática

Comité científico: 

Marta Batoréo (UFES) 

Henrique Bursztyn (IMPA) 

Brayan Ferreira (UFES)

Organização local:

Marta Batoréo (UFES)

Brayan Ferreira (UFES)

Mateus de Melo (UFES)

Contatos: 

marta.batoreo@ufes.br

brayan.ferreira@ufes.br

mateus.melo@ufes.br

Apoios e suporte: 

Programa de Pós-Graduação em Matemática/UFES

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Títulos e Resumos

Nome: Alejandro Cabrera (UFRJ)

Título: On symplectic geometry, microlocal analysis and quantization of Poisson manifolds 

Resumo: This talk can be seen as a continuation of one given in a previous instance of this workshop also in Vitória, 2017. The idea is to provide an unifying survey on different topics which are motivated by the problem of quantization of Poisson manifolds. These topics include a "de-singularization" of the Poisson geometry into a symplectic one, the corresponding Lie-theoretic aspects and a link to microlocal analysis of Fourier integral operators. We shall discuss recent results on this area at the end. 

Nome: Alexsandro Schneider (Unicentro)

Título: Transverse Foliations for Two-Degree-Of-Freedom Mechanical Systems (slides)

Resumo: In this talk, I will investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential is assumed to contain a finite number of saddle points. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. We apply the results to the Hénon-Heiles potential for energies slightly above $1/6$. We also apply the results to some decoupled mechanical systems including the frozen Hill's lunar problem with centrifugal force, the Stark problem, and the potential of a chemical reaction. This is a joint work with N. de Paulo, S. Kim, and P. Salomão. 

Nome: Bruno Costa (UFSC)

Título: Gauge Theories, Minimal Coupling and Utiyama's Theorem (slides)

Resumo: We present a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we developed the theory in the context of gauge theories and deal with minimal coupling and Utiyama's theorem.

Nome: Carolina Lemos de Oliveira (UERJ)

Título: Bifurcations of finite energy foliations for Reeb flows in dimension three (slides)

Resumo: Transverse foliations for three-dimensional flows are singular foliations whose singular set consists of finitely many periodic orbits, called binding orbits, and the regular leaves are transverse to the flow. We study one-parameter families of finite energy foliations that project to transverse foliations adapted to Reeb flows. We consider parameter values at which a binding orbit bifurcates into a finite set of Reeb orbits. Under regularity conditions on the family and local hypotheses on the bifurcation, we show that the finite energy foliations persist and the new Reeb orbits become binding orbits. This talk is based on a work-in-progress joint with Pedro Salomão and Alexsandro Schneider. 

Nome: Diego Otero (UFPR)

Título: Symplectic and projective geometry of higher order variational problems (slides)

Resumo: Given an one parameter higher order variational problem we associate a curve in a divisible Grassmannian space. This curve is fanning, a regular property given by its derivatives. With this regularity property, we show that the jet prolongations define an isotropic- Lagrangian-coisotropic flag in a symplectic space. The half prolongation, called Jacobi curve, gives local minimality information of the variational problem. Conjugacy conditions that obstruct minimality are also given by this curve. This is done by studying its self-intersections, associating with the invariants (index and nullity) of the second variation. 

Nome: Kennerson Lima (UFCG)

Título: The Notion of Riemannian λ_1-extremal Metric and its Application to Compact Homogeneous Kahler Cases (slides)

Resumo: In this talk will be present a sufficient and necessary condition for a Kahler-Einstein metric on a generalized flag manifold to be extremal for the functional that assigns for each Riemannian metric its first positive eigenvalue of the associated Laplacian. The notion of extremality presented here was studied by Panelli and Podest`a in their work [1] which provides the above mentioned condition in terms of the Lie algebra which defines the flag manifold. This result was applied in the same work in order to determine the maximal flag manifold whose K¨ahler-Einstein metric is extremal. A few cases involving partial flag manifolds as well as the relationship of this notion with minimal isometric immersions into spheres will be mentioned.
References

Nome: Leonardo Macarini (IMPA)

Título: Multiplicity, stability and symmetry of periodic orbits of Reeb flows (slides)

Resumo: Reeb flows form a prominent class of Hamiltonian systems on regular energy levels. Since these flows have no singularities, the most basic dynamical objects are the periodic orbits. In this talk, I will survey some results on the multiplicity, stability and symmetry of periodic orbits of Reeb flows without assuming generic assumptions. 

Nome: Lucas Castello Branco (PUC-Rio)

Título: Complex Lagrangians in the Moduli Space of Higgs Bundles (slides)

Resumo: Motivated by mirror symmetry, we explore two classes of complex Lagrangian subvarieties inside the moduli space of G-Higgs bundles on a curve, where G is a complex reductive Lie group. One class arises from real forms of G, while the other is associated with symplectic representations. We then focus on examples where the image of these Lagrangians lies in the discriminant locus of the base of the integrable system defined by the Hitchin map. In particular, the corresponding spectral curves in these cases are non-reduced. 

Nome: Mikhail Verbitsky (IMPA)

Título: TBA

Resumo: TBA

Nome: Naiara de Paulo (UFSC)

Título: Transverse foliations in the Euler problem of two fixed centers 

Resumo: The Euler problem of two fixed centers in the plane describes the motion of a massless body (the satellite) under the influence of two fixed bodies (the primaries) which attract the satellite according to Newton's law of gravitation. This mechanical system is a simplified version of the well-known circular planar restricted three-body problem.  In this talk, I will show some genus zero transverse foliations that can be obtained in the Euler problem of two fixed centers by means of the theory of pseudo-holomorphic curves. In particular, the binding set of such a singular foliation is formed by finitely many periodic orbits of Conley-Zehnder indices 2 or 3, and the regular leaves consist of planes and cylinders transverse to the flow. This is a joint work with Seongchan Kim, Pedro Salomão, and Alexsandro Schneider.  

Nome: Paula Balseiro (UFF) 

Título: Nonholonomic momentum map reduction for constrained systems (slides)

Resumo: In this talk, we study a reduction process via a momentum map for nonholonomic systems with symmetries admitting a special type of conserved quantities. Since nonholonomic systems are not Hamiltonian, many of the properties used in the classical reduction procedure are no longer applicable.  I will explain the new tools developed to perform this reduction, focusing on the relationship between the nonholonomic momentum map and the conserved quantities. Additionally, I will present examples to illustrate this theory.  This is a joint work with Danilo Machado (UFF). 

Nome: Renato Vianna (USP)

Título: On singular Lagrangian fibrations 

Resumo: We construct singular Lagrangian fibrations on domains cotangent bundles, mostly of spheres, given as the disk cotangent bundles with respect to some metrics.  This reinterpretation recover toric domains already constructed via integrable systems in the cotangent disk bundle of a disk (Lagrangian by disk) by Ramos, and on the complement of a cotangent fibre on the cotangent disk bundle of ellipsoids of revolution. Our method provides new examples in higher dimensions and can be used to find embeddings of balls, via Traynor trick, hence providing an estimate for Gromov-width. This is joint work with Santiago Achig-Andrango and Alejandro Vicente.

Nome: Vinicius Ramos (IMPA)

Título: Geodesics on ellipsoids of revolution and toric manifolds 

Resumo: It is well-know that the geodesic flow on ellipsoids of revolution is integrable. In this talk I will explain how we can use this fact to obtain a symplectomorphism between the unit disk bundle of such an ellipsoid and a toric manifold. Using that, we can compute the Gromov width of such unit disk bundles. This is joint work with Brayan Ferreira and Alejandro Vicente. 

Eventos anteriores

V MWGS (UFSC - Blumenau, 2020)

IV MWGS (UFF, 2019)

III MWGS (PUC-Rio, 2018)

II MWGS (UFES, 2017)

I MWGS (UFRJ, 2016)