II Mini-Workshop em Geometria Simplética

Auditório do edifício do PPGFis 

(cortesia do Programa de Pós-Graduação em Física)

Marta Batoréo (UFES) 

Henrique Bursztyn (IMPA) 

Umberto Hryniewicz (UFRJ)

Marta Batoréo (UFES)

Fábio Júlio Valentim (UFES)

marta.batoreo@ufes.br

ppgmat.ufes@gmail.com 

Sociedade Brasileira de Matemática

Centro de Ciências Exatas/UFES

Programa de Pós-Graduação em Matemática/UFES

I MWGS

Alejandro Cabrera (UFRJ)

Título: Pontes entre a geometria simplética e a de Poisson 

Resumo: Nesta palestra vamos descrever vários resultados nos quais a geometria de Poisson, de natureza degenerada, se relaciona com a simplética.

Começaremos com um 'overview' rápido sobre a geometria de Poisson. Logo, iremos apresentar a noção de 'realização simplética' de uma estrutura de Poisson e ver como ela se relaciona naturalmente a outras áreas da matemática como a teoria de grupoides de Lie e a de quantização. Alguns desses resultados foram obtidos em colaboração com M.A. Salazar, I. Marcut e B. Dherin.

Alessia Mandini (PUC-RJ) 

Título: Symplectic embeddings and infinite staircases 

Resumo: McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase related to the odd index Fibonacci numbers. Infinite staircases have been shown to exist also in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3).

This talk describes joint work with Cristofaro-Gardiner, Holm, and Pires, where we find new examples of symplectic toric 4-manifolds for which the graph of the embedding capacity function has an infinite staircase. I will briefly explain the proof of the existence of these infinite staircases, which uses ECH capacities and Ehrhart quasipolynomials as its main tools. I will also explain why we conjecture that these are the only such manifolds for which an infinite staircase can occur.

Daniele Sepe (UFF) 

Título: Rigidity of symmetric Lagrangian products 

Resumo: The problem of finding obstructions to symplectic embeddings is one of the driving questions in symplectic topology. Recently, some symplectic submanifolds of the cotangent bundle to Euclidean space, known as Lagrangian products, have come to the fore in symplectic topology, primarily because of their connection to billiards. For instance, Ramos has calculated the optimal symplectic embeddings of the 4-dimensional Lagrangian bidisc into a ball and an ellipsoid. The aim of this talk is to show that for a large class of Lagrangian products of any dimension, the corresponding symplectic embedding problem is rigid, i.e. the natural inclusion is the best possible embedding. The proof of the result is inspired by Ramos' techniques and combines ideas from the theory of integrable systems with two symplectic capacities, namely the Gromov width and the cube capacity recently introduced by Gutt and Hutchings. This is joint work with Vinicius G. B. Ramos. 

David Martínez-Torres (PUC-RJ) 

Título: Non-contractible loops in the diffeomorphism group of coadjoint orbits 

Resumo: A compact connected semisimple Lie group G acts in a Hamiltonian fashion on its coadjoint orbits, i.e. G maps into the group of Hamiltonian transformations of the coadjoint orbit. McDuff and Tolman showed that the induced map on fundamental groups is injective, answering a question of A. Weinstein. In this talk I shall show that this is not quite a symplectic phenomenon, but a topological one, since the (finite) fundamental group of G already injects in the fundamental group of the group of diffeomorphisms. The proof uses a combination of technniques from classical algebraic topology, together with Schubert calculus. This is joint work with I. Mundet (Barcelona).  

Hans-Christian Herbig (UFRJ) 

Título: Hilbert series as a tool for approaching the symplectomorphism problem 

Resumo: We address the symplectomorphism problem for symplectic quotients at zero level of quadratic moment maps. With collaborators G. Schwarz and C. Seaton we obtained some general results concerning symplectomorphisms to linear symplectic orbifolds. Apart from this the symplectomorphism problem is relatively poorly understood. For getting a rough picture of the situation, we investigate invariants. The Hilbert series of the graded algebra of regular functions on the symplectic quotient is merely invariant under graded regular diffeomorphism. But it has the advantage of being amenable to computations. Among other things, we present general formulas for the first Laurent coefficients of symplectic circle quotients and finite symplectic quotients. We digress and present similar results for Hilbert series of GIT quotients by finite groups, circles and SL(2).

The formulas for SL(2) generalize formulas of David Hilbert (1892) from the case of irreducible representations to the general case. We explain how Gorensteinness is visible in the Laurent expansion of the Hilbert series. 

Keon Choi (IME-USP)

Título: Embedded contact homology and symplectic embedding problems

Resumo: Embedded contact homology is an invariant of contact 3-manifolds by M. Hutchings. It is generated by the Reeb orbits and was used by Taubes to prove the 3-dimensional Weinstein conjecture. In 2010, Hutchings showed it is also useful for studying embeddings of symplectic 4-manifolds they bound. Many new embedding results followed but there are limitations due to difficulties in computing ECH. In this talk, we survey these results and discuss a joint work with P. Salomao to obtain more refined information about a new type of embedding.

Leonardo Macarini (UFRJ) 

Título: Multiplicity of periodic orbits for Reeb flows in dimension bigger than three

Resumo: The existence and multiplicity of periodic orbits on energy levels of Hamiltonian systems is an important and long-standing question. Strong results were established when the energy level has dimension three, but the problem is much harder in higher dimensions. I will survey some recent results on this subject obtained jointly with several collaborators, including M. Abreu, V. Ginzburg, B. Gurel, J. Gutt, U. Hryniewicz and J. Kang.

Lorenzo Díaz (PUC-RJ)

Título: Espetro de Lyapunov para cociclos de matrizes 

Resumo: O foco são os  expoentes de cociclos de matrizes 2X2. 

Neste contexto existem dois conjuntos abertos cuja união é densa no espaço dos cociclos, os hiperbólicos e os elípticos. Estudaremos estes últimos e descreveremos o espectro de Lyapunov de um  subconjunto aberto e denso dos cociclos elípticos. Veremos que estes cociclos verificam uma condição fraca de "certa hiperbolicidade" e explicaremos como os resultados seguem do estudo de certos produtos tortos e resultados de aproximação ergódica.

Trabalho em conjunto com K. Gelfert (UFRJ) e M. Rams (IMPAN)

Paula Balseiro (UFF) 

Título: Hamiltonization and conserved quantities of nonholonomic systems 

Resumo: It is well known that nonholonomic systems are not hamiltonian. The "hamiltonization problem", studies when a nonholonomic system can be made hamiltonian after a process of reduction by a Lie group of symmetries.  In this talk we will discuss the connection between the hamiltonization of nonholonomic systems and the existence of conserved quantities. 

Pedro Salomão (IME-USP) 

Título: A Brouwer's translation theorem for Reeb flows

Resumo: It is well know that an area preserving diffeomorphism of the open disk has at least one fixed point. This follows directly from Brouwer's translation theorem. In this talk I will present a version of this theorem for Reeb flows on the tight 3-sphere. Namely, if the Reeb flow admits a Hopf fiber P as a nondegenerate closed orbit and if its rotation number is greater than 1, then there must exist another closed orbit P' which is simply linked to P. To see the analogy with Brouwer's translation theorem, observe that if P is the boundary of a disk-like global surface of section for the Reeb flow, then P' is the closed orbit corresponding to any fixed point of the first return map. The proof uses a version of cylindrical contact homology in the complement of a link, as introduced in [1] and developed in [2].

Referências

1. Al Momin. Contact Homology of Orbit Complements and Implied Existence. J. Mod. Dyn. 5, no. 3 (2011), 409–472.

2. U. Hryniewicz, A. Momin and Pedro A. S. Salom ̃ao. A Poincar ́e-Birkhoff theorem for tight Reeb flows on S^3, Inventiones Mathematicae 199, 2 (2015), 333–422.

Coautores: U. Hryniewicz (UFRJ) e A. Momin.

Renato Vianna (UFRJ)

Título: Lifting Lagrangians from Donaldson divisors

Resumo: A classical construction due to Paul Biran [Bir, BK] allows to lift a Lagrangian submanifold L from a Donaldson Y divisor to a Lagrangian L' in an ambient symplectic manifold X. In [BK], it is shown that if the minimal Chern number of Y is greater than 1, then the count of Maslov index 2 holomorphic disks with boundary on the lifted Lagrangian L' is equivalent to the similar count of disks with boundary on L plus one extra disk. We study this enumerative geometry problem in the case when the minimal Chern number of Y is 1. This reveals several new, previously unexplored connections it has with relative closed- string Gromov-Witten theory of the pair (X,Y). We explore applications, in particular, we use that to distinghish (up to action of Symp(X)) lifts of previously known Lagrangians.

(Joint work with: Luís Diogo, Dmitry Tonkonog and Weiwei Wu)

References

[Bir] P. Biran. Lagrangian non-intersections. Geom. Funct. Anal., 16(2):279{326, 2006.

[BK] P. Biran and M. Khanevsky. A Floer-Gysin exact sequence for Lagrangian submanifolds. Comment. Math. Helv., 88(4):899{952, 2013.

Nome                                         Afiliação

Aaron Aragon Maroja                          Universidade Federal Fluminense 

Adriano Nunes Oliveira Pinto                 Instituto Federal do Espírito Santo - Cariacica

Alejandro Cabrera                            Universidade Federal do Rio de Janeiro 

Alessia Mandini                              Pontifícia Universidade Católica - RJ

Apoenã Passos Passamani                      Universidade Federal do Espírito Santo

Crislaine Kuster                             Universidade Federal do Espírito Santo 

Daniele Sepe                                 Universidade Federal Fluminense

Davi Cabral Rodrigues                        Departamento de Física, UFES 

David Martínez-Torres                        Pontifícia Universidade Católica - RJ

Deyze Carvalho                               Universidade Federal do Espírito Santo 

Diogo Bessam                                 Universidade Federal do Espírito Santo

Etevaldo dos Santos Costa Filho              Universidade Federal do Espírito Santo

Fábio Júlio Valentim                         Universidade Federal do Espírito Santo 

Gabriel Luchini                              Universidade Federal do Espírito Santo 

Hans-Christian Herbig                        Universidade Federal do Rio de Janeiro  

Henrique Bursztyn                            Instituto de Matemática Pura e Aplicada

Isaac Torres Sales                           Universidade Federal do Espírito Santo

Joelso Giovanelli                            Universidade Federal do Espírito Santo

José Eduardo Cordeiro                        Universidade Federal do Espírito Santo

José Gilvan de Oliveira                      Universidade Federal do Espírito Santo 

Keon Choi                                    Instituto de Matemática e Estatística - Universidade de São Paulo

Leonardo Meireles Câmara                     Universidade Federal do Espírito Santo

Leonardo Macarini                            Universidade Federal do Rio de Janeiro 

Lorenzo Diaz                                 Pontifícia Universidade Católica - RJ

Marcio F Cerqueira                           Universidade Federal do Espírito Santo

Marcos Mercandeli Rodrigues                  Universidade Federal do Espírito Santo 

Paula Balseiro                               Universidade Federal Fluminense 

Pedro Salomão                                Instituto de Matemática e Estatística - Universidade de São Paulo

Renata Pilon                                 Universidade Federal do Espírito Santo  

Renato Fehlberg Júnior                       Universidade Federal do Espírito Santo 

Renato Vianna                                Universidade Federal do Rio de Janeiro 

Ricardo Soares Leite                         Universidade Federal do Espírito Santo

Rosa Elvira Quispe Ccoyllo                   Universidade Federal do Espírito Santo

Wayner Moyses Marcelino                      Universidade Federal do Espírito Santo