Seminário Simplético
Conjunto no Rio

O Seminário Simplético Conjunto é uma iniciativa de pesquisadores do IMPA, PUC-RJ, UFF e UFRJ, com encontros mensais rotativos. Cada encontro conta com duas palestras, ministradas por pesquisadores locais ou convidados, em temas relacionados à geometria simplética (num sentido amplo). Alunos são particularmente encorajados a participar.

The Joint Symplectic Seminar is organized by IMPA, PUC-RJ, UFF and UFRJ, and consists of monthly meetings that alternate among these institutions. Each meeting features two talks on topics related to symplectic geometry (in a broad sense). Students are particularly encouraged to attend the seminars.

2020 2019 2018 2017 2016 2015 2014


Encontros 2014
# Sexta 21/nov @ IMPA.
15:30. Nicolas Puignau (UFRJ) Soma simpletica real e aplicacao à geometria enumerativa das variedades simpleticas reais.
Descreveremos uma versao real da soma simpleica de 4-variedades simpleicas reais que consiste em "simular" uma degeneracao nodal. Como aplicacao estudaremos os invariantes de Welschinger, ilustrando os resultados em superfiies racionais reais. Esses invariantes sao analogos reais dos invariantes de Gromov-Witten pontuais em genero zero.
17:00. Pedro Salomão (USP) Uma desigualdade sistolica para fluxos geodesicos na 2-esfera.
Considere uma metrica Riemanniana na esfera de dimensao 2. A razao sistolica desta metrica e' dada pela razao entre o quadrado do comprimento da geodesica mais curta e a area da esfera. Provamos entao o seguinte teorema: se a curvatura Gaussiana e' positiva e pincada por uma constante maior que 0.84, entao a razao sistolica e' menor ou igual a pi, sendo a igualdade valida se e somente se a metrica e' Zoll. Este resultado implica, em particular, a seguinte conjectura de Balacheff: a metrica standard e' um ponto de maximo local para a razao sistolica. O trabalho tem a colaboracao de A. Abbondandolo, B. Bramham e U. Hryniewicz.
# Sexta 10/out @ IMPA.
14:30. Roberto Rubio (IMPA) Generalized complex geometry guided tour.
We will start this 60-minute tour by introducing the basics of generalized complex geometry: the generalized tangent space, the B-field action, differential forms as spinors, generalized complex structures and the type change locus. We will then have the chance to appreciate the beauty of manifolds which are neither complex nor symplectic but are generalized complex. We will finish our tour by peeping at Bn generalized complex geometry, a recent twist on this theory. All the prerequisites are included in the tour. Meeting point: IMPA Möbius band sculpture.
16:00. Maria Amelia Salazar (IMPA) About multiplicative structures on Weinstein groupoids: integrating compatible tensors on Lie algebroids.
Some interesting geometric structures (e.g. Poisson and Jacobi manifolds) can be encoded using Lie algebroids together with a fixed tensor. For these geometries one can look for desingularisations in the following sense: as in the Lie algebra case, using the space of paths of the Lie algebroid, one constructs a groupoid and a multiplicative tensor integrating the Lie algebroid and its tensor, respectively. When this groupoid is smooth, one obtains the desired desingularisation, for example, the symplectic groupoid of a Poisson manifold. In this talk, I will explain in more detail how to integrate to a possible non smooth groupoid tensor on a Lie algebroid (which then will be a multiplicaitve tensor on the groupoid) and tell about some possible applications, for example how to use sprays to construct local Lie groupoids and local models for interesting geometries. This is an ongoing work with Alejandro Cabrera.
# Segunda 08/set @ UFF, Auditorio da Posgraduação, 7mo andar.
14:30. Daniele Sepe (UFF) On leaf spaces of completely integrable Hamiltonian systems
A celebrated theorem due to Delzant (who built on previous results due to Atiyah, Guillemin and Sternberg) states that a compact symplectic toric manifold is completely determined by the image of its moment map, which, in and of itself, has some remarkable combinatorial properties. Completely integrable Hamiltonian systems (on manifolds that are not necessarily compact) are a natural generalisation of symplectic toric manifolds, as they can be thought of as `well-behaved' Hamiltonian R^n-actions. This talk describes some properties of the leaf space associated to a completely integrable Hamiltonian system in some special cases: in the toric setting, this space can be identified with the image of the moment map and, in general, it shares some features with it. This is (very much!) ongoing work in progress with Rui Loja Fernandes and (partly) with Eugene Lerman.
16:00. Marta Batoreo (IMPA) On (non-contractible) hyperbolic periodic orbits and periodic points of symplectomorphisms.
We present a result on the existence of infinitely many (non-contratible) periodic orbits of certain symplectomorphisms isotopic to the identity if they admit at least one hyperbolic (non-contactible) periodic orbit. It holds for a certain class of closed symplectic manifolds and the main tool used in the proof is Floer-Novikov homology for (non-contractible) periodic orbits. This generalizes a result by Ginzburg-Gurel.
# Segunda 26/mai @ PUC-RJ , Sala 856L.
14:30. Rui Loja Fernandes (U. of Illinois at Urbana-Champaign) Poisson manifolds of proper type.
Lie algebras of compact Lie groups have very nice properties and can be completely described/classified. A Poisson manifold is a special kind of infinite dimensional Lie algebra and there is a group-like object associated with it, which can be obtained as a symplectic quotient. When this group-like object is "compact" one expects the Poisson manifold to have very nice properties. In this talk I will discuss these Poisson manifolds of proper type and some of its nice properties. Joint work with Marius Crainic and David Martinez-Torres.
16:00. Umberto Hryniewicz (UFRJ) Homologia de Morse, equivariante e invariante.
Dada uma variedade compacta e sem bordo onde age um grupo finito, discutiremos a possibilidade de encontrar um par Morse-Smale invariante. Em outras palavras, estudaremos um problema de transversalidade com simetrias. Esta construção nos permite definir o que chamamos de homologia de Morse invariante, a qual mostraremos ser isomorfa à homologia equivariante. Uma versão local deste tipo de problema aparece naturalmente na definição da homologia de contato local. Estes resultados foram obtidos em conjunto com Doris Hein e Leonardo Macarini.
# Segunda 28/abr @ UFRJ, Sala C-119.
14:30. Graham Smith (UFRJ) Special Lagrangian curvature.
We introduce the concept of special Lagrangian curvature of immersed hypersurfaces in Riemannian manifolds. We relate it to the almost-Calabi-Yau structure of the unitary bundle of the ambiant space. We obtain a compactness result for families of space-like special Legendrian hypersurfaces of the unitary, and we show how this projects to a compactness result for families of Weingarten hypersurfaces in the manifold. Together this allows us to prove the following nice (and utterly useless) result:
Theorem: Let $(M,g)$ be $4$-dimensional Riemannian manifold diffeomorphic to $2$-dimensional complex projective space and suppose that $g$ has positive sectional curvature. For generic perturbations of $g$, for all $\lambda>0$, there exist at least $3$ reparametrisation inequivalent locally strictly convex immersed hyperspheres in $M$ such that $K-\lambda H=0$, where $K$ and $H$ are respectively the extrinsic and mean curvatures of the hypersphere.
16:00. David Martinez Torres (PUC-RJ) Sympletic topology of b-manifolds.
In this talk we shall discuss a class of Poisson manifolds called b-symplectic manifolds. The adjective "symplectic" refers to the fact that if one modifies slightly the notion of k-form and works with the so called b-forms, b-symplectic structures are given by closed, non-degerate b-forms of degree 2. As we shall see many constructions in symplectic geometry and topology can be transferred to b-symplectic geometry.
# Segunda 31/mar @ IMPA, Sala 349.
14:30. Vinicius Gripp Ramos (U. Nantes) Homologia de contato e capacidades simpléticas.
As homologias de Floer sao invariantes que possuem diversas aplicações em geometria simplética. As homologias de contato, que são versões de contato da homologia de Floer, ajudam-nos a classificar certas variedades de contato e entender cobordismos, alem de ter inúmeras aplicações em topologia de baixa dimensão e teoria de nós. Nesta palestra, falarei sobre a homologia de contato mergulhada e de algumas de suas aplicações, como suas capacidades e consequentes obstruções a mergulhos simpléticos.
16:00. Alessia Mandini (U. Pavia) On the Gromov width of polygon spaces.
After Gromov's foundational work in in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold $(M,\omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in $(M,\omega)$. I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in $\mathbb{R}^3$ with edges of lengths $(r_1,\ldots,r_n)$. Under some genericity assumptions on lengths $r_i$, the polygon space is a symplectic manifold. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.