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.

Encontros 2018

# Sexta-feira 30/nov @ UFF, IME, Campus Gragoatá, Niterói.

There is well-known correspondence between linear vector fields on a vector bundle E (i.e. fields whose flow is given by bundle automorphisms) and on its dual E*. In this talk, we show that such correspondence extends to vector valued forms on E and on E*. In the case of (1,1) tensors K: TE --> TE, we show that the complex structures associated to holomorphic structures on E and on E* are dual to each other in this sense. Also, the tangent and cotangent lifts of (1,1) tensors of the base manifold. As an application, we revisit the theory of Poisson-Nijenhuis (PN) explaining the concomitant in light of such duality relation and also giving the infinitesimal description of PN groupoids.

# Sexta-feira 26/out @ UFRJ, sala C116, Instituto de Matemática, Fundão.

A classical theorem of Lefschetz asserts that if a smooth projective variety is intersected transversely by a hyperplane, the homotopy of the variety relative to the intersection vanishes up to the middle dimension. We shall discuss an appropriate generalization where projective submanifolds are replace by certain foliated manifolds, and where we seek to control the relative homotopy of every single leaf.

In this talk, I will describe how the Arnold-Liouville theorem can be used to describe many lagrangian products as toric domains. As an application, we can study the ridigity and flexibility of many symplectic embedding problems.

# Sexta-feira 28/set @ PUC-Rio, room L856, Mathematics Department.

Contact groupoid actions appear as a natural (and much richer) generalization of contact group actions. As in the symplectic case, where much of the geometry of a symplectic manifold with a Hamiltonian action is encoded in the Hamiltonian moment map to the dual of the Lie algebra, in contact geometry a contact manifold with a contact group gives rise to a (Hamiltonian version of the moment) map with target the projectivization of the dual of the Lie algebra -- a Jacobi manifold which plays the role of the linear Poisson structure of the dual of a Lie algebra, in the context of Jacobi geometry -- and much of the contact geometry is encoded in this map. Contact groupoid actions fits in the setting of contact dual pairs, and this gives a nice explanation of the relation of the contact geometry and the moment maps.

Knot contact homology (KCH) is an invariant of knots in the 3-sphere, and its definition combines the dynamics of the geodesic flow on the sphere and pseudoholomorphic curves. Recent developments in mathematics and theoretical physics have unveiled surprising relations between KCH, mirror symmetry and large N duality (the latter relates colored HOMFLY polynomials and open Gromov-Witten invariants in the resolved conifold). This talk will have the more modest goal of explaining a formula relating KCH (more specifically, the augmentation polynomial) and the Alexander polynomial of the knot. We will begin with an introduction to KCH. This is joint work with Tobias Ekholm.

# Terça-feira 14/agosto @ IMPA, Auditório 3.

A Weinstein domain is a symplectic thickening of a singular CW complex called its skeleton. Floer theory on a Weinstein domain offers a way of looking at the geometry of the loop space of the skeleton. While this story reduces to the celebrated Viterbo theorem in the case of cotangent bundles, it is much more interesting and less explored in the case when the skeleton is singular. For example, using this circle of ideas, Shende and Ekholm-Shende-Ng proved in 2017 that the Legendrian conormal of a knot detects the smooth isotopy class of the knot. I will present an overview of these results.

I will survey some results on the multiplicity and stability of periodic orbits of Reeb flows on the standard contact sphere. After that, I will discuss recent results, obtained in an ongoing joint work with V. Ginzburg, on the multiplicity of periodic orbits on symmetric strongly dynamically convex spheres and examples of dynamically convex spheres in $\R^{2n)$ ($n>2$) that are not equivalent to convex ones via symplectomorphisms that commute with the antipodal map.

# Quinta-feira 05/julho @ UFF, Sala 407 bloco H.

I will discuss a joint result with Mike Usher, showing that many toric domains $X$ in the 4-dimensional euclidean space admit symplectic embeddings $f$ into dilates of themselves which are knotted (i.e. non-equivalent to the inclusion) in the strong sense that there is no symplectomorphism of the target that takes $f(X)$ to $X$.

In this talk, we will review the notion of Dirac structure, which arised in the context of constraints in mechanical systems, and we will detail instances in which they provide the 'right' perspective on infinite dimensional Poisson geometry. We also describe a reduction procedure and apply it to obtain finite dimensional geometric structures out of infinite dimensional ones. Finally, we comment on applications to the description of symplectic/Poisson structures on moduli spaces of flat connections over surfaces. This is based on joint work with M. Gualtieri and E. Meinrenken.

# Quinta-feira 07/junho @ UFRJ, sala C116, Instituto de Matemática, Fundão.

Os sistemas Hamiltonianos integráveis (de dimensão finita) surgem naturalmente no estudo da mecânica clássica por ser modelos e aproximações de vários modelos físicos (e.g. spinning tops, átomos de hidrogénio em campos elétricos e magnéticos fracos, etc.). Do ponto de vista matemático, tais sistemas podem ser vistos como ações Hamiltonianas "integráveis" do grupo de Lie abeliano simplesmente conexo, isto é, os espaços reduzidos pela ação são de dimensão zero. Neste sentido, os sistemas Hamiltonianos integráveis generalizam as ações tóricas integráveis, que foram completamente classificadas em trabalhos do Delzant (no caso em que a variedade simplética é compacta) e de Karshon e Lerman (no caso geral), usando trabalhos fundamentais de Atiyah, Guillemin-Sternberg and Marle. A classificação dos sistemas Hamiltonianos integráveis é uma pergunta guia (e muito difícil!) em mecânica Hamiltoniana e geometria simplética, com conexões a muitas outras áreas da matemática, da mecânica quantíca, da física e da espectroscopia. O objetivo desta palestra é apresentar algumas perguntas básicas (abertas) sobre a classificação destes sistemas a nível local (isto é, perto de um ponto ou de uma órbita compacta), semiglobal (perto de uma fibra da aplicação momento) e global. Se houver tempo, alguns resultados novos de natureza semiglobal serão apresentados. O ponto de vista e os resultados apresentados são frutos de conversas com o Rui Loja Fernandes, Susan Tolman and San Vu Ngoc, entre outros.

In a recent work with R. Fernandes we show that a compact Hausdorff foliation over a compact connected manifold is rigid, in the sense that every one-parameter deformation of it is trivial. We study the foliation by means of its holonomy groupoid, a Lie groupoid integrating it, and combine some classical stability properties of foliations with our linearization results for Lie groupoids involving Riemannian metrics. In this talk I will present our theorem, relate it with classic results, and discuss cohomological aspects.

# Sexta-feira 11/maio @ PUC, Sala de reuniões do Decanato, andar 12 predio Leme.

In this talk, I will start by describing a linear flow which encapsulates the asymptotic dynamics of flows on polytopes along the heteroclinic network formed by polytope's edges. Using this method informations such as detection of chaotic behavior and existence of normally hyperbolic stable and unstable manifolds associated to heteroclinic cycles along the polytope's edges can be obtained. In the second part, I will introduce the polymatrix replicator systems which contains several important models in Evolutionary Game Theory. Studying these systems were the main motivation for this work. I will also talk about the Hamiltonian character of these systems which will include subjects in Poisson reduction, linear Dirac structures and linear big-isotropic structures. At the end, it will be shown that in the case of Hamiltonian Polymatrix games, the linear flow encapsulating the asymptotic dynamic inherits the Hamiltonian character of the Polymatrix game. I will illustrate the results with an example. This is a joint work with Pedro Duarte and Telmo Peixe From Lisbon University.

Nesta palestra vamos introduzir o espaço de moduli de conexões logarítmicas com $n$ polos e com resíduos fixados sobre uma curva de gênero $g$. Tal espaço admite uma estrutura de variedade algébrica, simplética de dimensão 2N, onde N=3g-3+n. Pretendemos descrever esta estrutura no caso n=2 e g=1. Em particular obtemos um resultado do tipo Torelli que diz que levando em consideração a estrutura simplética, podemos recuperar os dados iniciais: a curva com dois pontos marcados, o divisor de polos e os resíduos. Trabalho em colaboração com Frank Loray.

# Terça 3/abr @ IMPA, Auditório 1.

Local Lie groups is an old subject which has attracted much attention recently due to the solution of the local version of Hilbert's fifth problem by Golbring (see, e.g., the recent monograph by T. Tao). In this talk, I will give a survey of the theory of local Lie groupoids, which is the subject of the thesis of my PhD student Dan Michiels. One particularly important new aspect that will be discussed is the relationship between monodromy (obstructions to integrability of a Lie algebroid) and associators (obstructions to globalizability of a local groupoid).

I will start by reviewing the generalised Kahler geometry and explain the original idea behind the generalized Kahler potential. I will give the mathematical explanation for generalized Kahler potential when the generalized Kahler geometry of symplectic type. The construction is related holomorphic symplectic Morita equivalence. This talk is based on the joint work with Francis Bischoff and Marco Gualtieri.