Workshop

VII Workshop on Poisson Geometry

and Related topics

26-28 de Junho 2019 UFPR Curitiba

Sum´

ario

Palestras

Camilo Angulo (UFF)

T´ıtulo: Compactness in Jacobi Geometry. Resumo:

A Poisson manifold of compact type is an integrable Poisson manifold, whose symplectic groupoid is compact, s-proper or proper. These were introduced and studied by Crainic, Fernandes and Martinez Torres fairly recently and have many nice properties.

In this talk, we will define the analogous concept in Jacobi geometry, motivate how such manifolds could be useful in Poisson geometry and present some of their properties.

Cristian C´ardenas (UFF)

T´ıtulo: Deformations of Lie groupoid morphisms and Lie subgroupoids. Resumo:

In this talk I will introduce the deformation complexes of Lie groupoid morphisms and Lie subgroupoids and discuss some of their properties, I will also show how these complexes govern the deformations of morphisms and Lie subgroupoids. Moreover, I will explain some rigidity results for such structures by using these complexes, and I will comment about applications on multiplicative forms. This is a joint work with Ivan Struchiner

Elder de Moraes Correa (IMPA)

T´ıtulo: Principal Torus bundles over flag manifolds and applications in Hermitian Geometry with torsion.

Resumo:

A K¨ahler manifold with torsion (KT manifold) is a Hermitian manifold (M, g, J ) with a distinguished Hermitian connection ∇B (Bismut connection) which preserves the metric g and the complex structure J .

The KT structures have been recently studied by many authors due to their applications in type II string theory and their relations with generalized K¨ahler geometry.

In this talk, I will discuss some recent results related to the explicit construction of KT structures and Calabi-Yau structures with torsion (CYT) on principal torus bundles over complex flag manifolds.

Elizaveta Vishnyakova (UFMG)

T´ıtulo: On classification of simple modules of invariant subalgebras in a skew-group ring. Resumo:

I. Gelfand and M. Zeitlin constructed a basis in finite dimensional gln(C)-modules to-gether with explicit formulas for gln(C)-action. These formulas for gln(C)-action are cal-led classical Gelfand-Zeitlin formulas. Later it was noticed by I. Gelfand and M. Graev that the classical Gelfand-Zeitlin formulas may be used to obtain a family of infinite dimensional gln(C)-modules. A general theory of the so-called Gelfand-Zetlin modules was developed by Yu. Drozd, S. Ovsienko and V. Futorny. A classification of simple Gelfand-Zeitlin gln(C)-modules was obtained by Ben Webster in 2019.

In our most recent paper with V. Mazorchuk we obtained a generalization of the Gelfand-Zeitlin theory to any invariant subalgebras in a skew-group ring and constructed a series of simple modules which we call canonical simple modules. We will discuss our most recent result in this direction: a proof that under so called Sergel’s conditions any sim-ple module is isomorphic to a canonical one and (non-explicit) classification of simsim-ple modules for invariant subalgebras in a skew-group ring.

Fernando Studzinski (USP)

T´ıtulo: On the cohomology of a representation up to homotopy. Resumo:

When working with Lie groupoids, representations up to homotopy have shown to be very useful. Even though usual representations are Morita invariant, such a result for representations up to homotopy is still open.

In this talk I will present our work on the cohomological version of the problem, building over work in progress by M. del Hoyo and G. Trentinaglia, where representation up to homotopy are regarded as simplicial vector bundles.

Starting with a representation up to homotopy, I will describe how the simplicial vector bundle is defined, then I will define a cochain complex associated with it, and I will show that it computes the groupoid cohomology with coefficients in the representation up to homotopy. Finally, I will explain how this new model can be useful to set the Morita invariance. This is joint work with M. del Hoyo and C. Ortiz.

David Martinez Torres (PUC-Rio)

T´ıtulo: Matrix decompositions and linear coordinates for the Toda vector field. Resumo:

We will discuss a couple of results on how (non-linear) matrix decomposition interacts with (linear) constraints defined by the vanishing of appropriate entries.

What lies behind these results are the search for linear coordinates for the Toda vector field and the description of global properties of manifolds linked to the Toda vector field. This is joint work with C. Tomei (PUC-Rio).

Ivan Struchiner (IME-USP)

T´ıtulo: G-structure Groupoids and Algebroids. Resumo:

G-struture groupoids are Lie groupoids whose source fibers are G-structures. Their infinitesimal counterparts are known as G-structure algebroids.

In this talk I will discuss aspects of the theory of G-structure groupoids and algebroids with an aim towards the moduli spaces they control. The talk is based on joint work with Rui Loja Fernandes.

Josiney Alves de Souza (UEM)

T´ıtulo: Dispersive invariant control systems on the Heisenberg group. Resumo:

The notion of dispersiveness in control systems is characterized by the absence of recursive properties. It implies uncontrollability and Poisson instability.

In this talk we exhibit sufficient conditions for an invariant control affine system on the Heisenberg group to be dispersive. The results consequently yield necessary conditions for controllability.

Leonardo Soriani (UNICAMP)

T´ıtulo: T-Duality and branes on nilmanifolds. Resumo:

Topological T-duality can be undestood as a Courant algebroid isomorphism. This allows one to transport generalized complex structures between T-dual spaces. Sometimes this transport has a mirror symmetric behaviour, that is, it transports a symplectic structure to a complex one.

We show that this phenomena translates very well into a Lie algebraic setting, yielding the notion of infinitesimal duality. Under reasonable assumptions, infinitesimal duality of nilpotent Lie algebras can be upgraded to actual T-duality on corresponding nilmani-folds. We also show how this machinery is used to transport generalized complex branes between T-duals.

Maria Amelia Salazar (UFF)

T´ıtulo: Van Est differentiation and integration. Resumo:

Let G be a lie group with Lie algebra g. The van Est differentiation map is a cochain map from the the Lie group cohomology of G to the Lie algebra cohomology of g. Van Est also proved that the Lie group cohomology is isomorphic to the relative Lie algebra cohomology of g with respect to a maximal compact subgroup K of G.

Miquel Cueca Ten (IMPA)

T´ıtulo: Applications of graded manifolds to Poisson geometry. Resumo:

The works of Vaintrob, Voronov, Severa and Roytenberg show that Poisson manifolds, Lie algebroids and Courant algebroids can be expressed in terms of symplectic differential graded manifolds.

In this talk we will recall this viewpoint and show how we can use graded manifolds to solve some problems in Poisson and higher geometries.

Paula Balseiro (UFF)

T´ıtulo: On first integrals of nonholonomic systems. Resumo:

In this talk we will discuss geometric aspects of nonholonomic systems with symmetries. One of the main differences between hamiltonian systems and nonholonomic systems is that the existence of symmetries does not induce, in general, conserved quantities. However, we will see how, in certain cases, these symmetries generate first integrals. To illustrate our theory we will analyze classical examples of nonholonomic systems. This is a joint work with N. Sansonetto.

Posters

Clarice Netto (IMPA)

T´ıtulo: Dirac structures & Nijehuis Tensors. Resumo:

We will present the definition of Dirac-Nijenhuis structures, which encompasses the no-tions of Poisson-Nijenhuis and presymplectic-Nijenhuis structures introduced by Magri and Morosi in the context of integrable systems. Our definition differs from other defi-nitions found in the literature, and it is based on geometric tools developed by Bursztyn and Drummond in their study of multiplicative structures on groupoids.

We study several aspects of Dirac-Nijenhuis structures, including their integration to presymplectic-Nijenhuis groupoids (extending the integration of Poisson-Nijenhuis struc-tures due to Sti´enon-Xu) and the relations with holomorphic Dirac geometry.

Dan Aguero (IMPA)

T´ıtulo: Real index one Dirac structures. Resumo:

Generalized complex structures generalize both symplectic and complex structures. They are defined just in even dimensional manifolds.

We note that by changing the real index we obtain structures that generalize both presymplectic and CR structures.

We focus on the case of real index one. We note that in this case a new invariant appears, the subtype, which determines strongly the geometry of these structures. Moreover, we prove a splitting theorem for the case of subtype one.

Juan Sebasti´an Herrera Carmona e Cristian Ortiz (IME-USP)

T´ıtulo: Characteristic classes for principal bundles over Lie groupoids . Resumo:

In this work we present a notion of principal bundle with connection over a Lie groupoid, introduced by Laurent-Gengoux-Tu-Xu.

We explain how the Chern-Weil map can be extended to this setting. Naturality and examples are discussed. Finally, we will briefly discuss possible generalizations to the framework of principal 2-bundles over Lie groupoids.

Programmation

Main Program

Workshop Poisson 2019 – Curitiba

Time Wed (26/02) Thu (27/06) Fri (28/06)

09:00 -- 09:50 Fernando Studzinski – USP

10:00 -- 10:50 Coffee Break Eder Correa - IMPA

11:00 -- 11:50 Miquel Cueca Ten - IMPA

12:00 -- 12:50 Lunch Lunch Lunch

13:00 -- 13:50 Lunch Lunch Lunch

14:00 -- 14:50 Ivan Struchiner - USP Paula Balseiro - UFF

15:00 -- 15:50 Cristian Cardenas - UFF Camilo Angulo – UFF

16:00 -- 16:50 Coffee Break Poster Section

17:00 -- 17:50

19:00 – 22:00 Event Dinner

Talk Speaker’s Name

1 Maria Amélia Salazar - UFF x

2 Paula Balseiro - UFF x

3 Ivan Struchiner - USP x

4 Cristian Cardenas - UFF x

5 David Martínez Torres - PUC x

6 Paula Elizaveta Vishnyakova - UFMG x

7 Josiney Camilo Angulo - UFF x

8 David Miquel Cueca Ten - IMPA x

9 Cristian Eder Correa - IMPA x

10 Ivan Fernando Studzinski - USP x

Leonardo Soriani Alves CAMPINAS

Josiney Alves de Souza - UEM

Elizaveta Vishnyakova - UFMG

Maria Amélia Salazar -

UFF David Martínez Torres - PUC

*Preencher a tabela abaixo com os nomes dos palestrantes*