Abstracts

September 13

Ioan Marcut

Title: A local model around Poisson submanifolds

Abstract: The first jet of a Poisson tensor at a fixed-point encodes precisely the structure of a Lie algebra (i.e. the isotropy Lie algebra). The linear Poisson structure on the dual of this Lie algebra (aka the Kirillov-Kostant-Souriau Poisson structure) represents the first order jet approximation of the Poisson tensor. Much more intricate is the semi-global version of this construction due to Yuri Vorobiev, which provides a first order approximation for a Poisson structure around a symplectic leaf. Here the first order jet is encoded by a transitive Lie algebroid.In this talk I will explain a similar model for first order approximations of Poisson structures around Poisson submanifolds. This

model generalizes Vorobiev's construction, it depends only on the first jet of the Poisson structure, it is unique up to isomorphisms, but does

not always exist. I will also discuss existence criteria. This is joint work with Rui Loja Fernandes.

Cristian Cárdenas

Title: Rigidity of Lie groupoid morphisms and Lie subgroupoids from a cohomological approach

Abstract: Rigidity is a typical question when dealing with the deformation theory of mathematical structures. Intuitively, an object $X_0$ is rigid if for every deformation $X_\epsilon$ into which it fits, it is true that $X_\epsilon$ is isomorphic to $X_0$. In this talk, I will first make an overview on the deformation theory of Lie groupoids developed in [1]. I will then present the deformation cohomology of a Lie groupoid morphism and show how to use it to prove rigidity results of morphisms and Lie subgroupoids through Moser path arguments. Moreover, several properties of this deformation cohomology will also be explained. If time permits, I will explain how to obtain a deformation complex for multiplicative forms on a Lie groupoid from the deformation complex of morphisms.

[1] Crainic, M., Mestre, J. N., & Struchiner, I. (2020). Deformations of Lie groupoids. International Mathematics Research Notices, 2020(21), 7662-7746.

Elizabeth Gasparim

Title: Poisson structures on local Calabi-Yau threefolds

Abstract: I will discuss some aspects of deformation theory of noncompact manifolds, from both the commutative and the noncommutative viewpoints. I will then present explicitly new families of Poisson structures on local Calabi-Yau threefolds, including the resolved conifold. This is joint work with Bruno Suzuki. The main feature of our work is that for each manifold we construct not just one, all possible Poisson structures.

Clarice Netto

Title: Dirac-Nijenhuis structures

Abstract: We will introduce a notion of compatibility between Nijenhuis operators and Dirac structures extending the definition of Poisson-Nijenhuis and presymplectic-Nijenhuis structures introduced by Magri and Morosi in the context of integrable systems. The main goal is to discuss the integration of Dirac-Nijenhuis structures to presymplectic-Nijenhuis groupoids, and present the important case when the Nijenhuis tensor is a holomorphic structure. The talk is based on joint work with H. Bursztyn and T. Drummond.

September 14

Jiang Hua Lu

Title: On some examples of a theorem of D. Alvarez

Abstract: We discuss some examples of a theorem of Daniel Alvarez on symplectic groupoids inside Poisson groupoids.

Alejandro Barbosa

Title: Equivariant Cohomology for Differentiable Stacks

Abstract: We introduce the concept of equivariant cohomology in the smooth manifold case and the notion of differentiable stacks. Then we consider an action of a Lie group on a differentiable stack in the sense of Romagny and consider the stacky quotient associated to this action. Consequently, we construct an atlas that makes these stacky quotients a differentiable stack. Using that the nerve of the associated Lie groupoid of that stack gives its homotopy type, we provide a Borel model for equivariant cohomology in this context.

Dan Aguero

Title: New invariants and local description of complex Dirac structures.

Abstract: Complex Dirac structures are Dirac structures in the complexied generalized tangent bundle. The main examples of these structures include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal.

Chiara Esposito

Title: [Formality, Reduction]? First step

Abstract: In this talk we propose a reduction scheme for multivector fields phrased in terms of L-infinity-morphisms. First, using geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of DGLA’s. As a second step, we construct a Poisson analogue of the Cartan model for equivariant de Rham cohomology. As a consequence we prove the existence of a curved L-infinity morphism between equivariant multivector fields and multivector fields on the reduced manifolds that coincides with the standard Marsden–Weinstein reduction.