Differentiable Stacks, Poisson Geometry and related geometric structures

6 Feb 2022, 19:00 → 11 Feb 2022, 12:10 Europe/Zurich

Anton Alexeev (Universite de Geneve (CH)), Henrique Bursztyn (IMPA), Jiang-Hua Lu (University of Hong Kong,), Marco Gualtieri (University of Toronto), Rui Loja Fernandes (University of Illinois)

Recent days have seen a rapid developments on shifted Poisson and symplectic structures on (derived) differentiable or algebraic stacks. A differentiable stack is, roughly speaking, a Lie groupoid up to Morita equivalence, and the stack represented by a symplectic groupoid of a Poisson manifold naturally has a 1- shifted symplectic structure. There have also been remarkable recent advances in other geometries, such as Dirac geometry and generalized complex geometry, that generalize Poisson geometry and have Lie groupoids and Lie algebroids at their cores. Many basic concepts and constructions in these geometries can be rephrased using the language of differential stacks, and such reformulations put these geometric structures in vastly new perspectives and establish further connections with other fields of mathematics such as algebraic geometry, deformation theory and high category theory.

Specific topics covered :

•  Integrations of Poisson and Dirac structures
• Generalized complex geometry and mirror symmetry
• Multiplicative structures on Lie groupoids and stacks
• Shifted symplectic geometry
• Higher Lie groupoids and higher gauge theory

Recordings of first day (Passcode: h=bjn13D)

Monday 7 February

1 Minicourse on stacks in algebraic geometry vs differential geometry - session 1

Speaker: Pavel Safranov

2 What is a Poisson structure on a différentiable stack?

We will define what a Poisson structure on a differentiable stack is, the latter being seen as an equivalence class of Lie groupoids up to Morita equivalence, and explain why the notion makes sense, as well as those of vector fields and poly vector fields over a differentiable stack. Joint work with Bonechi, Ciccoli and Xu.

Speaker: Camille Laurent-Gengoux

11:00 Coffee Break

3 m-shifted symplectic Lie groupoids

Speaker: Miquel Cueca

12:20 Discussion and free time

4 Diffeological groupoids and their Lie algebroids

Speaker: Christian Blohmann

5 Lie groupoid cohomology relative to a Lie subgroupoid

Speaker: Maria Amelia Salazar

Tuesday 8 February

6 Minicourse on stacks in algebraic geometry vs differential geometry - session 2

Speaker: Pavel Safranov

7 Lie groupoids and differential equations

Speaker: Francis Bischoff

11:00 Coffee Break

8 Deformations of symplectic foliations

Speaker: Marco Zambon

12:20 Discussion and free time

9 Dirac reduction and shifted symplectic geometry

We introduce a notion of reduction of Dirac realizations induced by a submanifold of the base and give an interpretation in shifted symplectic geometry. It yields, in particular, to a notion of symplectic (resp. quasi-Hamiltonian) reduction where the level can be a submanifold of the dual of the Lie algebra (resp. the group) rather than a point, and explains some disparate constructions in symplectic geometry. This is joint work with Ana Balibanu and Peter Crooks.

Speaker: Maxence Mayrand

10 Classification of stacky vector bundles

This is report on a joint project with my student J. Desimoni, where we classify stacky vector bundles by the categorified Grassmanian, the differentiable 2-stack represented by the general linear 2-groupoid.

Speaker: Matias del Hoyo

Wednesday 9 February

11 Weil algebras for double Lie algebroids

Speaker: Eckhard Meinrenken

12 Differentiation of Lie n-groupoids

As a Lie n-groupoid is an atlas for an n-stack in differential geometry, one expects that their differentiation should be the tangent complex of the n-stack carrying a Lie n-algebroid structure. However, an explicit differentiation, like that for Lie groupoid, seems to be missing. Inspired by Severa's idea of an infinitesimal object, we perform (spending a lot of years fixing holes :) an explicit differentiation, and reach the tangent complex with a Lie n-algebroid structure. This is a joint work with Du Li, Rui Fernandes, Leonid Ryvkin and Arne Wessel.

Speaker: Chenchang Zhu

11:00 Discussion and free time

Thursday 10 February

13 Poisson structures from corners of field theories

Speaker: Alberto Cattaneo

14 Quantization and integrability

Speaker: Alejandro Cabrera

11:00 Coffee Break

15 The Fukaya category of the log symplectic sphere

Speaker: Charlotte Kirchhoff-Lukat

12:20 Discussion and free time

16 Some remarks on Lagrangian intersections in the algebraic case (Joint talk with Global Poisson Webminar)

Some years ago, in joint work with B. Fantechi, we constructed brackets on the higher structure sheaves of Lagrangian intersections, and compatible Batalin-Vilkovisky operators, when certain orientations are chosen (see our contribution to Manin’s 70th birthday festschrift). This lead to a de-Rham type cohomology theory for Lagrangian intersections. In the interim, much progress has been made on a better understanding of the origin of these structures, and some related conjectures have been proved. We will explain some of these results.

This is a joint talk with Global Poisson Webminar

Speaker: Kai Behrend

Friday 11 February

17 The linear model around Poisson submanifolds.

We built a local model around Poisson submanifolds, which we have shown to generalize Vorbojev's local model around symplectic leaves. A normal form theorem holds in many situations, e.g., Poisson manifolds integrable by proper groupoids, Hamiltonian quotients, etc. This is joint work with Rui Loja Fernandes.

Speaker: Ioan Marcut

18 Compatibility of Nijenhuis operators with various structures

This is a report on recent results involving the compatibility of Nijenhuis operators with various structures (e.g. Poisson groupoids, Dirac Structures, Courant algebroids) by means of an associated connection-like object. An interesting application is the study of holomorphic structures via their underlying real objects. Also, the investigation of Nijenhuis structures compatible in a suitably sense with Courant algebroids leads to a (not fully understood yet) relation with Kähler geometry.

Speaker: Thiago Drummond

11:00 Coffee Break

19 Symplectic gerbes or symplectic foliations

Speaker: Marius Crainic