Lie groupoid and Lie algebroid week in Coimbra III

School on modern aspects of the geometry and applications of Lie groupoids and Lie algebroids

This school will contain three mini-courses, aiming to introduce some modern advances  of the theory of Lie groupoids, Lie algebroids, and their applications. The goal is to have a leisurely schedule, allowing plenty of time for informal discussions.

It is geared toward PhD students, and other researchers interested in the topic, but previous exposure to Lie groupoids will not be assumed. 

The first day will contain either a small crash-course on Lie groupoids and Lie algebroids, or an open discussion with the participants, to introduce some definitions, examples, and ideas used in the other courses.

Mini-courses

• Karandeep Singh: Deformations, rigidity and stability through differential graded Lie algebras. 

• Maarten Mol: Topics revolving around Hamiltonian groupoid actions.

• Rui Loja Fernandes: Multiplicative Ehresmann connections. 

Other talks

• Giorgio Trentinaglia: The Adjoint Representation of a Higher Lie Groupoid.

• Lennart Obster: Crash-course on Lie groupoids and Lie algebroids. 

Location

Room 5.5 - Department of Mathematics, University of Coimbra

Registration

Participation is free and open to all, but registration is required for logistical reasons.  Please register by sending an email to jnmestre@mat.uc.pt saying that you want to participate.

Schedule and abstracts

In google calendar: link

Karandeep Singh

Title: Deformations, rigidity and stability through differential graded Lie algebras.

Maarten Mol

Title: Topics revolving around Hamiltonian groupoid actions.

Abstract: The notion of Hamiltonian action for symplectic groupoids unifies various momentum map theories in which the momentum map takes values in a Poisson manifold. In this series of talks we will discuss various aspects of the theory of such groupoid actions, such as an MGS-type normal form and groupoid versions of toric symplectic manifolds. Along the way we will come across several topics of independent interest, such as the linearization of Lie groupoids, integral affine geometry, orbifolds and equivariant sheaves for étale groupoids.

Rui Loja Fernandes 

Title:  Multiplicative Ehresmann connections

Abstract: I will present an introduction to the theory of multiplicative Ehresmann connections for Lie groupoids and their infinitesimal counterparts for Lie algebroids. The first were introduced by Laurent-Gengoux, Stienon & Xu, approximately 15 years ago in their study of non-abelian gerbes. They have been further generalized in two recent papers by Marcut and myself, where notable progress has been achieved, including a Lie algebroid version, establishing their existence for proper groupoids and unveiling new applications to local models in Poisson and Dirac geometry. Some of the topics covered in this course include: jet groupoids, multiplicative forms and IM forms (with coefficients), Cartan connections on groupoids and algebroids, bundle gerbes, and local normal forms

References:

• R.L. Fernandes and I. Mărcuţ, Multiplicative Ehresmann connections. Adv. Math.427(2023), Paper No. 109124.

• R.L. Fernandes and I. Mărcuţ, Poisson geometry around Poisson submanifolds, Preprint arXiv:2205.11457. To appear in JEMS.

• C. Laurent-Gengoux, M. Stiénon and P. Xu, Non-abelian differentiable gerbes, Adv. Math. 220 (5) (2009) 1357--1427.

Giorgio Trentinaglia 

Title:  The Adjoint Representation of a Higher Lie Groupoid

Abstract: I will explain how to extend the standard construction of the adjoint representation of a Lie groupoid to the case of an arbitrary higher Lie groupoid. As for a Lie groupoid, the adjoint representation of a higher Lie groupoid turns out to be a representation up to homotopy which is well defined up to isomorphism. Its existence and uniqueness are immediate consequences of a more general result in the theory of simplicial vector bundles: the representation up to homotopy obtained by splitting a higher vector bundle by means of a cleavage is, to within isomorphism, independent of the choice of the cleavage.

Organizers

• Raquel Caseiro

• João Nuno Mestre

Previous editions:
Lie groupoid week in Coimbra was held in September 2022.
Lie groupoid and algebroid week in Coimbra II was held in July 2023.