Lie groupoids: Definitions, examples and basic properties; actions and representations; principal fibrations; cohomology.

Lie algebroids: Abstract algebroids; the Lie algebroid of a groupoid; examples; connections and representations; relation with Poisson structures.

Integration: Lie Theorems I and II; Lie Theorem III – obstructions to integrability; simplicial groupoids and Poisson structures.

Linearization: Eigengroups; Haar systems; Riemannian structures on groupoids; Weinstein-Zung theorem; Relation to classical results; Rigidity.

Morita equivalences: Morita morphisms; principal bi-fibrates; constructions of categories of differentiable stacks; orbifolds.

VB-groupoids and VB-algebroids: Definitions; Representations less than homotopy; Differentiation and integration; Morita invariance.

References:
BURSZTYN, CABRERA, DEL HOYO. -Vector bundles over Lie groupoids and algebroids; Advances in Mathematics 290, 163-207, 2016.
CANNAS DA SILVA, WEINSTEIN. – Geometrics models for noncommutative algebras. Berkeley Math Lecture Notes series, AMS, 1999.
CRAINIC, FERNANDES. – Lectures on integrability of Lie brackets. Geometry and Topology Monographs, 2011.
DEL HOYO. – Lie groupoids and their orbispaces; Port. Math. 70, 161-209, 2013.
DEL HOYO, FERNANDES. – Riemannian metrics on Lie groupoids; Crelle’s Journal, in press, 2015.
MACKENZIE. – General theory of Lie groupoids and Lie algebroids. Cambridge University Press, 2016.
MOERDIJK, MRCUN. – Introduction to foliations and Lie groupoids. Cambridge University Press, 2003.

 

* Basic syllabus. The teacher has the autonomy to make any changes.