MAT 1341F Topics in Differential Geometry: Lie Groupoids and Lie Algebroids

Classes: M10-12, T2 BA 6180

Lie groupoids, and their infinitesimal counterparts, the Lie algebroids, arise in many areas of differential geometry, such a Poisson geometry, index theory, foliation theory, or symmetries of differential equations. In recent years, there has been a lot of progress towards understanding these structures, such as the Crainic-Fernandes integrability obstructions and various results on 'multiplicative structures' on Lie groupoids. Tentative topics include:

• Lie groupoids: Definitions, examples, basic properties
• Lie algebroids: Definitions, examples, basic properties
• The Crainic-Fernandes integration theorem
• Cohomology of Lie groupods and Lie algebroids, van Est map
• Double structures (?)
• The algebra associated to a Lie groupoid (?)

The course mark will be based on attendence, as well as a problem set that I hope to have ready around end of October. (I'll try to not make it too difficult.)

Course notes will be posted here.

Last update: October 1, 2017

Our main references are:

• Moerdijk-Mcrun: Introduction to Foliations and Lie Groupoids. Cambridge University Press
• Crainic-Fernandes: Lectures on Integrability of Lie Brackets. Available at this link.
• Cannas da Silva-Weinstein: Geometric models for noncommutative algebras, Berkeley Math. Lecture Notes 10, Amer. Math. Soc. (1999).

• K. Mackenzie: General theory of Lie groupoids and Lie algebroids
• E. Meinrenken: Poisson Geometry from a Dirac perspective. Available at the arXiv.