Lie groupoids appeared first in Ehresmann's global formulation of PDE's and nowadays play an important role in Differential Geometry, Foliation Theory, Poisson Geometry, Noncommutative Geometry, Index Theory, Orbifolds and Stacks and Generalized Geometry, among others. The infinitesimal counterpart of a Lie groupoid is given by a Lie algebroid, an object that unifies Lie algebras, regular foliations but also provides an appropriate setting to study singular geometric structures, e.g. Poisson structures.
Every Tuesday and Thursday from 10am to 12pm. Room 5-118.
Every Monday from 10:00 to 11:00 at Room 4-331A. Students are very welcome to ask questions about any subject related to the course. Please feel free to ask your questions, I will do my best to discuss all issues that might help us solve your doubts, I will be happy to do it. If you would like to talk about Math in general, I am also available and looking forward to discuss.
Differentiable manifolds.
This course is an introduction to Lie groupoids and Lie algebroids, paying special attention to geometric structures on Lie groupoids and their connections with Symplectic Geometry. The course is organized as follows:
I will provide notes containing an expanded version of the material presented during the lectures.
The following are notes and books that I recommend as very good sources of study for the first part of the course:
For the second part of the course, I will follow the papers:
but also you are encouraged to read
3. Arias Abad, C., Crainic, M., "Representations to homotopy of groupoids and the Bott spectral sequence", Adv. Math., 248, 416-452 (2013).
4. Arias Abad, C., Crainic, M., "Representations up to homotopy of Lie algebroids", Crelle, 2012 (663), 91-126 (2012)
where general representations up to homotopy are introduced and studied in detail.
For the third part of the course, the following references are recommended:
For the last part of the course, the following survey is highly recommended: