This course is an introduction to the theory of Lie groupoids and their infinitesimal counterparts, called Lie algebroids. This is a far reaching extension of the usual Lie theory, which finds application in many areas of Mathematics. Groups typically arise as the symmetries of some given object. The concept of a groupoid allows for more general symmetries, acting on a collection of objects rather than just a single one. Groupoid elements may be pictured as arrows from a source object to a target object, and two such arrows can be composed if and only if the second arrow starts where the first arrow ends. Just as Lie groups (as introduced by Lie around 1900) describe smooth symmetries of an object, Lie groupoids (as introduced by Ehresmann in the late 1950’s) describe smooth symmetries of a smooth family of objects. That is, the collection of arrows is a manifold G, the set of objects is a manifold M, and all the structure maps of the groupoid are smooth. Ehresmann’s original work was motivated by applications to differential equations. Since then, Lie groupoids have appeared in many other branches of mathematics and physics. These include:

- algebraic geometry: Grothendieck introduced stacks in the late 1960’s via fibered categories over a site. Fibered categories can be viewed as a type of sheaf of groupoids. More recently, this has led to the concept of a gerbe.
- foliation theory: Haefliger introduced transversal structures to foliations in the 1970’s, using the concept of a holonomy groupoid. This approach allows for a systematic study of transversal structures, and has been central to the subsequent development of the subject.
- noncommutative geometry and index theory: Lie groupoids made their appearance in noncommutative geometry through the monumental work of Connes in the 1980’s. He introduced the tangent groupoid of a space as a central ingredient in his approach to the Atiyah-Singer index theorem. This approach led to a number of refinements of the index theorem, such as the Connes-Skandalis index theory for foliations.
- Poisson geometry: motivated by quantization problems, Karasev and Weinstein introduced the symplectic groupoid of a Poisson manifold in the late 1980’s, as a way to “untwist” the complicated behavior of the symplectic foliation underlying the Poisson manifold.
- M. Crainic and R.L. Fernandes, Lectures on Integrability of Lie Brackets also available as
*Geometry & Topology Monographs***17**(2011) 1-107.

In this course I will not be following any particular book, but the following Lecture Notes should be useful: I will also provide notes of the lectures.

Students taking this course are assumed to know differential geometry at the level of Math 518 – Differentiable Manifolds. A knowledge of ordinary Lie theory at the level of Math 522 – Lie groups and Lie algebras is recommended but not strictly necessary.

**Email:** ruiloja (at) illinois.edu**Office:** 366 Altgeld Hall**Office Hours:** 11:30AM-12:30PM TR**Class meets:** 02:00-03:20PM TR, 345 Altgeld Hall**Prerequisites:** Math 518 or equivalent.

**In this page:**

- Announcements
- Syllabus
- Textbooks
- Grade Policy
- Lecture Notes
- Emergency information for students in Mathematics courses

**Announcements:**

- Class will meet for the first time on Tuesday, August 24.

**Lie groupoids****Lie algebroids****Lie functor and integrability****Differentiable stacks****Special Topics:**To be chosen from the interests of the students.

**Syllabus:**

**Textbooks:**

I will provide to participants some lecture notes as the course progresses, but the following references should also be very helpful:

- K. Behrend, Introduction to algebraic stacks, in Moduli Spaces, London Mathematical Society Lecture Note Series, 411, Cambridge University Press, 2014.
- A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, RI, 1999.
- M. Crainic and R.L. Fernandes, Lectures on Integrability of Lie Brackets, available as
*Geometry & Topology Monographs***17**(2011) 1-107. - K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society. Cambridge England ; New York : Cambridge University Press; 2005
- D. Metzler, Topological and Smooth Stacks, Preprint arXiv:math/0306176.
- I. Moerdijk and J. Mrcun, Introduction to Foliations and Lie Groupoids (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press, 2003.
- A. Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, 1104, Math. Surveys Monogr., 123, AMS Providence, RI (2005).

**Grading Policy**

**Expository Paper:**Students will be encouraged to write (in LaTeX) and present a paper. This is not mandatory. Following the tradition of topics courses, there will be no homework and no written exams.

**Lecture Notes:**

#### Part 1: General Theory

**Lecture 1. Why groupoids?****Lecture 2. Lie groupoids: definition and examples****Lecture 3. Lie groupoids: more examples and constructions****Lecture 4. Lie groupoids associated to foliations****Lecture 5. Classes of Lie groupoids. Lie algebroid of a Lie groupoid****Lecture 6. Lie algebroids: definition and first examples****Lecture 7. Lie algebroids: more examples and constructions; homological description****Lecture 8. Actions and representations of Lie groupoids****Lecture 9. Actions and representations of Lie algebroids****Lecture 10. Lie algebroid connections****Lecture 11. Curvature, parallel transport and A-path homotopy****Lecture 12. A-maps and A-homotopy****Lecture 13. Geodesics and exponential map. The Weinstein groupoid****Lecture 14. Weinstein groupoid and integrability****Lecture 15. Weinstein groupoid as a leaf space****Lecture 16. Main Theorem. Obstructions to integrability****Lecture 17. Computing the obstructions to integrability. Examples****Lecture 18. More examples. Proof of the main theorem**

#### Part 2: Singular Spaces

**Lecture 19. Orbifolds: definition and examples****Lecture 20. Orbifolds defined by proper actions and foliations****Lecture 21. Realization of an orbifold as a quotient M/G****Lecture 22. Equivalences of groupoids****Lecture 23. Morita equivalence****Lecture 24. Étale groupoids. Proper groupoids****Lecture 25. Linearization of proper groupoids. Pseudogroups vs étale groupoids****Lecture 26. Orbifolds as groupoids****Lecture 27. Principal groupoid bundles. Categories fibered in groupoids****Lecture 28. Stacks: definition and examples****Lecture 29. Differentiable stacks: stacks representable by Lie groupoids**

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**Emergency information for students in Mathematics courses**

For important emergency information related to fires, tornados or active threats, please look at the following leaflet.

*Last updated December 6, 2021*