Nigel Higson: The tangent groupoid in non-commutative geometry

This is a 22-lecture course, with each lecture being 90 minutes, given by Nigel Higson.

The tangent groupoid is a geometric construction that can be applied to any smooth manifold. Alain Connes pointed out its relevance to the Atiyah-Singer index theorem, and ever since he did so the tangent groupoid has appeared regularly in noncommutative geometry, often in ways related to index theory but usually illuminating other issues at the same time. Good examples of this are the elegant and simple ways of understanding pseudodifferential operators that have been developed recently by Claire Debord and Georges Skandalis, and by Erik van Erp and Bob Yuncken. I shall start with pseudodifferential operators, then introduce the tangent groupoid through them, and go on to examine applications in representation theory, hypoelliptic partial differential equations and elsewhere.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4
5. Lecture 5
6. Lecture 6
7. Lecture 7
8. Lecture 8
9. Lecture 9
10. Lecture 10
11. Lecture 11
12. Lecture 12
13. Lecture 13
14. Lecture 14
15. Lecture 15
16. Lecture 16
17. Lecture 17
18. Lecture 18
19. Lecture 19
20. Lecture 20
21. Lecture 21
22. Lecture 22

These videos were produced by the Fields Institute, as a graduate course (link to course page) belonging to the Thematic Program on Operator Algebras and Applications.