Lecturer: Nigel Higson (Pennsylvania State University)
Title: Index theory, groupoids and noncommutative geometry
Time: Monday, Wednesday 14:00-16:00
and Friday 10:30 - 12:30, two weeks starting from November 24
Abstract: This course will be about the Atiyah-Singer index theorem, its treatment from the perspective of noncommutative geometry, and various extensions of the index theorem that are made possible by noncommutative geometry. Most of the course will be organized around the concept of smooth groupoid, which will be a bridge between standard and noncommutative geometry. We shall present Connes' proof of the index theorem using the tangent groupoid, and then discuss equivariant index theory and the Baum-Connes conjecture. At the end we shall take a look at "local" approaches to index theory using cyclic cocycles.
Prerequisites include an acquaintance with Hilbert space theory, basic spectral theory, smooth manifolds, vector bundles and differential forms. Some prior contact with K-theory in one form or another will be very helpful, but not essential.