• Non-commutative Geometry and its Applications
• Summer School

• Schedule
• Participants
• Poster

In his book "Noncommutative Geometry" of 1994, Connes sketches a proof of the Atiyah-Singer index theorem based on a geometric construct called the tangent groupoid. The groupoid method has proven to be very versatile, and has been applied to index problems in more elaborate settings, for example to index theory for foliations, or more recently to Boutet-de Monvel's calculus for manifolds with boundary. In my talks I will review the construction of the tangent groupoid and its relation to the Fredholm index of elliptic (pseudo)differential operators. I will then explain how the groupoid method can be adapted to the index problem of a class of differential operators that are not elliptic.

I first discuss a free scalar field and their properties and mention Wightman axioms as well as Wightman functions and Schwinger functions. I mention the lattice regularization and the connection of euclidean QFT to critical phenomena and phase transitions. The triviality result and the Landau ghost problem are mentioned.

In order to improve this situation one defines fields over noncommutative spaces. The formulation is simple. One runs into new problems called infrared/ultraviolet mixing. Together with Raimar Wulkenhaar we managed to overcome it in a special model. Ward identities and Schwinger-Dyson equations together allow to construct the model and to solve it. The fixpoint model connects to a special matrix model.

In this series of four lectures I plan to survey some basic notions of noncommutative geometry. No prior knowledge of the subject will be assumed and I shall try to cover the necessary background. I shall use the context of a recent Gauss-Bonnet theorem in noncommutative geometry, and will gradually build all the machinery that is needed to formulate and prove such a result in a noncommutative setting. Topics to be discussed will include:

• Noncommutative spaces and where they come from,
• Tools from noncommutative algebraic topology, including cyclic cohomology and K-theory of topological algebras, Connes' Chern character map,
• Spectral triples, noncommutative spin and Riemannian manifolds, Connes' axioms, and reconstruction theorem,
• Heat kernel techniques and asymptotic expansion, local noncommutative geometric invariants, curvature in noncommutative geometry.

In what sense are Morita-Rieffel equivalences of C*-algebras "equivalences"? We will learn why categories are not enough to invert these equivalences properly and introduce the bicategories of Hilbert bimodules and C*-correspondences to achieve this. The bicategorical point of view gives a richer symmetry object for a C*-algebra: a bicategory instead of a category. This makes precise in which sense inner automorphisms are more trivial than other automorphisms. Actions of groups and groupoids by correspondences turn out to be equivalent to saturated Fell bundles, so we make precise in what sense Fell bundles are to be thought of as generalised actions.  If many participants are familiar with Cuntz-Pimsner algebras, we may do the same for semigroup actions, which are very close to product systems, and interpret their Cuntz-Pimsner algebras. We also explain in what sense a groupoid fibration should be viewed as an action of one groupoid on another one. This provides us with interesting proper actions of non-Hausdorff groupoids on C*-algebras.

Click to download:
Ralf Meyer: Actions of Higher Categories on C*-Algebras