Instructional Program

We will discuss some links between cyclic homology and the topology of loop spaces. The main part of the lectures will be the way in which the theory of iterated integrals of differential forms may be used to transfer methods back and forth between loops spaces and cyclic homology. We will discuss some applications to index theory and to topological field theory.

Nigel Higson proposes to talk about Connes' (old) theorem on the Godbillon-Vey invariant and the flow of weights. It is a nice illustration of the material that Adam Rennie (Masoud Khalkhali) will be presenting.

Download notes as PDF.

Spectral triples have come to play a central role in noncommutative geometry over the last ten years. They serve several purposes, all of which highlight their role as `geometric spaces' in noncommutative geometry. They have a metric space attached to them, they represent K-homology classes, and so are important in index theory, they have a natural differential structure and usually come equipped with an `integral calculus' of some sort. This course aims to introduce these various ideas and structures by presenting the basic motivating examples in detail. If time permits we will look at some more interesting examples in detail as well. We will conclude with a brief survey of the important examples being studied worldwide at the moment.
Prerequisites: The more differential geometry the better, but if you know what a `compact oriented manifold' is, you'll learn something. The more functional analysis the better, but if you are familiar with the functional calculus you'll learn something. Knowledge of unbounded operators on Hilbert space would be a plus. There will be problem sheets for the diligent.

Khalkhali's lecture notes are on the math archives, http://arxiv.org/abs/math.KT/0408416.