(joint with Ausoni, Baas, Richter and Rognes)

Two vector bundles give rise to a geometrically defined cohomology theory extrapolating past the theory of vector bundles (K-theory) and differential forms (de Rham cohomology), capturing information related to cobordisms of manifolds beyond K-theory and deRham cohomology's reach.

The analytic and differential geometric understanding of two vector bundles is still very much in its infancy. There was a hope that an "integration of determinants through loops" construction would give an integral functor from two vector bundles to quantum field theories. However, the fact that the commutative ring spectrum representing complex K-theory does not support a determinant rules this out.

The first obstruction has a geometric interpretation: the one-dimensional two vector bundle represented by the Dirac monopole over the three sphere splits virtually.

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Integration is sometimes said to be a solved problem in computer algebra, but integration problems are the source of a significant percentage of bug reports and complaints to Mathematica and Maple. The reasons for this are discussed and some remedies described.