My lecture this year is based on my collaboration with G. Landi and M. Dubois-Violette and has as its subject non-commutative varieties, of which many concrete examples are described. The essential problem is that of classifying spherical non-commutative varieties, and arose naturally in connection with Poincaré duality in K-homology.
The main result is the complete classification of non-commutative spherical varieties of dimension 3. We find a three-parameter deformation of the standard 3-sphere S3 and a corresponding deformation of the Euclidean spaceR4.
For generic values of the deformation parameter, we show that the resulting algebra (of polynomials on the deformation ofR4) is isomorphic to the algebra introduced by Sklyanin for the Yang-Baxter equations. Degenerate values of the deformation parameter do not give Sklyanin algebras and we extract a class of them, the θ-deformations, which we study in detail.
We show thanks to θ-deformations that any spinorial compact Riemannian variety whose group of isometries is of rank r ≥ 2 admits an isospectral deformation into a one-parameter family of noncommutative geometries, verifying all the axioms I introduced in 1996.