In my lecture this year, I developed the relationship between the " Differential Geometry " and " Algebraic Geometry " aspects of non-commutative geometry, building on my collaboration with M. Dubois-Violette in which we classified non-commutative spheres of dimension three. We develop the notion of central quadratic form for quadratic algebras, as well as the notion of positivity for central forms. We then give a general construction, independent of ϑ-function theory, of a homomorphism of a quadratic algebra into the C*-algebra cross-product of the characteristic variety by the canonical correspondence. We then apply these results to non-commutative spheres and compute the Jacobian of the branched covering thus obtained.
Lecture