Mathematics

 Following the duality between manifolds and its algebra of smooth coordinate functions, Hochschild cocycles correspond to de Rham currents (continuous linear functionals on differential forms).In the first part of the talk, we shall explain a 1-1 correspondence, for only surfaces, between extreme points of the convex set of positive representatives of a Hochschild 2-cocycle class (related to the fundamental class of the surface) and conformal structures on the surface.On the de Rham side, it corresponds to Lelong's notion of positive currents which unitizes the underlying complex structure.Such perspective allows us to understand complex structures on surfaces as solutions of a variational problem and could be an interesting starting point to develop complex geometry in the framework of noncommutative geometry. In the second part, we shall carry out the blueprint in detail on noncommutative two tori, which are the best known examples of noncommutative manifolds.It is kind of a reading seminar on Connes's book Noncommutative Geometry VI.2.