Mathematics

Following the duality between manifolds and its algebra of smoothcoordinate functions, Hochschild cocycles correspond to de Rhamcurrents (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 positiverepresentatives of a Hochschild 2-cocycle class (related to the fundamentalclass of the surface) and conformal structures on the surface.On the de Rham side, it corresponds to Lelong's notion of positive currentswhich unitizes the underlying complex structure.Such perspective allows us to understand complex structures onsurfaces as solutions of a variational problem and could be aninteresting  starting point to develop complex geometry in theframework of noncommutative geometry. In the second part, we shallcarry out the blueprint in detail on noncommutative two tori,whichare the best known examples of noncommutative manifolds.It is kind of a reading seminar on Connes's book Noncommutative Geometry VI.2.