An amazing discovery of physicists is that there are many seemingly quite different quantum field theories that lead to the same observable predictions. Such theories are said to be related by dualities. A duality leads to interesting mathematical consequences; for example, certain K-theory groups on the two spacetime manifolds have to be isomorphic. We will explain how some of these K-theory isomorphisms predicted by physics correspond to certain cases of the Baum-Connes Conjecture, or to equivalences of derived categories of twisted coherent sheaves.

We introduce pseudo-Riemannian calculus of modules over noncommutative algebras in order to investigate as to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this framework, it is possible to prove an analogue of the Levi-Civita theorem. It states that there exists at most one connection, which satisfies a torsion-free condition and a metric compatibility condition on a given smooth manifold. More significantly, the corresponding curvature operator has the same symmetry properties as in the classical curvature tensors. In my talk, I will discuss pseudo-Riemannian calculi over the noncommutative 4-sphere for a conformal class of the round metric and their corresponding scalar curvatures. Lastly, in this setting it is possible to prove a Gauss-Bonnet-Chern type theorem.

We will start by showing that generically for all holomorphic functions f on a planar simply connected domain every order's antiderivative is unbounded. This leads to consider multivalued antiderivatives of any order on every non simply connected domain. If on such a domain V a holomorphic function f has a one-valued antiderivative of any order in V, then f has a holomorphic extension on the simply connected envelop of V. On a non simply connected domain generically every holomorphic function deos not have a one-valued primitive.

Can the space H(V) of all holomorphic functions be replaced in the above results by other spaces of holomorphic functions?