I will survey on a recent joint work with M. Khalkhali and a paper by Connes and Moscovici on scalar curvature for noncommutative two-tori. The scalar curvature is computed by considering small time heat kernel expansions of the perturbed Laplacian which encodes the metric information of a general translation invariant conformal structure and a Weyl conformal factor on the noncommutative two-torus. There is an equivalent formulation for the scalar curvature in terms of special values of spectral zeta functions. I will also talk about our result on the Gauss-Bonnet theorem for noncommutative two-tori which extends the work of Connes and Tretkoff to the general conformal structures.

We will state some results and formulate some global problems related to the geometry of decomposition of CR manifolds into CR orbits.

After a brief introduction to braids I will discuss a conformal invariant and a dynamical invariant, the relation between them and some applications.

The study of the relations between various moduli spaces arising from connections on vector bundles over algebraic varieties is an interesting topic. The most intriguing question is the relation between the moduli space of connections and that of the underlying vector bundles. We illustrate this concept in considering a familly of rank 2 logarithmic connections over an elliptic curve. These rank 2 logarithmic connections are obtained as direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We provide an explicit parameterization of all such connections and determine their monodromy and differential Galois group. We address the local structure of the moduli space of connections and give an example of a moduli space of connections having a singularity. This singularity is solved by means of the toric geometry. I will finish this talk in saying a few words on what happens while one considers the genus-1 singular curves.