I will start from the definition of vector bundles over a manifold, basic operations on vector bundles, connections and curvature, Chern-Weil's approach to Chern classes of vector bundles, and basic properties of Chern classes.

It is well-known that the $\overline{\partial}$--Neumann operator on a product domain does not preserve smoothness up to the boundary; however, the canonical solution operator does preserve smoothness up to the boundary. We try to understand this phenomenon, and derive some estimates for the canonical solution in Sobolev spaces. (Joint work with Mei-Chi Shaw.)

My aim was to explain four formulas of Riemannian geometry due to Alain Connes which have analogues in NCG. So far we discussed a dual version formula of geodesic distance on a manifold which is stated in terms of algebra of smooth functions, Dirac operator in the Hilbert space of spinors. We also talked about Weyl's formula which gives us the integration against the volume form in this set up. Next we introduced Connes' differential forms and their inner product to bring in Yang-Mills action. In the upcoming talk, I shall try to reach the third formula which states the relation between Connes differential graded algebra and de Rham algebra of differential forms on the manifold.