Abstract: 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.

Abstract: 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.)