Topics: 

•  Forms and  Currents on manifolds, currents associated to algebraic cycles
•    equation and Green Currents
• Review of intersection theory on algebraic varieties
•  Arakelov’s intersection theory on surfaces
• Arithmetic Intersection theory in arbitrary dimension
• Chern classes, and Chern-Weil theory, Chern forms of Hermitian holomorphic vector bundles on complex manifolds
• Secondary characteristic classes and Bott-Chern forms
• Chern classes in arithmetic intersection theory
• Applications.  The amount of detail on applications will depend on the amount of time we have.  The goal would be to approach some results such as the arithmetic Hilbert-Samuel formula,  and results of Vojta and Faltings in Diophantine geometry.