Chern-Weil theory and complex differential Geometry
The first half of this course is about basic study on connections, curvature and characteristic classes. We start with connections for vector bundles including Levi-Civita connection for Riemannian manifolds and Chern connection for Hermitian vector bundles. Then we turn to connections for general principal bundles and the theory of characteristic classes (Chern-Weil theory). In the second half of the course some recent topics on complex/Kähler geometry will be discussed.
Basic knowledge on manifolds, Lie groups and Lie algebras.
 S.Kobayashi and K.Nomizu, Foundations of Differential Geometry I, II, 1996, Wiley Classics Library.
 S.S.Chern, Complex manifolds without potential theory, Van Nostrand Math. Series, 1967.
 A.Futaki, Kähler-Einstein metrics and integral invariants. Springer Lecture Notes, vol. 1314, 1988.