This is a continuation of the course 532. The topics include
1. De Rham Cohomology and Hodge Theorem
2. Vector Bundles, Connections, and Characteristic Classes
3. Complex Manifolds (Riemann Surfaces) and Holomorphic Vector Bundles
4. Basic Gauge Theory (Yang-Mills Theory and Narasimhan-Seshadri Theorem)
Depending on the students' demand, the topics can be changed during the semester.
References: My Lecture Notes, together with Kobayashi-Nozimu: Foundations of Differential Geometry and Warner: Foundations of Differentiable Manifolds and Lie Groups
Manifold theory, basic topology, and preliminary knowledge about partial differential equations