A short Review in Modern Mathematics

This is the lecture notes from Mat280. I typed the notes and Schwarz revised it. The lecture notes contain the following topics:

1. Categories
2. Metric and Topological Spaces
3. Differential Calculus
4. Lie algebra and Lie groups
5. Homotopy theory
6. Vector bundles and Connections
7. G-structure
8. Noncommutative Geometry
9. Algebraic Geometry and Arithmetic Geometry.

The lecture note is now available at http://frankmath.cc/UCD/ASMath.pdf

Hopefully you can enjoy it. If you have any question, please tell me.