762 - Differential Topology

This course will teach the fundamental techniques and theorems
on the topology of differentiable manifolds. The techniques and
theorems taught in this course form a prototype which
is used in many other related mathematics (e.g., geometric topology,
symplectic geometry and several complex variables and so on).
I will try to teach
the classical materials listed below in a way that they can be easily
adaptable to application to these areas. In addition, I will also
cover rudiments of the Morse theory and the Fredholm theory on
infinite dimensional manifolds.
At least the following topics will be covered:

1. Basics: Whitney embedding theorem,
Whiteny approximation theorem, Transversality theorem,
Intersection theory,

2. Elements of Morse theory: Morse inequality, Morse homology

3. Banach manifold theory: Fredholm theory, Sard-Smale theorem