Smooth manifolds are ubiquitous in the modern research of mathematics. This course is an introduction to smooth manifolds and their topology (differential topology).
In the second part, we will study differential topology, i.e. the topology of smooth manifolds. The topics include: Whitney immersion and embedding theorems, approximation theorem, Sard theorem, transversality, intersection numbers, Morse functions and Morse theory. If we have more time, I will talk about the h-cobordism theorem, which is the main part of Smale's proof of the Poincare conjecture of dimension at least 5.
John Lee, Introduction to smooth manifolds; Victor Guillemin & Alan Pollack, Differential topology
Mathematical analysis, linear algebra, general topology (equivalent to Math 412, Math 350 and Math 441)