In this course we will try to understand three aspects of finite-dimensional Morse theory. The first is the topological aspect, following part of the famous book of Milnor (Morse theory). The second is the differential geometric aspect, focusing a (linear) analytical proof of the Morse inequality following Witten’s insightful idea (Supersymmetry and Morse theory) and a chapter of the book of W. Zhang (Lectures on Chern-Weil theory and Witten deformation). The third is the homological aspects, which focuses on the rigorous construction of the Morse-Smale-Witten complex and the Morse homology using nonlinear analysis. There are a number of available references for the third aspect.

Prerequisites: Topology 1 (636), Differential Geometry 1 (622), or instructor approval