Junior Seminar: Morse Theory

Suggested references:

We will follow chapters 1–4 of the book by Audin and Damian; all of these books except for the one by Milnor should be freely available to the Princeton community via the above links. (Use the EZproxy links if you are off-campus.)

Material covered in class (notes are currently offline):

2/16 SS Review of differential geometry, critical points, the Morse lemma Notes
2/23 SS Existence and genericity of Morse functions Notes
3/2 RZ Pseudo-gradients, Morse charts, stable and unstable manifolds Notes
DD Pseudo-gradient trajectories, sublevel sets away from critical values Notes
3/9 EH Sublevel sets at a critical value Notes
MDG Transversality and the Smale condition (definition, examples) Notes
3/23 SS Existence and genericity for the Smale condition Notes
DD Definition of the Morse complex Notes
3/30 RZ Broken trajectories, compactness Notes
EH Broken trajectories form a manifold with boundary Notes
4/6 SS Orientations, homological algebra, invariance of Morse homology (outline) Notes
MDG Invariance of Morse homology (proof) Notes
4/13 DD Functoriality for diffeomorphisms and embeddings Notes
SS Functoriality and homotopy invariance: the general case Notes
4/20 RZ The Künneth formula Notes
EH Long exact sequence of a pair Notes
4/27 MDG HM0 and connectedness, homology of RPn Notes
SS Applications: Borsuk-Ulam, fundamental theorem of algebra Notes