Junior Seminar: Morse Theory

Spring 2015, Monday 7-9 PM in Fine 801
Office hours: by appointment, Fine 808

Suggested references:

Michèle Audin and Mihai Damian, Morse theory and Floer homology (EZproxy)
Augustin Banyaga and David Hurtubise, Lectures on Morse homology (EZproxy)
John Milnor, Morse theory
Liviu Nicolaescu, An invitation to Morse theory (EZproxy)
Matthias Schwarz, Morse homology (EZproxy)

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