Room 208
Vivatsgasse 7
53111 Bonn
Vivatsgasse 7
53111 Bonn
Steven Sivek
Imperial College Department of MathematicsJunior 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 | HM_{0} and connectedness, homology of RP^{n} | Notes |
SS | Applications: Borsuk-Ulam, fundamental theorem of algebra | Notes |