Prof. Alexander F. Ritter
Associate Professor, University of Oxford
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MORSE HOMOLOGY, WITH A VIEW TOWARDS FLOER THEORY
(Oxford graduate course, TCC 2024)
(Thursdays 9-11 via Microsoft Teams, see here)
LECTURE NOTES
I would be extremely grateful for your feedback because these notes will be published as a book. Ideas, comments, corrections, anything.
Hand-written notes from the Teams meetings:
Lecture 1: Overview about Morse functions and Morse homology
Lecture 2: Overview about Morse homology, intro to symplectic manifolds
Lecture 3: Overview about Floer homology, symplectic action functional, almost complex structures, Morse-Bott homology, Morse-Bott-Floer homology
Lecture 4: Example of Morse-Bott homology, Example of Morse-Bott-Floer homology (cotangent bundle of the sphere), Viterbo's theorem, symplectic cohomology, energy in Morse theory, energy in Floer theory
Lecture 5: Monotonicity lemma, Gromov's trick, Energy quantisation in Floer theory
Lecture 6: Remarks about ODE theory, Arzela-Ascoli theorem, Elliptic bootstrapping, Discussion of a Banach space topology on the moduli space of gradient flowlines, Energy consumption, Exponential convergence at the ends, Compactness theorem in Morse theory
Lecture 7: Compactness proofs in Floer theory versus Morse theory, bubbling of holomorphic spheres, removal of singularities theorem, aspherical and semipositive symplectic manifolds, an explicit example of bubbling (degeneration to a node)
Lecture 8: Gluing theorem, Novikov field coefficients, invariance proofs using Floer's continuation method, a map from Morse to Floer cohmology (PSS map)
My contact details: find the contacts here
My homepage: here
HOMEWORKS (PDF): you have no obligation to do these, but it may help you understand the material to check these out. There are lots of hints so you don't end up spending too much time on them.
Lecture 1 (PDF): A broad overview: Morse functions, topology of sublevel sets, Morse homology, number of critical points of a generic function, geometry vs functional analysis, Poincare duality via Morse homology
I'm currently updating the notes from HERE, so please in the meantime look at those. This course will be quite similar to that one in the first few lectures, after which we may take a different route (the hope is to have a few lectures on Floer theory).
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