Student Seminar on Morse Homology
BMS Student Seminar on Morse Homology (WS 2021/22)
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For a closed smooth manifold, we aim to construct the Morse chain complex via the Floer approach discussed in the lecture notes by Alexander Ritter and the book Morse Homology by Schwarz.
The seminar is organized by students for students, if you would like to attend or give a lecture on some topics of your choice from the lecture notes, please write to us on symplecticg@gmail.com. This seminar is conducted for educational purposes and you won't get any credits for your master or bachelor's degree by attending this seminar.

Talk 01: What do we expect to cover in this seminar?
Abstract:We introduce the Morse homology using the "old" approach, i.e. stable and unstable manifolds, and how the moduli spaces are obtained from them. We sketch the Floer approach what we also call the functional analytic approach to Morse homology .
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Location: Via Zoom

Talk 01: Some fundamentals I
Abstract: Connections, exponential map, Sard’s theorem, transversality, stability and genericity, and sections of vector bundles.
Reference: Lectures 2, 3 and the first part of lecture 4 here .
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Location: Via Zoom

Talk 03: Some fundamentals II
Topics: Banach manifolds, Morse functions, ArzelaAscoli theorem (just statement)
Reference: End of lecture 4 and lecture 5 here .
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Talk 04: Fredholm theory
Topics: Fredholm operators, SardSmale theorem (infinitedimensional version), Regularity of the zero sets of Fredholm sections.
Reference: Lecture 6 here .
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 Talk 05: Gradient flowlines
Topics: Negative gradient flowlines, Convergence at the ends, Transversality for the moduli space of flow lines connection any two fixed critical points: outline of the proof
Reference: Lecture 7 and the beginning of lecture 10 here .
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Location: Via Zoom
 Talk 06: Analytic setup
Topics: Sobolev spaces on Euclidean space (brief intro), Sobolev embedding theorems (no proofs), Sobolev spaces for manifolds, Sobolev spaces of sections of vector bundles, Sobolev theorems for manifolds.
Reference: Lectures 11, 12 here
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Location: Via Zoom.
 Talk 07: Proof of transversality, Part 1
Topics: Sobolev setup for the transversality theorem
, Transversality theorem, Hilbert spaces tricks, Claim 1 ⇒ Claim 2.
Reference: Lectures 13 here
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Location: Via Zoom.
 Talk 08: Proof of transversality, Part II
Topics: Claim 1, the section is Fredholm, Index computationfollowing the book by Schwarz
Reference: Lectures 14 and 15 in here
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Location: Via Zoom.
 Talk 09: Compactness
Topics: Motivation, The topology of \(M(p, q)), \(C^0_{loc}\)Convergence, Convergence to broken trajectories, Compactness theorem, Gluing theorem (just a sketch of the proof)
Reference: Lectures 17 and 18 in here
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Location: Via Zoom.
 Talk 10: Gluing
Topics: Gluing theorem
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 Talk 11: Morse Homology
Topics: Definition, Morse Homology is isomorphic to cellular homology
Reference: Lecture 19 in here
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 Talk 12: Invariance
Topics: Invariance theorem,
Reference: Lecture 20 in here
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 Talk 13: Applications
Topics: Poincaré duality, Künneth’s theorem, Morseinequalities, Products.
Reference: Lecture 21 in here
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Location: Via Zoom.