For a closed smooth manifold, we aim to construct the Morse chain complex via the Floer approach discussed in the and the book Morse Homology by Schwarz and the lecture notes by Alexander Ritter.
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.
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Talk 01: What do we expect to cover in this seminar?
Speaker: Yoanna Kirilova
When: 3:00 pm, 25.10.2021
Location: Via Zoom
Lecture notes: link
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Talk 02: Some fundamentals
Abstract: Connections, exponential map, Sard’s theorem, transversality, stability and genericity, and sections of vector bundles, Banach manifolds, Morse functions, Arzela-Ascoli theorem (just statement)
Speaker: Dominik Gutwein
When: 3:00 pm, 08.11.2021
Location: Via Zoom
Lecture notes: link
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Talk 03: Fredholm theory
Topics: Fredholm operators, Sard-Smale theorem (infinite-dimensional version), Regularity of the zero sets of Fredholm sections.
Speaker: Jacek Rzemieniecki
When:3:00 pm, 15.11.2021
Location: Via Zoom
Lecture notes: link
- Talk 04: 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
Speaker:Naageswaran Manikandan
When: 3:00 pm, 22.11.2021
Location: Via Zoom
Lecture notes: link
- Talk 05: 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
Speaker: Gorapada Bera
When: 3:00 pm , 29.11.2021
Location: Via Zoom
Lecture notes: link
- Talk 06: Proof of transversality, Part 1
Topics: Sobolev setup for the transversality theorem
, Transversality theorem, Hilbert spaces tricks.
Speaker: Yoanna Kirilova
When: 3:00 pm, 06.12.2021
Lecture notes: link
- Talk 07: Proof of transversality, Part II
Topics: Claim 1, the section is Fredholm, Index computation-following the book by Schwarz
Speaker: Yoanna Kirilova
When: 3:00 pm, 13.12.2022
Lecture notes: link
- Talk 08: Compactness
Topics: Motivation, Convergence to broken trajectories, Compactness theorem
Speaker:Dominik Gutwein
When: 3:00 pm, 17.01.2022
Lecture notes: here
- Talk 09: Gluing theorem
Speaker:Gorapada Bera
When: 3:00 pm, 31.01.2022
Location: Via Zoom
Lecture notes: link
- Talk 10: Morse homology mod 2
Abstract::
Using the data developed by my colleagues in the previous talks, we construct the Morse-complex mod 2. We prove the resultant homology is independent of the choice of the Morse function and Riemannian metric used to construct the complex. As an application, we prove the Poincaré duality theorem.
Speaker:Shah Faisal
When: 3:00 pm, 07.02.2022
Location: BMS Seminar Room
- Talk 11: Equivalences for Morse homology: a paper by Matthias Schwarz
Abstract::
We construct an explicit natural isomorphism between Morse homology and singular homology via the technique of pseudo-cycles. Given a Morse cycle, we glue a suitable compactification of the unstable manifolds of critical points contributing to the cycle to obtain a well-defined pseudo-cycle. An isomorphism between Morse homology and singular homology is obtained by sending any Morse cycles to the singular homology class, which corresponds to the pseudo-cycle obtained from the Morse cycle.
Speaker:Shah Faisal
When: 3:00 pm, 18.02.2022
Location: BMS Seminar Room
Reference:link