Janko Latschev
Online course Floer homology, Sommersemester 2020
This is a course on Floer homology, and it will fully take place online. As none of us has ever done this before, it will take some effort on everyone's part to make it a success. My current plans involve some independent reading, supplemented by live lecture sessions, problem sheets and live problem sessions. Much of the details is up for discussions, we will try out different things.
If you are interested in this course, please register for it in STiNE, since this will be our channel of communication outside of class until we find a better solution. The official meeting times for this course are:
Tuesdays 12:15-1:45 (online),
Thursdays 8:30-10(online), and
Wednesdays 11:50-1:20 (online) for the exercise sessions.
Apparently Chrome or Chromium as browsers give the best performance.
Here are the notes I use(d) in my presentations:
April 21 (1-3)
April 23 (4-13)
April 28 (14-20)
April 30 (21-26)
May 5 (27-34)
May 7 (35-42)
May 12 (43-51)
May 14 (52-56)
May 19 (57-64)
May 26 (65-75)
May 28 (75-82)
June 9 (83-90)
June 11 (89-95)
June 16 (96-103)
June 18 (104-111)
June 23 (112-119)
June 25 (120-126)
June 30 (127-133)
July 2 (134-143)
July 7 (144-152)
July 14 (153-159)
The aim of this course is to give an introduction to Floer theory in symplectic geometry. There are two main versions, namely Hamiltonian Floer homology and Lagrangian intersection Floer homology, which are related but somewhat different. All other versions essentially derive from these two. An excellent survey of the subject was written a few years ago by Alberto Abbondandolo and Felix Schlenk. Covering both versions in detail is too much for the time we have, so we will cover Hamiltonian Floer homology in more depth.
The exercise sheets will be posted here:
Sheet 1
Sheet 2
Sheet 3 (corrected May 15)
Sheet 4
(Links have been tested from inside the UHH network.)
The following books and lecture notes are useful study material for various parts of the course.
For background on symplectic and contact topology:
D. McDuff, D. Salamon | Introduction to symplectic topology | Oxford University Press |
A. Canas da Silva | Lectures on Symplectic Geometry | Springer Lecture Notes in Mathematics 1764 |
H. Hofer, E. Zehnder | Symplectic Invariants and Hamiltonian dynamics | Birkhäuser |
L. Polterovich | The Geometry of the Group of Symplectic Diffeomorphisms | Birkhäuser |
H. Geiges | An introduction to contact topology | Cambridge University Press |
For holomorphic curves in symplectic geometry:
D. McDuff, D. Salamon | J-holomorphic curves in symplectic topology | AMS Colloquium Series |
C. Wendl | Lectures on holomorphic curves |
M. Audin, J. Lafontaine (eds.) | Holomorphic curves in symplectic geometry | Birkhäuser Progress in Math. 117 |
References for functional analytic setup and Morse theory
Standard references for Hamiltonian Floer homology:
A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 1989, 575-611 and references therein
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D. Salamon, Lectures on Floer homology, published in Symplectic Geometry and Topology, IAS/Park City Mathematics Series vol. 7, AMS, 1999 |
M. Audin, M. Damian, Morse theory and Floer homology, Springer Verlag, 2014
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for background on the Conley-Zehnder index in particular:
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J. Gutt, The Conley-Zehnder index for a path of symplectic matrices
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References for Lagrangian Floer homology:
A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom. 28, 1988, 513-547 and references therein
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P. Seidel, Fukaya categories and Picard-Lefschetz theory, EMS Publishing House, 2008 |
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, Lagrangian intersection Floer homology - anomaly and obstruction, AMS, 2009
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Vin de Silva, Joel Robbin, Dietmar Salamon, Combinatorial Floer homology, Memoirs of the AMS 230, 2014
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Denis Auroux, A beginner's introduction to Fukaya categories, chapter in the book Contact and Symplectic Topology, Springer, 2014
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P. Seidel, Graded Lagrangian submanifolds, Bull. SMF, vol. 128, p. 103-149, 2000
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