This is an introductory course in symplectic geometry. We will cover some foundations of symplectic geometry (such as local theory, Lagrangian submanifolds and Hamiltonian flows). The last part of the course will be devoted to symplectic capacities and some rigidity results.
Prerequisites: Familiarity with differential geometry (in particular, differential forms and vector fields on manifolds) and with topology (including elementary algebraic topology) will be assumed.
Handwritten lecture notes will be posted here (the password was emailed to all registered students on 19.09.2023).
Here is an errata for both, lecture notes and exercise sheets: Errata
Below we will post a tentative plan for the upcoming lectures.
| Date | Content |
|---|---|
| 19.09. | Motivation from physics, definition of a symplectic manifold, statements of highlights of the course (Gromov's Non-squeezing Theorem, Eliashberg–Gromov Theorem). |
| 21.09. | Arnol'd conjecture, Examples of symplectic manifolds (symplectic vector spaces, cotangent bundles, oriented surfaces, complex projective space) |
| 26.09. | Complex projective space, Symplectomorphisms, Hamiltonian diffeomorphisms |
| 28.09. | The group of Hamiltonian diffeomorphisms, autonomous Hamiltonians |
| 3.10. | Autonomous Hamiltonians, Moser stability |
| 5.10. | Darboux Theorem, equivalence of symplectic structures, linear complex structures |
| 10.10. | Linear complex structures, compatible triples |
| 12.10. | Compatible triples, almost complex structures |
| 17.10. | Integrable complex structures, symplectic capacities |
| 19.10. | Gromov width, rigidity of symplectomorphisms |
| 24.10. | Characterisation of symplectomorphisms via capacities |
| 26.10. | Characterisation of symplectomorphisms via capacities, Hofer-Zehnder capacity |
| 31.10. | Outline of proof of non-triviality of Hofer-Zehnder capacity, action functional |
| 2.11. | Action principle, action functional |
| 7.11. | Minimax principle, extending the action functional to a Hilbert space |
| 9.11. | Analysing the action functional |
| 14.11. Different room: F26.3 | Applying the minimax principle to the action functional |
| 16.11. | End of proof of non-triviality of Hofer-Zehnder capacity |
| 21.11. | Lagrangian submanifolds, the group of symplectomorphisms is locally path-connected |
| 23.11. | Lagrangian neighbourhood theorem, the group of symplectomorphisms is locally path-connected |
| 28.11. | Symplectic isotopies in exact symplectic manifolds |
| 30.11. | Flux homomorphism |
| 5.12. | Flux homomorphism and symplectic isotopies |
| 7.12. | Hamiltonian isotopies |
| 12.12. | (Non-)simplicity results, Calabi homomorphism |
| 14.12. | Q&A |
| 19.12. | Exam |
| 21.12. | A quick look into Floer homology |
A new exercise sheet will be posted every week on Tuesday or Wednesday. We will discuss the sheet in the exercise class on Friday, so it is recommended that you have a look at it beforehand.
The deadline for handing in solutions is 23:59 on Tuesday the following week. If you wish to hand in your solutions, please do so using the SAM Upload Tool. Corrections will be uploaded there a few days after the deadline.
If there are any issues with the SAM Upload Tool or if you notice any mistakes or typos in the exercises or in the solutions, please email Ana, the course organiser.
| Exercise sheet | Due by | Solutions |
|---|---|---|
| Sheet 1 | 26.09. | Solutions 1 |
| Sheet 2 | 03.10. | Solutions 2 |
| Sheet 3 | 10.10. | Solutions 3 |
| Sheet 4 | 17.10. | Solutions 4 |
| Sheet 5 | 24.10. | Solutions 5 |
| Sheet 6 | 31.10. | Solutions 6 |
| Sheet 7 | 07.11. | Solutions 7 (corrected version 14.12.23) |
| Sheet 8 | 14.11. | Solutions 8 |
| Sheet 9 | 21.11. | Solutions 9 |
| Sheet 10 | 28.11. | Solutions 10 |
| Sheet 11 | 05.12. | Solutions 11 |
| Sheet 12 | 12.12. | Solutions 12 |
| Sheet 13 | - | Solutions 13 |
Occasional notes related to what was discussed in exercise classes will be posted here (the password is same as the password for the lecture notes Polybox folder and was emailed to all registered students on 19.09.2923).
The exam will take place on Tuesday 19.12.2023. from 10:00 to 13:00 in HG D 1.2.
PhD students who wish to obtain credits for this course do not have to take the exam, but they have to hand in correct solutions to at least 50% of the exercises.