Vorlesung: Introduction to Symplectic Field Theory, Sommersemester 2020

Announcements

older announcements (no longer relevant)

Lecture notes etc.

The current version of the notes includes Lectures 1-13, Appendices A-C and the bibliography. (last update: 15.07.2020 at 0:45)
(Note: The file behind this link will be updated routinely throughout the semester, so make sure you have pressed the "refresh" button if necessary to get the most recent update.)

As this course progresses, I will be revising the lecture notes I produced while teaching a similar course at UCL four years ago, and making the revision available at the link above. (The complete unrevised version is available on the arXiv, in case you're curious.) The aim will be to cover approximately one "lecture" in the notes every week (hence 2 actual class lectures, plus Übung), so you can expect the link to be updated weekly. These notes are to be published as a book in the EMS Series of Lectures in Mathematics series when they are finished. For that reason and others, I warmly welcome all information about typos or mathematical errors you may notice, or any other suggestions that occur to you as you read them. Just send me an e-mail.

I will also post my whiteboard notes from the online lectures here as they become available:
Week 1: April 21 April 22 April 22 (Übung)
Week 2: April 28 April 29 April 29 (Übung)
Week 3: May 5 May 6 May 6 (Übung)
Week 4: May 12 May 13
Week 5: May 19 May 20 May 20 (Übung)
Week 6: May 26 May 27 May 27 (Übung)
Week 7: June 2 June 3 June 3 (Übung)
Week 8: June 9 June 10 June 10 (Übung)
Week 9: June 16 June 17
Week 10: June 23 June 24 June 24 (Übung)
Week 11: June 30 July 1 July 1 (Übung)
Week 12: July 7 July 8 July 8 (Übung)
Week 13: July 14 July 15 (lecture + Übung)

Exercises


General information

Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage)

Moodle: https://moodle.hu-berlin.de/course/view.php?id=95257
IMPORTANT: You will need to join the moodle for the course in order to obtain the Zoom links for online lectures. HU students can access moodle using their HU username and password. Non-HU users can access it by following the above link and then setting up a HU Moodle Account with their external e-mail address as a username. You will need an enrolment key, which was announced during the first lecture; if you have not yet joined the moodle but would still like to attend the class, contact me for the enrolment key.

Time and place: Lectures on Tuesdays 13:00-15:00 (c.t.) in room 1.115 (RUD25), and Wednesdays 11:00-12:30 (s.t.) in room 1-1304 (RUD26), plus Problem Class (Übung) Wednesdays 13:30-15:00 (s.t.) in room 1.011 (RUD25).
All lectures and problem classes will take place via ZOOM until a return to campus is possible. The required links for Zoom meetings will be posted on the moodle (see above).

Language: The course will be taught in English.

Prerequisites: I will assume that all students are comfortable with the essentials of differential geometry (smooth manifolds, vector fields and Lie bracket, differential forms and Stokes' theorem, de Rham cohomology, connections on vector bundles), as well as some basic algebraic topology (fundamental group, singular homology and cohomology) and functional analysis (continuous linear operators on Banach spaces, the standard Lp-spaces). Some previous knowledge of additional topics from topology (homological intersection theory, the first Chern class) and functional analysis or PDE theory (Fourier transforms, distributions, Sobolev spaces) will sometimes be helpful. For students who have not seen any symplectic or contact geometry before, I will give a concise overview of the subject in the first problem class. Some prior familiarity with more directly related topics such as holomorphic curves and Floer homology may be helpful for grasping the big picture, but will not be explicitly assumed.

Contents:
Symplectic manifolds were invented around the turn of the 20th century as the natural geometric setting in which to study Hamilton's equations of motion from classical mechanics. The subject of symplectic geometry has developed considerably since then, and it retains a close connection with theoretical physics despite being technically a branch of “pure” mathematics. In particular, the subfield known as symplectic topology, which deals with “global” rather than “local” properties of symplectic manifolds, has witnessed an explosion of activity since the introduction of techniques from elliptic PDE theory in the 1980s. The most spectacular advances came from Gromov's theory of pseudoholomorphic curves, which has led to a wide assortment of algebraic invariants of symplectic manifolds, some of them related to structures that physicists study in quantum field theory or string theory. Symplectic field theory (SFT) is one such invariant, or more accurately, a large framework for extracting algebraic invariants of contact manifolds (the odd-dimensional analogue of symplectic manifolds) and symplectic cobordisms between them by counting punctured holomorphic curves.

After a brief introduction and survey of the structure of SFT, the goal of this course will be to establish the basic analytical underpinnings of the theory (moduli spaces, Fredholm theory, index calculation, transversality and compactness), and then explore a few of its applications to natural questions in contact geometry such as distinguishing non-diffeomorphic contact structures, obstructing symplectic fillings of contact manifolds, and proving the existence of periodic orbits of Hamiltonian systems on contact-type hypersurfaces (AKA the “Weinstein conjecture”).

As some students may already have heard, giving a fully rigorous definition of SFT requires very recent and subtle analytical ideas (e.g. polyfolds) which would fill an entire course in themselves, thus we will not attempt this, but instead focus on the “classical” portion of the theory (i.e. everything that can be proved without needing multiply covered holomorphic curves to be regular). This is sufficient for the applications mentioned above.

References: The course will mostly follow my lecture notes which are posted (with regular updates) at the top of this page.

If you are new to symplectic geometry, you will find most of the basics explained nicely in the book Lectures on Symplectic Geometry by Ana Cannas da Silva (Springer LNM, 2008). If at some point you decide to get really serious about symplectic topology and/or holomorphic curve theory, then you'll eventually also need to buy both of the books by Dusa McDuff and Dietmar Salamon:

Homework: I will assign exercises sometimes. Sometimes I will discuss them in the problem class. They will not be graded.

Grades: Since this is an advanced course, I have a fairly relaxed attitude about grades. If you stay with me for the whole semester, you can come to my office / meet with me on Zoom for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular topic from the course to focus on (typically the contents of one week, i.e. one lecture in the notes), and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.

Chris Wendl's homepage