Vorlesung: Introduction to Symplectic Field Theory, Sommersemester 2020

General information

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

Time and place: Lectures on Tuesdays 13:00-15:00 (c.t.) in room 1.115 (RUD25), and Wednesdays 11:00-13:00 (c.t.) in room 1-1304 (RUD26), plus Problem Class (Übung) Wednesdays 13:00-15:00 (c.t.) in room 1.011 (RUD25).

Note: Since people generally prefer to have more than 20 minutes for lunch, we will discuss shifting the Wednesday lecture and/or problem class by 15 minutes. Please e-mail me if you have an opinion on this but can't be there for the first lecture.

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 soon-to-be-published book, Lectures on Symplectic Field Theory. Please send me corrections if you notice any errors! I plan to post more up-to-date versions of the book on this page as the course progresses.

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 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. two lectures), 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