Symplectic Field Theory
Informal graduate course at University College London

This is the archived homepage for a course that is now finished, but you may be interested in the lecture notes, which have now been assembled into a single file and uploaded to the arXiv, and will be appearing in book form in the EMS Lectures in Mathematics series. See the expository section of my publications page for the relevant links and further updates.

Quick links:
schedule and lecture notes   literature


Chris Wendl
UCL Mathematics, office 802a (25 Gordon Street)
c dot wendl at ucl dot ac dot uk

Time and place

Lectures in the 2015 Spring term will take place on most Wednesdays 4-6pm (starting on 3 February), usually in UCL Maths Room 505 (see schedule below for details).

Lectures in the 2015 Autumn term took place on select Wednesdays 4-6pm in UCL Maths Room 505 and Mondays 4-6pm in Drayton B06, with occasional exceptions.


Symplectic Field Theory is a general framework for defining invariants of contact manifolds and symplectic cobordisms between them, using Gromov's theory of pseudoholomorphic curves. First introduced in a paper by Eliashberg, Givental and Hofer in 2000, the ideas behind SFT go back at least as far as Gromov's famous 1985 paper on holomorphic curves and Floer's subsequent solution to the Arnold conjecture on symplectic fixed points, which created a major industry in the development of "Floer type" theories. After Hofer's introduction in 1993 of finite-energy J-holomorphic planes to study the Weinstein conjecture on periodic orbits of Hamiltonian systems, it became clear that one should try to develop a Floer type theory based on punctured pseudoholomorphic curves in symplectisations of contact manifolds. The algebraic structure of the resulting theory turned out to be significantly more elaborate than in the original Floer homology, and its name derives partially from a certain similarity to topological quantum field theories. Its analytical difficulties are also formidable and, indeed, not all of them have yet been satisfactorily dealt with.

The goal of this course will be to explain the algebraic structure of SFT and the analytical and geometric phenomena that underlie that structure, and to illustrate it with a few sample applications in which computations can be carried out and lead to rigorously provable results. A large portion of the course will deal with "standard" topics on the analysis of pseudoholomorphic curves, and we will deal with these topics in a rigorous way to the extent that time allows. An important caveat to understand from the beginning is that even 15 years after the structure of SFT was first sketched, its analytical foundations remain work-in-progress, and one could base an entire course on the development of the rather non-standard methods required for these analytical foundations. This is not that course, and as a consequence, it will not be within our power to provide complete proofs that SFT has all the structure it is meant to have, nor indeed that it is a well-defined theory at all, except in a few very special cases. At the present moment in its development, SFT should be thought of less as a theory to be applied than as a source of inspiration: it can often provide valuable intuition and suggest conjectures that then turn out to be rigorously provable using more standard techniques. We will illustrate this principle via applications to the classification of contact structures and obstructions to symplectic fillings and cobordisms between certain contact 3-manifolds.

Target audience

The course is aimed mainly at PhD students in differential geometry or related fields who are not afraid of analysis. Some knowledge of the following topics in particular will be assumed:

The following topics are not considered prerequisites, but some knowledge of them is likely in any case to be helpful, and you will want to have a good reference for them within arm's reach (such as the ones recommended below):

Lecture schedule and notes

The following schedule is tentative and subject to change. I will attempt to produce lecture notes for each lecture, but experience suggests that this may not be sustainable.

(Update November 2016: The lecture notes have now been removed from this page but remain available in updated form via the expository section of my publications page.)
Term 1:
Wednesday, 14 October Introduction: history (Gromov '85, Floer homology, Weinstein conjecture), contact manifolds and symplectic cobordisms, sketch of the algebraic formalism
Wednesday, 21 October Some basics on pseudoholomorphic curves: linearizations, elliptic regularity, similarity principle, unique continuation
Monday, 26 October Nonlinear regularity, asymptotic operators and spectral flow
Wednesday, 28 October The Conley-Zehnder index, the Fredholm property on surfaces with cylindrical ends
Wednesday, 18 November Riemann-Roch on surfaces with cylindrical ends
Monday, 23 November Stable Hamiltonian structures, symplectic cobordisms with stable boundary, moduli spaces of asymptotically cylindrical pseudoholomorphic curves
Monday, 30 November Nonlinear functional analytic setup, implicit function theorem, transversality for generic J in cobordisms
Wednesday, 2 December
in room 500!
Proof of generic transversality in cobordisms and symplectizations
Monday, 7 December Finite energy and asymptotics, bubbling and breaking, sketch of the SFT compactness theorem
SFT compactness "animation"
Wednesday, 9 December Cylindrical contact homology and an application to distinguishing tight contact structures on the 3-torus
Term 2
Wednesday, 3 February Determinant line bundles, coherent orientations, good vs. bad orbits
Wednesday, 10 February lecture postponed to 17 February
Wednesday, 17 February The SFT generating function, grading and signs, combinatorics of gluing
Wednesday, 24 February Full contact homology, rational and full SFT, BV-infinity algebra formalism, cobordism maps and algebraic torsion
Wednesday, 2 March Automatic transversality in dimension four, normal Chern number and "wind-pi", consequences for orientations
(lecture notes will appear someday)
Wednesday, 9 March Intersection numbers and adjunction formula for punctured holomorphic curves
(lecture notes will appear someday)
Wednesday, 16 March no lecture
Wednesday, 23 March lecture postponed to 5 April
Tuesday, 5 April
in room 500!
Vanishing of contact homology and algebraic torsion computations
(lecture notes will appear someday)


To supplement the lecture notes for the course, the following is a list of papers and books on relevant topics. It will probably grow as the term progresses.

Chris Wendl's homepage