LecturerChris WendlUCL Mathematics, office 802a (25 Gordon Street) c dot wendl at ucl dot ac dot uk Time and placeLectures 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.
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:
Term 1: | |
Wednesday, 14 October | Introduction: history (Gromov '85, Floer homology, Weinstein conjecture), contact manifolds and symplectic cobordisms, sketch of the algebraic formalism |
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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) |