Lecturer 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.

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:

• Differential geometry: manifolds and vector bundles, differential forms and Stokes' theorem, connections, basic familiarity with symplectic manifolds
• Functional analysis: linear operators on Banach spaces, basics of Sobolev spaces, Fredholm operators
• Differential topology: smooth mapping degree, intersection numbers, Sard's theorem
• Algebraic topology: fundamental group, homology and cohomology of manifolds, Poincaré duality, first Chern class, homological intersection numbers

• Contact manifolds (Geiges, An introduction to contact topology)
• Differential calculus on Banach spaces and Banach manifolds (two books by Lang: Real and Functional Analysis and Fundamentals of Differential Geometry)
• Closed pseudoholomorphic curves (McDuff-Salamon J-holomorphic Curves and Symplectic Topology, or my book in preparation)
• Floer homology (Salamon, Lectures on Floer homology; here's a free pdf version)

• Bourgeois, Introduction to Contact Homology, lecture notes (2003)
• Bourgeois, A Morse-Bott approach to contact homology, PhD thesis (2002)
• Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder, Compactness results in symplectic field theory; arXiv version (2003)
• Eliashberg, Invariants in contact topology (1998)
• Eliashberg, Symplectic field theory and its applications (2007)
• Eliashberg, Givental and Hofer, Introduction to symplectic field theory; arXiv version (2000)
• Geiges, An introduction to contact topology, Cambridge University Press 2008
• Gromov, Pseudoholomorphic curves in symplectic manifolds (1985)
• Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three (1993)
• Latschev and Wendl (with an appendix by Hutchings), Algebraic torsion in contact manifolds; arXiv version (2011)
• McDuff and Salamon, J-holomorphic Curves and Symplectic Topology, AMS 2012
• Salamon, Lectures on Floer homology; free pdf version (1997)
• Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders; arXiv version (2007)
• Siefring, Intersection theory of punctured pseudoholomorphic curves; arXiv version (2009)
• Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, book in preparation

Chris Wendl's homepage