Symplectic Field Theory (SFT) is a very large project designed to describe in a unified way, the theory of invariants of symplectic and contact manifolds. The project was initiated by Eliashberg, Givental and Hofer. and since then has found many striking applications in symplectic geometry and beyond. It is, probably, impossible to complete this project because new and new connections and directions of research being discovered. However, currently there are several distinct and active parts of this project. The analytic foundations of the theory brought to life a new type of functional analysis, called the polyfold theory.
The previous, 2005 Leipzig workshop, was devoted to this theory, and though still a lot to be done to build the firm foundations of the theory, this work is on the way to its successful completion.
The workshop will begin with a 2-day pre-course (Saturday-Sunday, August 5-6) in which the better known relevant material is being recalled for the benefit of young researchers entering the area.
Currently we plan 3 two-hour mini-lecture series:
Floer homology theory (2 lectures, M. Schwarz)
Definition of Floer Homology as a relative Morse theory
Floer Homology with the structure of a topological field theory
Gromov-Witten invariants (2 lectures, U. Frauenfelder)
Definitions of Gromov-Witten invariants in the monotone case
The Pair of pants product in Floer homology
The PSS-isomorphism
The quantum cup length estimate.
Differential algebra of a Legendrian knot (2 lectures, J. Latschev)
After a brief introduction to the classical theory of Legendrian knots, the main point of these two lectures is to discuss Chekanov's combinatorial description of the DGA of a Legendrian knot in R^3.
Here is the list of currently planned main lecture series:
Algebraic structures arising in SFT (2 lectures, Y. Eliashberg)
SFT brought to life new algebraic structures whose properties need to be thoroughly studied. One of the examples of such structures is the category of graded infinite-dimensional differential graded Weyl algebras and Fourier integral operators between them. One of the important questions here is to find computable homotopy invariants of these algebras. In some sense, the situation here is similar to Dennis Sullivan minimal model theory for differential graded algebras. However, as far as we know, nobody studied this in the context of Weyl algebras.
SFT and quantum integrable systems (1 lecture, Y. Eliashberg)
There was a discovered a link between SFT and the theory of infinite dimensional quantum integrable systems. Roughly speaking, SFT associates to a contact manifold V an infinite sequence of commuting differential operators. An explicit computation of these operators is currently known only in a few examples. However, even in the simplest example when V = S1 these operators turns out to be quantized Poisson commuting integrals of the classical integrable hierarchy of Burgers equation. Better understanding of this mechanism would be very important both, for SFT and the theory of integrable systems.
Contact homology: Computations and applications (3 lectures, F. Bourgeois)
SFT is such a rich and large theory that it is often difficult to compute. Contact homology and cylindrical contact homology are simpler invariants extracted from SFT, and some techniques can be used to simplify their computation. These are still very powerful invariants, since they can be used to distinguish contact structures, study homotopy groups of the space of contact structures and prove some non-squeezing theorems in contact topology.
Floer homology and loop product (1 lecture, M. Schwarz)
Floer homology for Hamiltonian systems had first been designed to find a proof for the non-degenerate Arnold-Conjecture. The key is that on closed symplectic manifolds Floer homology computes the classical homology. This fails, in general, completely on noncompact manifolds. Floer Homology can be infinite-dimensional. In this lecture it is shown that Floer homology on cotangent bundles with fibrewise asymptotically quadratic Hamiltonians is isomorphic to the homology of the free loop space. Moreover, the pair-of-pants product becomes isomorphic to the loop product.
SFT and string topology (3 lectures, K. Cieliebak)
To a closed oriented smooth manifold one can associate two types of invariants: the SFT of its unit cotangent bundle, and the string topology of its unparametrized loop space. Recent joint work with J.Latschev shows that these invariants, when suitably interpreted, are isomorphic. This isomorphism can be used to compute the SFT of unit cotangent bundles by topological methods, which in turn may have applications to the study of Lagrangian embeddings.
Invariants of knots and links via SFT (2 lectures, L. Ng)
I will describe what sorts of invariants can be associated to a knot or link in three-space by studying the SFT (or, more specifically, the contact homology) of the Legendrian unit conormal bundle. These invariants can be described in two ways: algebraically, through a direct count of holomorphic curves (joint work with T. Ekholm, J. Etnyre, and M. Sullivan), and topologically, in a way which seems closely related to string topology.
SFT and symplectic capacities (1 lecture, R. Hind)
Using some ideas from SFT, we will show that there does not exist a symplectic embedding of B^2(1) x R^4 into B^4(R) x R^2 for any R>0. This leads to ideas of higher order symplectic capacities and is joint work with Ely Kerman.
Categorification of Gromov-Witten theory and SFT ( 1 lecture, A. Voronov)
An algebraic formalism extending the structure of Gromov-Witten theory as a Topological Conformal Field Theory to Symplectic Field Theory will be discussed.
A version of rational symplectic field theory for Legendrian submanifolds (2 lectures, T. Ekholm)
In the first lecture we introduce a version of rational symplectic field theory for a Legendrian submanifold. More precisely, we associate to any many component Legendrian submanifold L of a contact manifold V a graded vector space over Z/2 with a differential which counts punctured holomorphic disks in the symplectization of V with boundary on the R-invariant Lagrangian submanifold associated to L and asymptoptic to Reeb chords at punctures. The corresponding homology is invariant under Legendrian isotopy. In particular, if L is a Legendrian submanifold of V then the differential on the vector space associated to the Legendrian link obtained from L by taking many parallel copies can be computed from the moduli spaces of holomorphic disks with boundary on L x R together with moduli spaces of Morse flow trees in L. In the second lecture we present applictaions of the theory described in the first lecture.