Reduced to its essence, Symplectic Field Theory (SFT) is the study of holomorphic curves in symplectic manifolds with cylindrical ends. It contains Gromov-Witten theory as the special case when the manifold has no ends, and symplectic Floer homology as the special case of holomorphic cylinders in the product of the closed symplectic manifold and a twice-punctured sphere; but the general case is far more subtle than either of these examples. In their seminal "Introduction to Symplectic Field Theory", Eliashberg, Givental and Hofer outlined the geometric phenomena which should occur in SFT. They also sketched a rich algebraic formalism, in terms of infinite-dimensional Poisson algebras, which captures these geometric phenomena. (The formalism is so rich, for example, as to lead to remarkable new examples of infinite hierarchies of integrable systems.) Since then, several other formalisms capturing parts of the SFT package have also emerged. These have lead to striking relationships with other classes of invariants.
To date, most of the applications of SFT have used only small parts of the theory's full potential, for two reasons. The first reason is that many of the foundational results required to define SFT invariants in general are only now being proved. The necessary compactness results were only established in 2003. Trying to prove the necessary transversality results has led to the invention of polyfolds, a broad new class of objects the fundamental nature of which is still being actively explored, but which should have many applications outside of SFT. In the important case of "open'' SFT (i.e., SFT relative to a Lagrangian submanifold), even less is known. Indeed, it is not yet known what general algebraic structures should arise in this case, though progress is being made.
The second reason is that in general SFT remains remarkably difficult to compute. The full SFT has been computed in only a very limited number of cases (e.g., S1× R). To date, there are two main methods of computation: working in a highly degenerate setting an using the Morse-Bott definition of SFT, or relating SFT to another theory which is more readily computable (e.g., relative Gromov-Witten invariants, or string topology).
Despite the still limited understanding of its general technical underpinnings, SFT and the associated philosophy has already had substantial practical impact. SFT and its immediate ancestors provided the first modern tools for distinguishing contact manifolds and their Legendrian submanifolds; answers to classical questions about the existence of embedded Lagrangian submanifolds in symplectic manifolds; information on the contactomorphism group of contact manifolds and the symplectomorphism group of symplectic manifolds; results on closed Reeb orbits in contact manifolds and many cases of the Weinstein conjecture; new topological invariants of three-manifolds and knots inside them; results on the contact analogues of the symplectic non-squeezing theorems; and many others. Related fields include gauge theory and the many gauge-theoretic Floer homologies; Lagrangian intersection Floer homologies and cluster homology; enumerative invariants in algebraic geometry including Gromov-Witten theory and Donaldson-Thomas theory; quantum topology and string topology; and many other fields.
Symplectic field theory, thus, touches on many other fields of modern geometry and topology. This conference will focus on both symplectic field theory itself and related fields like enumerative algebraic geometry, Floer homology, symplectic and contact topology, and gaug theory. While some talks will emphasize foundational aspects in various of these areas, many are expected to focus on the deep connections between the different subjects.
