Principal Investigator: Helmut H. Hofer and Dusa McDuff
This instructional workshop, co-sponsored by the DFG (the German equivalent of the NSF) will bring together junior and senior researchers in the field of symplectic geometry. The field has its origins in Hamiltonian dynamics and geometric optics and has many fundamental applications. For example, numerical methods based on symplectic ideas (symplectic integrators) are used to compute the orbits of satellites and other celestial bodies. Historic highlights are KAM-Theory (Kolmogorov-Arnold-Moser), which leads to a proof of the stability of our solar system, and the theory of infinite-dimensional integrable systems, which is the mathematical underpinning of fiber optics, a technology that lies at the heart of all of our communication networks. More recently, the rapidly developing theory of pseudoholomorphic curves has led to deep new insights into the structure of three and four-dimensional space, and to unexpected new connections between geometry and physics. It is a surprising fact that the same underlying ideas apply to geometric optics, to dynamics and to the most abstract physics (i.e. string theory).
It is the purpose of this instructional workshop to describe the construction of symplectic field theory (SFT) in detail. SFT is a comprehensive theory of symplectic invariants, including such well-known theories as Floer theory, Gromov-Witten theory and contact homology. It is constructed by measuring moduli spaces of pseudoholomorphic curves. The richness of its structure comes from the fact that the moduli spaces have boundaries and singularities and that infinitely many moduli spaces interact with each other. These structures can be captured by a novel nonlinear Fredholm theory which is distinguished by two facts. The ambient spaces do not carry smooth structures in the usual sense and even have locally varying dimensions. However there is a notion of transversality (or regularity), and at regular points the solution sets are smooth orbifolds with boundaries and corners. Moreover the theory makes precise what it means for infinitely many Fredholm operators to interact with each other, which leads to a so-called "Fredholm Theory with Operations". This abstract Fredholm Theory is developed and illustrated by its application to SFT. The issues addressed are pertinent to a variety of important problems, and one can expect that the ideas presented at the workshop will have implications in a number of mathematical fields and their applications.
