Workshop on Symplectic Field Theory; May 14-20, 2005; Leipzig, Germany

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AbstractAward: DMS-0505968Principal Investigator: Helmut H. Hofer and Dusa McDuffThis instructional workshop, co-sponsored by the DFG (the Germanequivalent of the NSF) will bring together junior and seniorresearchers in the field of symplectic geometry. The field hasits origins in Hamiltonian dynamics and geometric optics and hasmany fundamental applications. For example, numerical methodsbased on symplectic ideas (symplectic integrators) are used tocompute the orbits of satellites and other celestialbodies. Historic highlights are KAM-Theory(Kolmogorov-Arnold-Moser), which leads to a proof of thestability of our solar system, and the theory ofinfinite-dimensional integrable systems, which is themathematical underpinning of fiber optics, a technology that liesat the heart of all of our communication networks. More recently,the rapidly developing theory of pseudoholomorphic curves has ledto deep new insights into the structure of three andfour-dimensional space, and to unexpected new connections betweengeometry and physics. It is a surprising fact that the sameunderlying ideas apply to geometric optics, to dynamics and tothe most abstract physics (i.e. string theory).It is the purpose of this instructional workshop to describe theconstruction of symplectic field theory (SFT) in detail. SFT is acomprehensive theory of symplectic invariants, including suchwell-known theories as Floer theory, Gromov-Witten theory andcontact homology. It is constructed by measuring moduli spaces ofpseudoholomorphic curves. The richness of its structure comesfrom the fact that the moduli spaces have boundaries andsingularities and that infinitely many moduli spaces interactwith each other. These structures can be captured by a novelnonlinear Fredholm theory which is distinguished by twofacts. The ambient spaces do not carry smooth structures in theusual sense and even have locally varying dimensions. Howeverthere is a notion of transversality (or regularity), and atregular points the solution sets are smooth orbifolds withboundaries and corners. Moreover the theory makes precise whatit means for infinitely many Fredholm operators to interact witheach other, which leads to a so-called "Fredholm Theory withOperations". This abstract Fredholm Theory is developed andillustrated by its application to SFT. The issues addressed arepertinent to a variety of important problems, and one can expectthat the ideas presented at the workshop will have implicationsin a number of mathematical fields and their applications.