Fukaya Categories and Applications

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AbstractAward: DMS-0405516Principal Investigator: Paul SeidelThe main aim of the project is to harness the power of Fukayacategories for use in symplectic topology. This includesdeveloping computational tools, building a repertoire ofnontrivial examples, and also studying the relationship betweenFukaya categories and Gromov-Witten invariants. Concreteexamples that will be looked at include: Fukaya categories ofRiemann surfaces; Lagrangian submanifolds in cotangent bundles;and the symplectic geometry of Khovanov's linkinvariants. Various aspects of mirror symmetry will be importantsources of intuition, and the experience gained in this wayshould ultimately feed back into a better understanding of mirrorsymmetry itself, especially its homological aspects.From a wider perspective, the project is part of the reaction ofthe scientific community to recent advances in theoreticalphysics. The contribution that mathematics can make is to verifythe soundness and consistency of physical intuition, and toprepare the general ground on which further development canoccur. This is particularly important in those situations wheredevelopments in physics suggest the presence of deep and complexstructures, which are difficult to detect by direct experiment. Anice aspect is that the same circle of ideas allows one to answersome purely mathematical (geometric) questions inspired by the"old" physics of classical mechanics.