Holomorphic curves in symplectic manifolds and integrable systems

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AbstractAward: DMS-0707150Principal Investigator: Todor MilanovThe long term goal of these projects is to understand thetopology of moduli spaces of holomorphic curves in symplecticmanifolds in terms of the theory of integrable systems. Thefirst project aims to construct an integrable hierarchy whichgoverns the topology of the moduli spaces of degree d stableholomorphic maps from a genus-g, nodal Riemann surface to avariety X which is assumed to be a toric complete intersection.The topological information is encoded in a formal power seriescalled the total descendant potential of the toric variety, andthe goal is to show that this potential is a solution to anintegrable hierarchy. The second project in this researchprogram is dedicated to an interaction of integrable systems withsymplectic field theory of a compact Kaehler manifold Y,developing a potential related to the total descendant potentialdescribed above. Relationships are expected to emerge betweenGromov-Witten invariants for Y and for a projective line bundleover Y, with the relationships described via transformations ofintegrable systems.Symplectic geometry is the structure underlying the Hamiltonianformalism of classical mechanics, in which the behavior of amechanical system is determined by an energy-like function.Geometric spaces carrying such structures can be high-dimensionaland structurally complicated, and much recent work in the area isdevoted to exploring symplectic structures through associatedspaces of well-imbedded two-dimensional subsurfaces of them.Two-dimensional surfaces have been studied for more than 150years and our detailed understanding of them leads to constraintsupon the associated invariants of high-dimensional symplecticmanifolds. The principal investigator's work explores newconstraints of this kind that take the form of well-behaveddifferential equations, the integrable systems of the proposaltitle.