Holomorphic curves in symplectic manifolds and integrable systems

AbstractAward：DMS-0707150首席研究员：Todor Milanov这些项目的长期目标是根据可积系统的理论来理解辛流形中全纯曲线的模空间的拓扑。 第一个项目的目标是构造一个可积族，它控制从亏格g，结点Riemann曲面到各种复曲面X的d次稳定全纯映射的模空间的拓扑，并将拓扑信息编码在一个称为复曲面簇的全后代势的形式幂级数中，目的是证明这个势是一个可积族的解。 本研究计划的第二个项目致力于可积系统与紧致Kaehler流形Y的辛场论的相互作用，开发与上述总下降势相关的势。 通过可积系统的变换描述了Y的Gromov-Witten不变量和Y上的投影线的Gromov-Witten不变量之间的关系.辛几何是经典力学的Hamilton形式主义的基础结构，其中力学系统的行为由类能量函数决定.承载这种结构的几何空间可以是高维的和结构复杂的，近年来，人们对二维曲面的研究已经有150多年的历史，对二维曲面的深入理解导致了对高维辛流形的相关不变量的限制。 首席研究员的工作探索了这种新的约束，这种约束采取了良好的微分方程的形式，即命题标题中的可积系统。

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.