Holomorphic curves in symplectic manifolds and integrable systems

项目负责人：Todor milanov这些项目的长期目标是利用可积系统理论来理解辛流形中全纯曲线的模空间的拓扑结构。第一个项目旨在构建一个可积的层次结构，该层次结构控制从属g，节点黎曼曲面到变种X的d次稳定全纯映射的模空间的拓扑结构，假设变种X是一个环完全交。拓扑信息被编码在一个正式的幂级数中，称为环变的总后代势，目标是证明这个势是一个不可积层次的解。本研究计划的第二个项目致力于可积系统与紧化Kaehler流形Y的辛场论的相互作用，开发与上述总后代势相关的势。期望在Y和Y上的射影线束的engromov - witten不变量之间出现关系，这种关系通过可积系统的变换来描述。辛几何是经典力学哈密顿形式化的基础结构，在哈密顿形式化中，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能是高维的，结构复杂的，最近该领域的许多工作都致力于通过它们的良好嵌入的二维亚表面的相关空间来探索辛结构。二维曲面已经被研究了150多年，我们对它们的详细理解导致了对高维辛流形相关不变量的约束。首席研究员的工作探索了这类新的约束，这些约束以行为良好的微分方程的形式出现，即提案标题中的可积系统。

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.