Holomorphic curves in symplectic manifolds and integrable systems

摘要奖：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理论理解辛流形中全纯曲线的模空间的拓扑学。第一个项目的目的是构造一个可积族，它控制从亏格-g，节点Riemann曲面到被假定为环面完全交的多样性X的d次稳定全纯映射的拓扑，拓扑信息被编码在一个形式的幂函数级数中，称为环面簇的总后代势，目的是证明这个势是一个可积族的解。这个研究项目的第二个项目致力于可积系统与紧致Kaehler流形Y的辛场理论的相互作用，产生与上述总派生势有关的势。关于Y的Gromov-Witten不变量和Y上的射影线丛的Gromov-Witten不变量之间的关系预计会出现，这些关系是通过不可积系统的变换来描述的。辛几何是经典力学的哈密顿形式主义的基础结构，其中力学系统的行为是由类能量函数决定的。包含这种结构的几何空间可以是高维的和结构复杂的，该领域的许多最近的工作致力于通过嵌入良好嵌入的二维子曲面的关联空间来探索辛结构。二维曲面已经被研究了150多年，我们对它们的详细了解导致了对高维辛流形的相关不变量的约束。主要研究人员的工作探索了这类新的约束，其形式为行为良好的微分方程组，即提议标题的可积系统。

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.