Moduli Spaces and Integrable Systems

模空间和可积系统

• 批准号: 9971966
• 负责人: Emma Previato
• 金额: $ 8.28万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971966&HistoricalAwards=false
• 关键词: Moduli; Spaces; Integrable; Systems

AbstractAward: DMS-9971966 Principal Investigator: Emma PreviatoThe project consists of three parts and concerns constructions inalgebraic geometry and solutions to partal differentialequations. Part I uses differential algebra to investigate openproblems of the KP hierarchy: Ritt's theory of constructivedifferential algebra will be applied to characterize maximalcommutative rings of ordinary/partial differential operators;differential resultants will be implemented to expresshigher-dimensional reciprocity laws. In part II, a connection isestablished between rank-2 vector bundles with canonicaldeterminant over an algebraic curve, and varieties containing thecanonical curve. By this technique, equations for important lociof bundles will be sought, as well as the significance of theseloci in 2-Theta space, where the moduli space of bundles embeds,with applications to the Schottky problem, for the 2-Theta spaceof a principally polarized abelian variety. These equations formoduli spaces of bundles have potential applications to explicitintegration of systems of Hitchin type. Part III will linkintegrable systems and Frobenius manifolds, by producing asymplecto-geometric interpretation of canonical and flatcoordinates on Hurwitz spaces.The proposed work spans the areas of algebraic geometry anddifferential equations. While algebraic geometry has been afield of basic research since the nineteenth century, in recentyears it has found applications to mathematical physics. Thisproject explores new techniques to give answers to open problemsof classical algebraic geometry such as equations for certainmoduli spaces, or the number of polynomials of a given type thatdefine a canonical curve; it also explores new applications ofthese algebraic structures to the exact solution of partialdifferential equations that model physical phenomena, from waterwaves to quantum-field-theory. One new component is the use ofcomputer algebra systems, or symbolic computation, to implementclassical theories. The computational focus will be an explicitdefinition of algebraic varieties via differential equations; byrunning the program through several classes of examples, whichused to be unfeasible manually, conjectures will be tested andproperties sought to characterize unknown features of thesespectral varieties. Applications of technology to fields such asalgebraic geometry has attracted great recent interest, includingprograms like "Symbolic Computation in Geometry and Analysis" atMSRI (Fall 1998). This project will produce MAPLE packages whichwill be made publicly available to other researchers.

项目负责人：Emma previato项目由三个部分组成，涉及代数几何的构造和偏微分方程的解。第一部分使用微分代数研究KP层次的开问题：Ritt的构造微分代数理论将被应用于刻画常/偏微分算子的极大交换环；微分结果将用于表示高维互易律。在第二部分中，建立了代数曲线上具有标准行列式的秩2向量束与包含标准曲线的变体之间的联系。通过这种技术，将寻求束的重要位置的方程，以及束的模空间嵌入的2-Theta空间中这些位置的意义，并将其应用于Schottky问题，用于主要极化阿贝尔变体的2-Theta空间。这些束模空间的方程对希钦型系统的显式积分具有潜在的应用价值。第三部分将通过在赫维茨空间上给出正则坐标和平面坐标的无渐几何解释，将可积系统和Frobenius流形联系起来。提议的工作跨越代数几何和微分方程的领域。虽然代数几何自19世纪以来一直是基础研究领域，但近年来它已经在数学物理中找到了应用。该项目探索了新的技术，以给出经典代数几何开放问题的答案，如某些模空间的方程，或定义正则曲线的给定类型的多项式的数量；它还探索了这些代数结构在模拟物理现象的偏微分方程的精确解中的新应用，从水波到量子场论。一个新的组成部分是使用计算机代数系统或符号计算来实现经典理论。计算重点将是通过微分方程明确定义代数变量；通过运行程序，通过几个类别的例子，这些过去是不可行的手动，推测将被测试和性质寻求表征这些光谱品种的未知特征。技术在代数几何等领域的应用最近引起了人们的极大兴趣，其中包括msri（1998年秋季）的“几何与分析中的符号计算”等项目。该项目将产生MAPLE软件包，这些软件包将公开提供给其他研究人员。