Moduli Spaces and Integrable Systems

AbstractAward：DMS-9971966首席研究员：Emma Previato该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求解。 第一部分使用微分代数调查KP层次的开放问题：Ritt的理论constructivedifferential代数将被应用于刻画极大交换环的普通/偏微分算子;微分结果将被实现来表达高维互易律。 第二部分建立了代数曲线上具有标准行列式的秩-2向量丛与包含标准曲线的簇之间的联系。 通过这种技术，方程的重要lociof束将寻求，以及theseloci在2-Theta空间的意义，其中包的模空间嵌入，与应用程序的肖特基问题，为2-Theta空间的一个主要极化阿贝尔品种。 这些丛模空间方程对于希钦型系统的显式积分有潜在的应用。 第三部分将通过对Hurwitz空间上的正则坐标和平坦坐标的渐近几何解释，将可积系统和Frobenius流形联系起来。 虽然代数几何自世纪以来一直是基础研究的一个领域，但近年来它已被应用于数学物理。 这个项目探索新的技术来回答经典代数几何的开放问题，例如某些模空间的方程，或者定义一条标准曲线的给定类型的多项式的数量;它还探索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，以实现经典理论。 计算的重点将是一个明确的定义代数簇通过微分方程;通过运行程序通过几个类的例子，这是不可行的手动，结构将被测试和属性寻求表征未知的特点，这些频谱品种。 应用技术领域，如代数几何吸引了极大的兴趣，包括最近的程序，如“符号计算几何和分析”在MSRI（1998年秋季）。 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。

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.