Moduli Spaces and Integrable Systems

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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.