Mathematical Sciences: Periods and Moduli Spaces

本研究将围绕两个基本问题展开。 主要 一个涉及周期空间的几何学。 第一部分 这个问题的私人侦探将试图构建一个平滑的 周期空间的任意权解析紧化 边界分量应解释为周期 混合霍奇结构的空间和胶合应反映 关于Hodge结构退化的几个基本结果。 这种紧凑化应该可以重新塑造大部分的 霍奇理论在一个更几何的形式，并帮助研究 志村品种。 在这个问题的第二部分， 外正则积分元的Cartan-Kahler理论 微分系统将被应用于获得维数界 和周期图的图像的局部不变量，以及 它与志村品种的交叉。 第二个问题 研究Lax结构的模空间。 这里的研究 将集中研究Lax结构的模空间， 一个给定的可积哈密顿系统作为基本代数- 系统的几何不变式取代了不太自然的 虽然光谱曲线的常见概念。 应用 一系列经典系统的形式， 函数参数化是完全自然的， 系统，而不是工作，只有到ISO9001。 这项研究将集中在代数几何问题 偏微分方程的应用 不仅 那么几何对象将作为解的集合而出现 同时多项式方程的研究，为自己的 但将尝试将这一深刻的理论应用于其他

This research will center on two basic problems. The main one involves the geometry of the period space. In the first part of this problem the P.I. will attempt to construct a smooth analytic compactification of period space in arbitrary weight. The boundary components should have an interpretation as period spaces for mixed Hodge structures and the gluing should reflect several fundamental results on degeneration of Hodge structures. This compactification should make it possible to recast much of Hodge theory in a more geometric form and help in the study of Shimura varieties. In the second part of this problem the Cartan-Kahler theory of regular integral elements for exterior differential systems will be applied to obtain dimension bounds and local invariants for the image of the period map as well as its intersection with Shimura varieties. The second problem deals with moduli spaces of Lax structures. Here the research will focus on the study of the moduli space of Lax structures for a given integrable Hamiltonian system as the fundamental algebra- geometric invariant of the system replacing the less natural though common notion of spectral curve. Applications are envisaged to a series of classical systems in the form of theta- function parametrizations which are completely natural to their systems rather than working only up to isogeny. This research will focus on problems in algebraic geometry and applications to partial differential equations. Not only then will the geometric objects arising as the set of solutions of simultaneous polynomial equations be studied for their own sake but attempts will be made to apply this deep theory to other