Mathematical Sciences: Periods and Moduli Spaces

本研究将围绕两个基本问题展开。主要的一个涉及周期空间的几何。在这个问题的第一部分，P.I.将尝试构造任意权值下周期空间的光滑解析紧化。边界分量应被解释为混合Hodge结构的周期空间，胶合应反映Hodge结构退化的几个基本结果。这种紧化可以使霍奇理论的许多内容以更几何的形式重新塑造，并有助于志村变体的研究。在这个问题的第二部分，将应用外微分系统正则积分元的Cartan-Kahler理论来求周期映射图像及其与Shimura变元的交的维限和局部不变量。第二个问题处理Lax结构的模空间。本文将研究给定可积哈密顿系统的Lax结构的模空间作为该系统的基本代数-几何不变量，以取代谱曲线这一不太自然但普遍的概念。应用设想了一系列经典系统的形式的函数参数化，这是完全自然的系统，而不是只工作到等同源。本研究将集中于代数几何中的问题及在偏微分方程中的应用。这样一来，作为联立多项式方程的一组解而产生的几何对象不仅会因其本身的缘故而得到研究，而且还会尝试将这一深奥的理论应用于其他领域

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