Mathematical Sciences: Periods and Moduli Spaces

点击翻译按钮获取中文摘要

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