Mathematical Sciences: Geometry of Period Mappings and Abelian Varieties

这项研究将在代数几何中进行，特别是对曲线、超曲面和阿贝尔变体的模空间的几何定义子变体的研究，以及对更一般的周期空间的研究，以及对周期映射的应用。特别是，他们计划将超曲面族的判别轨迹的几何分析扩展到映射和代数循环族。一个主要的应用将是研究射影三倍上的一个环和阿贝尔变体上的两个环的Hodge问题。

This research will be in algebraic geometry, especially the study of geometrically defined subvarieties of moduli spaces of curves, hypersurfaces, and abelian varieties, and of more general period spaces, and applications to period maps. In particular they plan to extend their analysis of the geometry of discriminant loci for families of hypersurfaces to mappings and families of algebraic cycles. One principal application will be to the study of the Hodge problem for one cycles on projective threefolds and two-cycles on abelian varieties.