CAREER: Advances in Hodge Theory and Moduli

这项建议涉及模空间几何学的研究和研究它们的工具的发展。特别是，PI计划将抽象霍奇理论中的一些新开发的技术应用于模空间的几何背景。PI将专注于研究特殊类别的品种，如Calabi-Yau三重和高维Hyperkahler流形。在一个不同但相关的方向上，PI旨在通过KSBA方法为某些类别的曲面构造几何紧化，该方法受到最小模型程序的启发。PI积极参与本科生和研究生的培训，近年来，他组织了几次具有很强教育成分的活动。他将扩大这些活动。特别是，他将为本科生举办暑期研究活动。此外，作为将在菲尔兹研究所（2013年秋季）举行的卡拉比-丘品种专题方案的一部分，PI将为研究生组织一个入门学校和其他培训活动。其他计划的活动包括为本科生和教师教育项目的学生开发新课程，组织霍奇理论和模空间研讨会，并撰写专著。 该提案的一般领域是代数几何，与复几何和算术几何有关。 代数几何学研究由代数（或等价多项式）方程定义的对象的几何特性。在代数几何中，PI对模空间的研究感兴趣。模空间是一个几何对象，它参数化了给定拓扑类中对象的形状。通过研究模空间，人们可以获得关于给定类型的几何对象的重要信息，特别是关于具有指定特殊性质的对象的存在或不存在的信息。这项研究有许多应用代数几何和其他相关领域，包括数学物理。

This proposal is concerned with the study of the geometry of moduli spaces and the development of tools for studying them. In particular, the PI plans to apply and adapt some newly developed techniques in abstract Hodge theory to the geometric context of moduli spaces. The PI will focus on the investigation of special classes of varieties, such as Calabi-Yau threefolds and higher dimensional Hyperkahler manifolds. In a different, but related direction, the PI aims to construct geometric compactifications for certain classes of surfaces via the KSBA approach inspired by the minimal model program. The PI is actively involved with the training of undergraduate and graduate students, and in recent years he has organized several activities with a strong educational component. He will expand these activities. In particular, he will run a summer research activity for undergraduates. Also, as part of the thematic program on Calabi-Yau varieties to be held at the Fields institute (Fall 2013), the PI will organize an introductory school for graduate students and other training activities. Additional planed activities include developing new courses for undergraduates and students in the teacher education program, organizing a workshop on Hodge theory and moduli spaces, and writing a monograph. The general area of the proposal is algebraic geometry with connections to complex geometry and arithmetic geometry. Algebraic Geometry studies the geometric properties of objects defined by algebraic (or equivalently polynomial) equations. Within Algebraic Geometry, the PI is interested in the study of moduli spaces. A moduli space is a geometric object that parameterizes the shapes of objects within a given topological class. By studying a moduli space, one obtains important information about geometric objects of a given kind, in particular about the existence or non-existence of objects with prescribed special properties. This study has numerous applications to algebraic geometry and other related fields including mathematical physics.