CAREER: Advances in Hodge Theory and Moduli

职业：霍奇理论和模数的进展

• 批准号: 1254812
• 负责人: Radu Laza
• 金额: $ 41.4万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1254812&HistoricalAwards=false
• 关键词: CAREER; Advances; Hodge; Theory; Moduli

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.

该提案涉及模空间几何的研究以及研究它们的工具的开发。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和调整到模空间的几何背景中。 PI 将重点研究特殊类别的簇，例如 Calabi-Yau 三重流形和更高维的 Hyperkahler 流形。在一个不同但相关的方向上，PI 的目标是通过受最小模型程序启发的 KSBA 方法为某些类别的表面构建几何压缩。 PI积极参与本科生和研究生的培训，近年来组织了多项具有浓厚教育意义的活动。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，作为菲尔兹研究所（2013 年秋季）举办的 Calabi-Yau 品种主题项目的一部分，PI 将为研究生组织一所入门学校和其他培训活动。其他计划的活动包括为教师教育项目中的本科生和学生开发新课程、组织霍奇理论和模空间研讨会以及撰写专着。 该提案的总体领域是代数几何，与复杂几何和算术几何有联系。 代数几何研究由代数（或等效多项式）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，它参数化给定拓扑类内对象的形状。通过研究模空间，人们可以获得有关给定类型的几何对象的重要信息，特别是有关具有规定特殊属性的对象是否存在的信息。这项研究在代数几何和包括数学物理在内的其他相关领域有广泛的应用。