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.

该建议与模量空间的几何形状以及研究它们的工具的开发有关。特别是，PI计划将抽象理论中的一些新开发的技术应用于模量空间的几何环境。 PI将集中于对特殊类别的特殊类别的研究，例如Calabi-yau三倍和更高的尺寸Hyperkahler歧管。在不同但相关的方向上，PI旨在通过受到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积极参与对本科生和研究生的培训，近年来，他组织​​了几项具有强大教育成分的活动。他将扩大这些活动。特别是，他将为本科生开展夏季研究活动。此外，作为将在田野学院（Fields Institute）（2013年秋季）举行的有关卡拉比YAU品种的主题计划的一部分，PI将组织一所研究生和其他培训活动的入门学校。其他计划的活动包括在教师教育计划中为大学生和学生开发新课程，组织有关霍奇理论和模量空间的研讨会，并撰写专着。 该提案的一般区域是代数几何形状，与复杂的几何形状和算术几何形状有连接。 代数几何形状研究代数（或同等多项式）方程定义的对象的几何特性。在代数几何形状中，PI对模量空间的研究感兴趣。模量空间是一个几何对象，它参数化给定拓扑类中对象的形状。通过研究模量空间，人们获得了有关给定类型的几何对象的重要信息，尤其是有关具有规定特殊属性的对象的存在或不存在。这项研究在代数几何形状和其他相关领域（包括数学物理学）上有许多应用。