Moduli Spaces - Geometry and Arithmetic

模空间 - 几何和算术

• 批准号: 1200875
• 负责人: Radu Laza
• 金额: $ 14.29万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200875&HistoricalAwards=false
• 关键词: Moduli; Spaces; Geometry; Arithmetic

The PI investigates several questions regarding moduli spaces by using techniques of algebraic and analytic nature. One main direction of Laza's research program is the construction of geometric compactifications for moduli spaces and the development of tools for studying them. In particular, he is interested in the study of the behavior of the period map at the boundary of the moduli spaces. Another direction is the study of moduli spaces of certain special varieties (K3, Calabi-Yau, and Hyperkahler). In this case, the interplay between geometry, algebra, and arithmetic is the strongest, and consequently one expects a good understanding of the moduli of these varieties. Some of the questions that arise here have direct relevance to string theory. This is a proposal in algebraic geometry, with connections to the related fields of Complex Geometry and Arithmetic Geometry. Algebraic Geometry is concerned with the study of geometric properties of objects defined by algebraic equations. Within Algebraic Geometry, the PI is interested in the study of moduli spaces, which parameterizes the shapes of objects within a given topological class. This study has connections and applications to several other branches of mathematics and to mathematical physics. The PI is involved in a number of activities that broaden the impact of the proposed research: He is advising graduate and undergraduate students. He is (co)organizing several conferences, as well as a thematic semester on the geometry of Calabi-Yau varieties.

PI 通过使用代数和分析性质的技术来研究有关模空间的几个问题。 Laza 研究计划的一个主要方向是构建模空间的几何紧化以及开发研究它们的工具。他特别对研究模空间边界处的周期图的行为感兴趣。另一个方向是研究某些特殊品种（K3、Calabi-Yau 和 Hyperkahler）的模空间。在这种情况下，几何、代数和算术之间的相互作用是最强的，因此人们期望对这些变量的模有很好的理解。这里出现的一些问题与弦理论直接相关。 这是代数几何中的一项提议，与复几何和算术几何的相关领域有联系。代数几何涉及由代数方程定义的对象的几何性质的研究。在代数几何中，PI 对模空间的研究感兴趣，模空间参数化给定拓扑类中对象的形状。这项研究与数学的其他几个分支和数学物理学有联系和应用。 PI 参与了许多扩大拟议研究影响的活动：他为研究生和本科生提供建议。他正在（共同）组织多个会议，以及一个关于 Calabi-Yau 品种几何学的主题学期。