Studies in Moduli Theory and Birational Geometry

模理论与双有理几何研究

• 批准号: 1500525
• 负责人: Dan Abramovich
• 金额: $ 34.78万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500525&HistoricalAwards=false
• 关键词: Studies; Moduli; Theory; Birational; Geometry

The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. Algebraic geometry has significant applications in coding, industrial control, and computation. But the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as a piece of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not just a metaphor but a rigorous and quite useful fact. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry.The investigator will continue studying problems in moduli theory; the main foci of the project are Moduli spaces of stable logarithmic maps and Artin fans. Additional topics include the degeneration formula for KKO invariants, the birational geometry of torus quotients, and logarithmic Kodaira dimensions of fibered powers in relation to uniformity of stably integral points.

这个项目的研究领域是代数几何，这是数学的一个分支，致力于研究被称为代数簇的几何形状，由多项式方程定义。代数几何在编码、工业控制和计算中有着重要的应用。但这个项目的主题与理论物理中的应用更密切相关，在理论物理中，物理学家认为代数族是我们宇宙精细结构的一部分。对于第一个主题，模理论尤其如此。这一理论研究了一个值得注意的现象，在这个现象中，所有相同类型的代数簇的集合本身就表现为一个代数簇，称为模空间。因此，在代数几何中，把一个“有机体”群落看作是一个“有机体”的隐喻，不仅是一种隐喻，而且是一个严谨而又相当有用的事实。本课题研究的另一个主题是二元几何，它致力于研究代数簇之间的一种抽象关系，称为二元等价，这种关系是代数几何的基础。研究者将继续研究模理论中的问题；本课题的主要焦点是稳定对数映射和Artin扇的模空间。其他主题包括kko不变量的退化公式，环面商的二次几何，以及与稳定积分点的一致性有关的纤化幂的对数Kodaira维度。