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.

该项目的研究领域在于代数几何形状，这是数学的分支，用于多项式方程定义的几何形状，称为代数品种。代数几何形状在编码，工业控制和计算中具有重要的应用。但是，该项目的主题与理论物理学中的应用更加紧密相关，在理论物理学中，物理学家将代数品种视为我们宇宙良好结构的一部分。第一个主题，模量理论尤其如此。该理论研究了一种了不起的现象，在这种现象中，同一类型的所有代数品种的收集被表现为代数品种，称为模量空间本身。因此，在代数几何形状中，思考“生物体”本身是“有机体”的隐喻不仅是一个比喻，而且是一个严格且相当有用的事实。该项目中研究的另一个主题是Birational几何形状，它致力于代数品种中的某种抽象关系，称为Birational等价，这是代数几何形状的基础。研究人员将继续研究Moduli理论中的问题。该项目的主要焦点是稳定对数地图和Artin粉丝的模量空间。其他主题包括KKO不变性的退化公式，圆环商的生育几何以及相对于稳定积分点的均匀性，纤维功率的对数Kodaira尺寸。