Studies in Moduli Theory and Birational Geometry

该项目的研究领域在于代数几何，数学的分支致力于几何形状称为代数簇，由多项式方程定义。代数几何在编码、工业控制和计算中有重要的应用。但这个项目的主题与理论物理学的应用更密切相关，物理学家认为代数簇是我们宇宙精细结构的一部分。第一个主题，模量理论，尤其如此。这个理论研究了一个显著的现象，其中所有相同类型的代数簇的集合表现为一个代数簇，称为模空间。因此，在代数几何学中，把一个“有机体”群落看作是一个“有机体”的隐喻不仅是一个隐喻，而且是一个严格而非常有用的事实。该项目的另一个研究课题是双有理几何，它致力于代数簇之间的某种抽象关系，称为双有理等价，这是代数几何的基础。研究人员将继续研究模理论中的问题;该项目的主要焦点是稳定对数映射和Artin fans的模空间。其他议题包括退化公式KKO不变量，双有理几何环面contricents，和对数科代拉尺寸的纤维化权力的均匀性稳定积分点。

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.