Studies in Moduli Theory and Birational Geometry

这个项目的研究领域属于代数几何，这是数学的一个分支，专门研究几何形状，称为代数变异，由多项式方程定义。代数几何在编码、工业控制和计算方面有重要的应用。但这个项目的主题与理论物理的应用更密切相关，物理学家认为代数变体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。这个理论研究了一个显著的现象，在这个现象中，所有相同类型的代数变体的集合表现为一个代数变体，称为模空间，在它自己的权利。因此，在代数几何中，把一群“有机体”看作是一个“有机体”的比喻不仅仅是一个比喻，而是一个严谨而非常有用的事实。本课题研究的另一个课题是二元几何，它研究代数变量之间的某种抽象关系，称为二元等价，这是代数几何的基础。研究者将继续研究模理论中的问题；项目的主要焦点是稳定对数映射的模空间和Artin扇形。其他的主题包括KKO不变量的退化公式，环面商的两族几何，以及纤维幂的对数Kodaira维数与稳定积分点的均匀性的关系。

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.