Studies in Moduli Theory and Birational Geometry

该项目的研究领域属于代数几何，这是数学的一个分支，致力于研究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、工业控制和计算中具有重要的应用。但该项目的主题与理论物理学的应用更为密切相关，物理学家将代数簇视为宇宙精细结构的一部分。对于第一个主题，模理论尤其如此。该理论研究了一个显着的现象，其中同一类型的所有代数簇的集合表现为一个代数簇，称为模空间，其本身就是这样的。因此，在代数几何中，将“有机体”共同体视为其本身就是一个“有机体”的隐喻不仅是一个隐喻，而且是一个严格且非常有用的事实。本项目研究的另一个主题是双有理几何，它致力于代数簇之间的某种抽象关系，称为双有理等价，它是代数几何的基础。研究者将继续研究模论中的问题；该项目的主要焦点是稳定对数映射的 Moduli 空间和 Artin 粉丝。其他主题包括 KKO 不变量的退化公式、环面商的双有理几何以及与稳定积分点均匀性相关的纤维幂的对数 Kodaira 维数。

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.