Studies in Moduli Theory and Birational Geometry

这位研究人员和他的同事研究了模理论和双曲几何中的问题。在模理论中，他们研究与扭曲稳定映射的模堆相关的问题，把它变成驯服的Deligne-Mumford堆栈。讨论的问题包括：堆栈的Gromov-Witten不变量的构造，扭曲稳定映射到高维和野生堆栈的推广，以及在可容许覆盖、能级结构、自旋结构和高维变元理论中的应用。研究人员和同事还将研究二次几何中的问题：强因式分解猜想，它说光滑射影簇的二次映射应该因式为一系列光滑的凸起，然后是一系列的光滑吹扫；环面猜想，它说，给定复杂射影流形的满射态射，有一种方法沿着光滑中心炸掉源和目标，从而得到的映射是环状态射。这个项目的研究领域是代数几何，数学的一个分支，被称为代数簇，由多项式定义。虽然代数几何在编码、工业控制和计算方面做出了贡献，但这个项目的主题与理论物理的应用更密切相关，在理论物理中，物理学家认为代数变体是我们宇宙精细结构的组成部分。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通常表现为一个代数簇，称为模空间，就其本身而言。因此，在代数几何中，把一个“有机体”群落想象成一个“有机体”本身就是一个“有机体”的比喻，不仅仅是一个比喻，而是一个严谨而又相当有用的事实。有时，代数变体的集合表现为稍微更通用的对象，称为堆栈，而不是变体。这样的堆栈是该项目研究的中心对象。这个项目中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某种抽象关系，称为二元等价，这是代数几何的基础。这个项目的一个目标是理解这个抽象的概念和一个更具体的关系之间的联系，这种关系是由一种名为“爆炸”的代数“手术”过程给出的。

The investigator and his colleagues study problems in moduli theory and birational geometry. In moduli theory they study questions related to the moduli stacks of twisted stable maps into a tame Deligne-Mumford stack. Questions addressed include constructions towards Gromov-Witten invariants of stacks, generalizations of twisted stable maps to higher dimensions and to wild stacks, and applications to the theory of admissible covers, level structures, spin structures and higher dimensional varieties. The investigator and colleagues will also study problems in birational geometry: the strong factorization conjecture, which says that a birational map of smooth projective varieties should factor as a sequence of smooth blowings up followed by a sequence of smooth blowings down; and the toroidalization conjecture, which says that, given a surjective morphism of complex projective manifolds, there is a way to blow up both source and target along smooth centers so that the resulting map is a toroidal morphism.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. While algebraic geometry has contributed applications in coding, industrial control, and computation, the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as components 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 often 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 a just a metaphor but a rigorous and quite useful fact. Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety. Such stacks are a central object of study of this project. 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. A goal of this project is to understand the connection between this abstract notion, and a much more concrete relationship given by an algebraic "surgery" procedure called "blowing up".