Studies in Moduli Theory and Birational Geometry

阿布拉莫维奇将继续研究问题的模理论，特别是（1）的模堆栈扭曲稳定的地图，减少模的主要捆绑在特征p，Gromov-Witten理论的堆栈，基本问题的堆栈，和（2）模空间的Bridgeland-Douglas半稳定对象的衍生类别的各种。 阿布拉莫维奇还将继续研究双有理几何中的问题，特别是（1）强因式分解猜想和超环面化猜想，以及（2）反常点层及其模作为构造和研究某些Mori翻转和触发器的工具。该项目的研究领域在于代数几何，数学的一个分支，致力于研究由多项式方程定义的称为代数簇的几何形状。虽然代数几何在编码，工业控制和计算中的应用做出了贡献，但这个项目的主题与理论物理学中的应用更密切相关，物理学家将代数簇作为我们宇宙精细结构的组成部分。这在第一个主题，模论中尤其如此。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通常表现为一个代数簇，称为模空间。因此，在代数几何学中，把“有机体”的能力看作其自身是一个“有机体”的隐喻，不仅是一种隐喻，而且是一个严格而非常有用的事实。有时代数簇的集合表现为稍微更一般的对象，称为栈，而不是簇。这样的书库是本项目研究的中心对象。该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，这是代数几何的基础。

Abramovich will continue studying problems in moduli theory, in particular (1) the moduli stacks of twisted stable maps, reductions of moduli of principal bundles in characteristic p, Gromov-Witten theory of stacks, foundational problems on stacks, and (2) moduli spaces of Bridgeland-Douglas semistable objects in the derived category of a variety. Abramovich will also continue studyingproblems in birational geometry, in particular (1) the strongfactorization conjecture and the toroidalization conjecture, and (2) perverse point sheaves and their moduli as a tool for construction and study of certain Mori flips and flops.The area of study of this project lies within algebraic geometry, thebranch of mathematics devoted to geometric shapes called algebraicvarieties, defined by polynomial equations. While algebraic geometryhas contributed applications in coding, industrial control, andcomputation, the topics of this project are more closely related toapplications in theoretical physics, where physicists consideralgebraic varieties as components of the fine structure of ouruniverse. This is especially true with the first topic, modulitheory. This theory studies a remarkable phenomenon in which thecollection of all algebraic varieties of the same type is oftenmanifested as an algebraic variety, called a moduli space, in its ownright. Thus in algebraic geometry, the metaphor of thinking about acommunity of "organisms" as itself being an "organism" is not just ametaphor but a rigorous and quite useful fact. Sometimes a collectionof algebraic varieties manifests itself as a slightly more generalobject, called a stack, rather than a variety. Such stacks are acentral object of study of this project. The other topic studied inthis project is birational geometry, which is devoted to a certainabstract relationship, called birational equivalence, among algebraicvarieties, which lies at the foundation of algebraic geometry.