Studies in Moduli Theory and Birational Geometry

Abramovich将继续研究模理论中的问题，特别是(1)扭曲稳定映射的模堆，特征p中主束的模约化，Gromov-Witten堆理论，堆上的基本问题，以及(2)各种派生范畴中bridgland - douglas半稳定对象的模空间。Abramovich还将继续研究几何中的问题，特别是(1)强分解猜想和环化猜想，以及(2)作为构造和研究某些Mori翻转和翻转的工具的反常点束及其模。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，称为代数变量，由多项式方程定义。虽然代数几何在编码、工业控制和计算方面的应用做出了贡献，但这个项目的主题与理论物理的应用更密切相关，物理学家认为代数变化是我们宇宙精细结构的组成部分。对于第一个主题，模块理论，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通常表现为一个代数变体，称为模空间，在其本身。因此，在代数几何中，把一群“有机体”看成本身就是一个“有机体”的比喻，不仅是一个隐喻，而且是一个严谨而非常有用的事实。有时，代数变体的集合表现为稍微更一般的对象，称为堆栈，而不是变体。这样的堆栈是本项目研究的中心对象。本课题研究的另一个课题是二元几何，它致力于代数变体之间的某种抽象关系，称为二元等价，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.