Studies in Moduli Theory and Birational Geometry

阿布拉莫维奇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中主丛的模约化，格罗莫夫-威腾堆栈理论，关于堆栈的基本问题，以及(2)派生范畴中Bridgeland-Douglas半稳定对象的模空间。阿布拉莫维奇还将继续研究二次几何中的问题，特别是(1)强因式分解猜想和环面猜想，以及(2)作为构造和研究某些Mori翻转和翻转的工具的倒点轮及其模。这个项目的研究领域是代数几何，致力于几何形状的数学分支，称为代数簇，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，即所有相同类型的代数族的集合往往表现为一个代数族，称为模空间。因此，在代数几何中，把“有机体”的群体看作“有机体”本身就是一个“有机体”的比喻，不仅是一种非比喻，而且是一个严谨而相当有用的事实。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。这样的堆栈是本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.