Studies in moduli theory and birational geometry

阿布拉莫维奇将继续研究问题的模理论，包括几个方面的格罗莫夫-维滕理论，特别是（1）广义模的相对稳定的地图;（2）比较虚拟基本类模的相对和orbifold稳定的地图;和（3）进一步研究模的非常扭曲的曲线和他们的地图。阿布拉莫维奇将继续研究双有理几何中的问题，特别是（1）坎帕纳星座，它们的地图和算术性质;（2）高维orbifolds的双有理几何和模，以及它们在高维簇模紧化中的应用。本项目的研究领域是代数几何，这是数学的分支，专门研究被称为代数簇的几何形状，由多项式方程定义。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，物理学家认为代数簇是我们宇宙精细结构的组成部分。第一个主题，模量理论，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，把一个“有机体”群落看作是一个“有机体”的隐喻，不仅是一个隐喻，而且是一个严格而非常有用的事实。有时候代数簇的集合表现为一个稍微更一般的对象，称为栈，而不是一个簇。这种书库是本项目研究的中心对象。在这个项目中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, including several aspects of Gromov--Witten theory, in particular (1) generalized moduli of relative stable maps;(2) comparison of virtual fundamental classes on moduli of relative and orbifold stable maps; and (3) further study of moduli of very twisted curves and their maps. Abramovich will continue studying problems in birational geometry, in particular (1) Campana constellations, their maps and arithmetic properties; (2) Birational geometry of and moduli of higher dimensional orbifolds, and their application in the compactification of moduli of higher dimensional varieties.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.