Studies in moduli theory and birational geometry

Abramovich将继续研究模理论中的问题，包括Gromov—Witten理论的几个方面，特别是(1)相对稳定映射的广义模；(2)相对稳定映射与轨道稳定映射模的虚基类比较；(3)对极扭曲线及其映射的模的进一步研究。阿布拉莫维奇将继续研究二分几何中的问题，特别是(1)坎帕纳星座及其映射和算术性质；(2)高维轨道的双几何和模及其在高维变体模紧化中的应用。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，称为代数变异，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。这个理论研究了一个显著的现象，在这个现象中，同一类型的所有代数变体的集合通常表现为一个代数变体，称为模空间，在其自身的权利。因此，在代数几何中，把一群“有机体”看作是一个“有机体”的比喻不仅仅是一个比喻，而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，它研究代数变量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, 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.