Studies in moduli theory and birational geometry

阿布拉莫维奇将继续研究模理论的问题;该项目的主要重点是稳定对数映射的模空间和相关的Gromov-Witten理论。其他主题包括变形的野生p-覆盖，和问题，也承担双理性几何的品种，包括研究的伪理想作为微分分级计划，和热带化的模空间。该项目的研究领域是在代数几何，数学的分支致力于几何形状称为代数簇，由多项式方程定义。代数几何在编码、工业控制和计算中有重要的应用。但这个项目的主题与理论物理学的应用更密切相关，物理学家认为代数簇是我们宇宙精细结构的一部分。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。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; the main focus of the project is moduli spaces of stable logarithmic maps and associated Gromov-Witten theory. Additional topics include deformations of wild p-covers, and problems which also bear on birational geometry of varieties, including the study of pseudoideals as differential graded schemes, and the tropicalization of moduli space.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. Algebraic geometry has significant applications in coding, industrial control, and computation. But the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as a piece 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 just a metaphor but a rigorous and quite useful fact. 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.