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；该项目的主要焦点是稳定对数映射的模空间和相关的Gromov-Witten理论。其他主题包括野生p-盖的变形，以及与品种的双几何有关的问题，包括伪理想作为微分梯度格式的研究，以及模空间的热带化。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。代数几何在编码、工业控制和计算方面有重要的应用。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一部分。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。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; 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.