Degenerations and Moduli in Algebraic Geometry

该奖项支持代数几何的研究，代数几何是数学的一个中心分支。它旨在从实践和概念上理解多变量多项式方程组的解。代数几何在数学的其他领域有重要的应用，如数论、拓扑学和分析，以及物理学、生物学、密码学和工程学。本研究项目所研究的特定问题涉及模空间，模空间是几何分类问题的参数化解的集合。一些正在研究的主题起源于物理学中的弦理论，而这项工作的结果可能反过来在那里找到应用。研究生也参与了这个项目。研究者将研究代数几何中的一系列问题，包括退化、稳定变量和对的紧模空间以及与最小模型程序的连接。一个中心项目是研究K3曲面模空间的函子紧化及其显式组合描述。研究者还将研究阿贝尔六倍的双几何。

This award supports research in algebraic geometry, a central branch of mathematics. It aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. Algebraic geometry has important applications to other fields of mathematics, such as number theory, topology, and analysis, as well as to physics, biology, cryptography, and engineering. The particular questions under study in this research project involve moduli spaces, which are sets that parameterize solutions of geometric classification problems. Some of the topics under study originated in string theory in physics, and results of this work may in turn find application there. Graduate students are involved in the project. The investigator will work on a range of questions in algebraic geometry centering on degenerations, compact moduli spaces of stable varieties and pairs, and connections to the minimal model program. A central project is the study of functorial compactifications of moduli spaces of K3 surfaces and their explicit combinatorial description. The investigator will also study the birational geometry of abelian six-folds.