Galois Groups and Fundamental Groups

首席研究员研究代数和算术几何中的伽罗瓦群和基本群。特别地，他研究了Shafarevich和Abhyankar猜想在簇上Galois群上的推广；覆盖的提升问题，从有限特征到混合特征；以及一元函数域在有理数域上的Galois扩张。目标包括高维伽罗瓦理论的发展，对覆盖的良好约化的理解，以及覆盖模的计算域。实现这些目标将增加对曲面几何的理解，以及数论和拓扑学之间的联系。方法包括形式补丁、嵌入问题、上同调、双态变换和群论。这个项目将代数和几何的各个方面联系在一起，使之有可能获得仅使用这两个领域中的任何一个都无法获得的研究结果。正在研究的几何空间可以用方程来代数定义，该项目利用了这些空间的代数和几何方面。该项目的一个特别焦点涉及这些空间的对称性，这些对称性反过来对应于涉及这些空间上的代数函数的对称性。该项目研究什么类型的对称可以发生，什么类型的对称扩展是可能的，以及为了获得具有给定类型的对称的空间需要什么类型的代数数。

The Principal Investigator studies Galois groups and fundamental groups in algebraic and arithmetic geometry. In particular, he studies generalizations of the conjectures of Shafarevich and Abhyankar on Galois groups over varieties; lifting problems for covers, from finite to mixed characteristic; and Galois extensions of the one-variable function field over the field of rational numbers. Goals include the development of Galois theory in higher dimensions, understanding good reduction of covers, and computing fields of moduli of covers. Achieving these goals would increase the understanding of the geometry of surfaces and the link between number theory and topology. Methods include formal patching, embedding problems, cohomology, birational transformations, and group theory.This project relates aspects of algebra and geometry in a way that makes it possible to achieve research results that could not be obtained using either of these two fields alone. The geometric spaces being studied can be defined algebraically using equations, and the project makes use of both the algebraic and geometric aspects of these spaces. A particular focus of the project concerns the symmetries of these spaces, which in turn correspond to symmetries involving the algebraic functions on these spaces. The project studies what types of symmetries can occur, which types of extensions of symmetry are possible, and what types of algebraic numbers are needed in order to obtain a space with a given type of symmetry.