Galois Groups and Fundamental Groups

主要研究者研究伽罗瓦群和基本群的代数和算术几何。特别是，他研究的概括abetures的Shafarevich和Abhyankar伽罗瓦集团的品种;解除问题的涵盖，从有限的混合特征;和伽罗瓦扩展的一元函数领域的领域的合理数。 目标包括在更高的维度伽罗瓦理论的发展，理解良好的覆盖减少，和计算领域的覆盖模。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。 方法包括形式修补、嵌入问题、上同调、双有理变换和群论。本项目以一种能够实现单独使用这两个领域中的任何一个都无法获得的研究结果的方式将代数和几何方面联系起来。 所研究的几何空间可以用代数方程来定义，该项目利用了这些空间的代数和几何方面。 该项目的一个特别重点是这些空间的对称性，这反过来又对应于涉及这些空间上的代数函数的对称性。 该项目研究了什么类型的对称性可以发生，什么类型的对称性的扩展是可能的，以及需要什么类型的代数数才能获得具有给定类型对称性的空间。

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.