Fundamental Groups and Absolute Galois Groups

主要研究基本群和簇的绝对伽罗瓦群，特别是有限特征群。 特别是，他调查的基本群体的仿射曲线在代数封闭领域的有限特征; absoluteGalois群体的功能领域的曲线在非代数封闭领域的有限特征;和基本群体和absoluteGalois群体的高维品种在代数封闭领域的任意特征。 这项研究在很大程度上是由Abhyankar的曲线猜想，其中主要研究者已经证明了可能的推广。 一个主要的目标是确定基本群和绝对伽罗瓦群的性质，并确定哪些有限群是这些有限群的子群，以便更深入地了解这些空间的代数和几何。 方法包括形式修补，嵌入问题，上同调，赋值理论和群论。这个项目的领域涉及代数和几何的各个方面，使得解决单独在任何一个领域都难以解决的问题成为可能。 代数和几何之间的联系来自于由代数方程定义的几何空间。这些空间可以有复杂的对称模式，这在代数和几何中都有体现。 这个项目旨在了解什么类型的对称性可以发生在这些空间的背景下，以及如何与一种类型的对称性空间可以与另一种类型的对称性给定的空间。 这些空间在许多情况下是相对于一个代数系统定义的，其中一个特定的素数起着特殊的作用;这些空间的性质可以被解释为给出关于涉及素数的方程的可解性的信息。

The Principal Investigator studies fundamental groups and absoluteGalois groups of varieties, especially in finite characteristic. Inparticular, he investigates the fundamental groups of affine curves overan algebraically closed field of finite characteristic; the absoluteGalois groups of function fields of curves over non-algebraically closedfields of finite characteristic; and the fundamental groups and absoluteGalois groups of higher dimensional varieties over algebraically closedfields of arbitrary characteristic. This study is in large part guidedby possible generalizations of Abhyankar's Conjecture for curves, whichthe Principal Investigator has proven. A major goal is to determine theproperties of the fundamental groups and absolute Galois groups inquestion, and to determine which finite groups are quotients of theseprofinite groups, in order to obtain greater insight into the algebraand geometry of these spaces. Methods include formal patching,embedding problems, cohomology, valuation theory, and group theory.The area of this project relates aspects of algebra and geometry in away that makes it possible to solve problems that would be intractiblein either field alone. The linkage between the algebra and geometryarises from geometric spaces that are defined by algebraic equations. These spaces can have complicated patterns of symmetry, which manifestthemselves both in the algebra and the geometry. This project seeks tounderstand what types of symmetries can occur in the context of thesespaces, and how spaces with one type of symmetry can relate to a givenspace with another type of symmetry. These spaces are in many casesdefined with respect to an algebraic system in which a particular primenumber plays a special role; and the properties of those spaces can beinterpreted as giving information about the solvability of equationsinvolving prime numbers.