Fundamental Groups and Absolute Galois Groups

主要研究品种的基本群和绝对伽罗瓦群，特别是有限特征的伽罗瓦群。特别地，他研究了有限特征代数闭域上的仿射曲线的基本群；有限特征非代数闭域上曲线函数场的绝对伽罗瓦群；以及任意特征的代数闭域上高维变异的基群和绝对伽罗瓦群。这项研究在很大程度上是由Abhyankar的曲线猜想的可能推广指导的，首席研究员已经证明了这一点。一个主要的目标是确定所讨论的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商，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、估值理论和群论。这个项目的领域涉及代数和几何的各个方面，这使得解决单独在任何一个领域都无法解决的问题成为可能。代数和几何之间的联系源于由代数方程定义的几何空间。这些空间可以具有复杂的对称模式，这在代数和几何中都表现出来。该项目旨在了解在这些空间的背景下会发生什么类型的对称，以及具有一种对称类型的空间如何与具有另一种对称类型的给定空间相关联。在许多情况下，这些空间是根据一个代数系统定义的，在这个代数系统中，一个特定的素数起着特殊的作用；这些空间的性质可以解释为给出了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.