Mathematical Sciences: Groups as Galois Groups

9623199 Voelklein该奖项是对在有理数域Q上将群实现为伽罗瓦群的研究，即著名的伽罗瓦逆问题。Belyi，Malle，Matzat，Thompson等人已经广泛地研究了Q上的一元正则Galois实现。这位研究人员的新书对这一理论进行了介绍，即将出版的Malle和Matzat的书对所有已知的结果进行了完整的描述。另一方面，当r大于1时，对r元的这种实现知之甚少。本研究将进一步探讨r大于1的情形。从几何上讲，这意味着研究射影r-空间中超曲面W的补的基本群。更具体地说，这意味着研究这些基本群的有限商，并决定何时在有理数上定义W的补。所提出的工作的主要内容是当W是(仿射空间中的超平面的)Coxeter排列或其相应Coxeter群的商的情况。为了证明这种W的补集的覆盖是在有理数上定义的，有两种方法可用。第一种方法是对黎曼球面的覆盖进行模空间的干预，第二种方法是利用高维的刚性准则。这个项目的另一部分与其他部分有很大的不同。这是与M.Fry关于非零特征p的伽罗瓦实现的联合项目。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

9623199 Voelklein This award is for an investigation on realizing groups as Galois groups over the rational field Q; i.e., the famous Inverse Galois Problem. Regular Galois realizations in one variable over Q have been studied extensively by Belyi, Malle, Matzat, Thompson, and others. The investigator's recent book gives an introduction to that theory, and the forthcoming book of Malle and Matzat gives a complete description of all known results. On the other hand, very little is known about such realizations in r variables when r is bigger than 1. This investigation will further explore the case of r greater than 1. Geometrically, this means studying fundamental groups of complements of hypersurfaces W in projective r-space. More specifically, it means looking at finite quotients of such fundamental groups and deciding when the complements of W are defined over the rational numbers. The main body of proposed work is in the case where W is a Coxeter arrangement (of hyperplanes in affine space) or its quotient by the corresponding Coxeter group. To show that a covering of the complement of such W is defined over the rationals, two methods are available. The first is to interpert such a covering as a moduli space for covers of the Riemann sphere, and the second involves using a higher-dimensional rigidity criterion. Another part of this project is quite different from the rest. This is a joint project with M. Fried about Galois realizations in non-zero characteristic p. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communicat ion systems.