Mathematical Sciences: Groups as Galois Groups

9623199 volelklein这个奖项是为了表彰在有理域Q上对实现群作为伽罗瓦群的研究；也就是著名的逆伽罗瓦问题。Belyi、Malle、Matzat、Thompson等人对Q上一个变量的正则伽罗瓦实现进行了广泛的研究。这位研究者最近的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，我们对r变量的实现知之甚少。本研究将进一步探讨r大于1的情况。在几何上，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，它意味着观察这些基本群的有限商，并决定W的补何时在有理数上被定义。所提议的工作的主体是在W是一个科克塞特排列（仿射空间中的超平面）或其商对应的科克塞特群的情况下。为了证明在有理数上定义了W的补的覆盖，有两种方法可用。第一种方法是将这种覆盖层作为黎曼球覆盖层的模空间进行干涉，第二种方法涉及使用高维刚度准则。这个项目的另一部分与其他部分完全不同。这是与M. Fried关于非零特征p中的伽罗瓦实现的合作项目。本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，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。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各种应用领域不可或缺的工具。

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.