Mathematical Sciences: Groups as Galois Groups

9623199 Voelklein该奖项是为了研究在理性域Q上实现群作为伽罗瓦群;即，著名的伽罗瓦逆问题正则伽罗瓦实现在一个变量超过Q已经广泛研究了贝利，马勒，Matzat，汤普森，和其他人。调查员最近的书介绍了这一理论，即将出版的书马勒和Matzat给出了完整的描述所有已知的结果。另一方面，当r大于1时，对r变量的这种实现知之甚少。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，这意味着要研究这些基本群的有限同分元，并决定W的补数何时定义在有理数上。建议的工作的主体是在W是一个考克斯特安排（超平面在仿射空间）或其商的情况下，由相应的考克斯特组。为了证明这样的W的补的覆盖被定义在有理数上，有两种方法可用。第一个是interpert这样的覆盖作为一个模空间覆盖的黎曼球面，第二个涉及到使用一个高维刚性标准。这个项目的另一部分与其他部分大不相同。这是一个与M。关于非零特征p的伽罗瓦实现。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

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.