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.
