This award supports work on the Inverse Galois problem. The principal investigator will improve the criteria for the realization of groups as Galois groups that he has recently found. He will also search for new criteria and apply these criteria to realize further interesting classes of groups, especially simple groups, as Galois groups over the rationals and over cyclotomic fields. Most of this work is based on the use of moduli spaces for covers of the Riemann sphere. A group is an algebraic structure with a single operation. It appears in many areas of mathematics, as well as, physics and chemistry. One of the major problems in algebra is concerned with identifying those groups which can be represented as the Galois group over the rationals. This involves a blend of algebra, number theory and algebraic geometry.
