The object of this proposal is to pursue new approaches to some fundamental problems in Galois theory, following recent breakthroughs in this subject. The focus of research will be on lifting finite (possibly ramified) Galois covers of smooth projective curves over a commutative ring and on finding small rings of definition for such lifts. These problems pertain to the classical Inverse Galois Problem as well as to the question of finding the relations between etale fundamental groups of various kinds.
This proposal deals with fundamental questions about the solutions of algebraic equations. The study of the symmetries of such equations has been the motivation for the development of much of abstract algebra. Advances in abstract algebra have been essential to the development of modern technology. Some particular examples of applications of abstract algebra include the compression and transmission of data, cryptography, search algorithms and solving optimization problems in the physical sciences, engineering and economics.