This project concerns Galois covers and fundamental groups of affine varieties, especially in finite characteristic. The main goal is to generalize and strengthen the result of Abhyankar's Conjecture, which was proven by Raynaud and the Principal Investigator. That result classified the Galois groups of unramified covers of affine curves over an algebraically closed field of finite characteristic. By strengthening this conjecture, more understanding would be achieved of how the Galois covers of a given curve fit together, and thus about the structure of the fundamental group. By generalizing the conjecture, results would be obtained about Galois groups over affine varieties in higher dimensions, and over curves defined over more general base fields. Methods will include formal patching, specialization, cohomology, and the theory of profinite groups.
The subject area of this project brings together two areas of mathematics that each concern symmetry -- symmetry in algebra, in the case of Galois theory; and symmetry in geometry, in the case of fundamental groups. In each of these two situations, mathematical objects can be studied by examining the forms that their symmetries can take. The connection between the two settings arises from the fact that geometric spaces can be described by algebraic equations, and those equations can be studied by Galois theory. The symmetries of the equations then relate to fundamental groups of geometric spaces. A problem that is posed in either of these two settings can then be translated into a problem in the other setting, where other techniques can be applied in order to provide a solution. This project concerns how these two situations can interact, so that algebra can be used in the service of geometry, and vice versa, in order to study problems that would otherwise be intractable.
