Galois Groups and Fundamental Groups

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9970481HarbaterThis project concerns Galois covers and fundamental groups of affinevarieties, especially in finite characteristic. The main goal is togeneralize and strengthen the result of Abhyankar's Conjecture, which wasproven by Raynaud and the Principal Investigator. That result classifiedthe Galois groups of unramified covers of affine curves over analgebraically closed field of finite characteristic. By strengtheningthis conjecture, more understanding would be achieved of how the Galoiscovers of a given curve fit together, and thus about the structure of thefundamental group. By generalizing the conjecture, results would beobtained about Galois groups over affine varieties in higher dimensions,and over curves defined over more general base fields. Methods willinclude formal patching, specialization, cohomology, and the theory ofprofinite groups.The subject area of this project brings together two areas of mathematicsthat each concern symmetry -- symmetry in algebra, in the case of Galoistheory; and symmetry in geometry, in the case of fundamental groups. Ineach of these two situations, mathematical objects can be studied byexamining the forms that their symmetries can take. The connectionbetween the two settings arises from the fact that geometric spaces can bedescribed by algebraic equations, and those equations can be studied byGalois theory. The symmetries of the equations then relate to fundamentalgroups of geometric spaces. A problem that is posed in either of thesetwo 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 thatalgebra can be used in the service of geometry, and vice versa, in orderto study problems that would otherwise be intractable.