Recognition Theorems of Group Theory with Applications to Modular Galois Theory and Desingularization

Purdue Research Foundation, West Lafayette IN

Investigators

Abstract

Jung's method of complex desingularization does not adopt to nonzero characteristic because in the complex case the local fundamental group above a normal crossing of the branch locus is Abelian whereas in nonzero characteristic it need not even be solvable. This is shown by constructing an unsolvable surface covering of degree six in characteristic five. Taking a plane section of this unsolvable surface covering, leads to a conjecture about the structure of the algebraic fundamental group of an affine algebraic curve over an algebraically closed ground field of nonzero characteristic. This Affine Curve Conjecture was settled affirmatively by Harbater and Raynaud. Various Recognition Theorems of group theory, which played a crucial role in the initial explorations of this Conjecture, have lead to some progress in a refined version of this Conjecture over finite ground fields. These Recognition Theorems, have also lead to some progress in Conjectures about Local and Global algebraic fundamental groups above normal crossings of branch loci of higher dimensional algebraic varieties. Another fertile application area for the Recognition Theorems has been in the direction of Permutation Polynomials and Exceptional Polynomials. This area, together with Guralnick's very recent work on genus zero coverings, as well as the theory of Moore-Carlitz-Drinfeld Modules is a rich source for finding explicit equations with prescribed Galois groups. The proposer intends to continue his investigations into the application of the Recognition into these areas of Galois theory. This research is in the combination of the fields of algebraic geometry and group theory. Although they are amongst the oldest parts of modern mathematics, both these fields have had a revolutionary flowering in the past fifty years. In its origin, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Likewise, group theory had its origin in the study of symmetries. Nowadays, both these fields make use of methods not only from algebra, but also from analysis and topology, and conversely they are finding applications in those fields as well as in physics, theoretical computer science and robotics. Moreover, interplay between algebraic geometry and group theory continues to enrich both these disciplines.