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

Jung的复去奇化方法不适用于非零特征，因为在复情况下，分支轨迹法线交点以上的局部基本群是可解的，而在非零特征下，它甚至不需要是可解的。这可以通过构造特征五中的六次不可解曲面覆盖来说明。取此不可解曲面覆盖的平截面，提出了关于具有非零特征的代数闭地场上仿射代数曲线的代数基本群结构的猜想。这个仿射曲线猜想被Harbater和Raynaud肯定地解决了。群论的各种认识定理在这个猜想的最初探索中起到了至关重要的作用，它已经导致了这个猜想在有限基场上的精化版本的一些进展。这些认识定理也促进了关于高维代数簇分支轨迹正规交点之上的局部和整体代数基本群的猜想的一些进展。识别定理的另一个丰富的应用领域是排列多项式和例外多项式。这个领域，再加上Guralnick最近关于亏格零覆盖的工作，以及Moore-Carlitz-Drinfeld模的理论，是寻找具有指定Galois群的显式方程的丰富来源。提名人打算继续调查承认在伽罗瓦理论的这些领域中的应用。本研究是代数几何与群论相结合的领域。尽管它们是现代数学中最古老的部分之一，但这两个领域在过去的50年里都取得了革命性的成就。在它的起源，代数几何处理的图形可以定义在平面上的最简单的方程，即多项式。同样，群论起源于对对称性的研究。如今，这两个领域不仅使用了代数的方法，还使用了分析和拓扑学的方法，相反，它们在这些领域以及物理、理论计算机科学和机器人学中都得到了应用。此外，代数几何和群论之间的相互作用继续丰富着这两个学科。

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.