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

荣格的复去奇异化方法不适用于非零特征，因为在复情况下，分支轨迹法向交叉上方的局部基本群是阿贝尔的，而在非零特征中，它甚至不需要可解。这是通过构造一个特征为5的6度不可解曲面覆盖来证明的。取这个不可解曲面覆盖的一个平面截面，得到一个关于非零特征的代数闭基域上仿射代数曲线的代数基本群结构的猜想。Harbater和Raynaud肯定地解决了这个仿射曲线猜想。群论中的各种识别定理在这个猜想的最初探索中起了关键作用，并导致了有限基域上这个猜想的改进版本的一些进展。这些识别定理也导致了高维代数簇的分支轨迹的正规交叉上的局部和整体代数基本群猜想的某些进展。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。这一领域，连同Guralnick最近的工作属零覆盖，以及理论的摩尔-Carlitz-Drinfeld模块是一个丰富的来源，寻找明确的方程与规定的伽罗瓦群。提议者打算继续他的研究，以应用到这些领域的伽罗瓦理论的承认。 本研究是代数几何与群论相结合的研究。虽然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五十年里都有了革命性的发展。在它的起源，代数几何处理的数字，可以定义在平面上的最简单的方程，即多项式。同样，群论也起源于对称性的研究。如今，这两个领域不仅使用代数方法，而且还使用分析和拓扑学方法，相反，它们在这些领域以及物理学，理论计算机科学和机器人学中找到了应用。此外，代数几何和群论之间的相互作用继续丰富这两个学科。

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.