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

Jung的复去广泛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群是阿贝尔的，而在非零特征下，它甚至不需要可解。这是通过构造特征五的六次不可解表面覆盖物来证明的。取这一不可解曲面覆盖的平面截面，对非零特征的代数闭场上仿射代数曲线的代数基群的结构进行了推测。这个仿射曲线猜想被哈波特和雷诺肯定地解决了。群论的各种识别定理在这个猜想的初步探索中起了至关重要的作用，导致了这个猜想在有限地面场上的改进版本的一些进展。这些识别定理也使得关于高维代数变体分支轨迹正交点上的局部和全局代数基群的猜想取得了一些进展。识别定理的另一个丰富的应用领域是置换多项式和例外多项式。这个领域，连同Guralnick最近关于属零覆盖的工作，以及Moore-Carlitz-Drinfeld模理论，是寻找具有规定伽罗瓦群的显式方程的丰富来源。作者打算继续研究伽罗瓦理论在这些领域中的应用。本研究是代数几何和群论领域的结合。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50年里都有了革命性的发展。在其起源中，代数几何处理的图形可以用最简单的方程，即多项式，在平面上定义。同样，群论也起源于对对称性的研究。如今，这两个领域不仅使用代数的方法，而且还使用分析和拓扑的方法，反过来，它们在这些领域以及物理学，理论计算机科学和机器人技术中也得到了应用。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.