Recognition Theorems of Group Theory and Various Inverse Galois Problems

小行星9732592 群论的各种识别定理为计算伽罗瓦群提供了有力的工具。他们还提供了建议性的指导方针，用于构造具有指定伽罗瓦群的显式方程，即，解决各种各样的Galois逆问题。建议继续使用这种方法来获得关于代数簇和算术簇的代数基本群的信息。其中一些识别定理是基于有限单群的分类。有些人使用极空间的分类。还有一些人使用较老的技术来寻找及物性的极限。另一个肥沃的应用领域，这些识别定理是在希尔伯特的第十三个问题的复合功能。另一个这样的应用领域在于置换多项式和例外多项式的方向。 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。我也是群论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。

9732592 Abhyankar Various Recognition Theorems of Group Theory provide powerful tools for computing Galois groups. They also provide suggestive guidelines for constructing explicit equations with assigned Galois groups, i.e., for solving various Inverse Galois Problems. It is proposed to continue using this method to get information on the algebraic fundamental groups of algebraic and arithmetical varieties. Some of these Recognition Theorems are based on the Classification of Finite Simple Groups. Some others use the Classification of Polar Spaces. Still some others use the older techniques of finding Limits of Transitivity. Another fertile application area for these Recognition Theorems is in Hilbert's Thirteenth Problem about composite functions. Yet another such application area lies in the direction of Permutation Polynomials and Exceptional Polynomials. 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 from analysis and topology, and conversely they are finding applications in those fields as well as in physics, theoretical computer science and robotics. Moreover, an interplay between algebraic geometry and group theory continues to enrich both these disciplines.