Recognition Theorems of Group Theory and Various Inverse Galois Problems

9732592阿比扬卡尔群论的各种识别定理为计算伽罗瓦群提供了强有力的工具。它们还为构造具有指定伽罗瓦群的显式方程，即求解各种伽罗瓦逆问题提供了指导性的指导。建议继续使用这种方法来获得关于代数变种和算术变种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的。其他一些人使用极地空间的分类。还有一些人使用寻找传递性极限的旧技术。这些识别定理的另一个丰富的应用领域是希尔伯特关于复合函数的第十三个问题。另一个这样的应用领域是排列多项式和例外多项式。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。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有应用。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。

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.