Recognition Theorems of Group Theory and Various Inverse Galois Problems

9732592 Abhyankar 群论的各种识别定理为计算伽罗瓦群提供了强大的工具。它们还为使用指定的伽罗瓦群构建显式方程（即解决各种逆伽罗瓦问题）提供了建议性指导。建议继续使用该方法来获取代数和算术簇的代数基本群的信息。其中一些识别定理基于有限简单群的分类。其他一些使用极地空间分类。还有一些人使用较旧的技术来寻找传递性的极限。这些识别定理的另一个丰富的应用领域是关于复合函数的希尔伯特第十三问题。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。 这项研究结合了代数几何和群论领域。尽管它们是现代数学最古老的部分之一，但这两个领域在过去五十年中都取得了革命性的繁荣。在其起源中，代数几何处理可以通过最简单的方程（即多项式）在平面中定义的图形。同样地。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，还使用分析和拓扑方法，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技术中都得到了应用。 此外，代数几何和群论之间的相互作用继续丰富了这两个学科。

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.