Galois Groups and Matrices

9970357这个建议是关于正则反Galois问题与其他两个主题之间的联系：第一，N.Katz在其1996年出版的《刚性局部系统》一书中发展的中间卷积理论；第二，Frey和Kani提出的关于椭圆曲线的亏格2覆盖和椭圆曲线的相关算术性质的某些问题。Galois理论，研究高次代数方程的解之间的对称性，数百年来一直是代数的中心。最近理论和应用性质的发展-有限单群的分类以及强大的计算机代数系统的出现-重新点燃了伽罗瓦理论中的许多研究活动，围绕着伽罗瓦理论的反问题，这是数学的主要开放问题之一：(在代数方程的解中)可以出现哪种类型的对称的问题。

9970357This proposal is about connections between the Regular Inverse Galois Problem and two other topics: Firstly, the theory of the middle convolution developed by N. Katz in his book " Rigid Local Systems", published in 1996; and secondly, certain questions raised by Frey and Kani about genus 2 covers of elliptic curves and related arithmetic properties of elliptic curves.Galois theory, the study of symmetries between solutions of high degree algebraic equations, has been a centerpiece of algebra for hundreds of years. Recent developments of both theoretical and applied nature --- namely, the classification of finite simple groups as well as the advent of powerful computer algebra systems ---have rekindled a lot of research activity in Galois theory, centering around the Inverse Problem of Galois Theory, one of the major open problems of mathematics: The question which type of symmetries can occur (among the solutions of algebraic equations).