Galois Groups and Matrices

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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).