Recruitment Talk -- Stefan Patrikis

Title: Inverse Galois problems

 

Abstract: The classical inverse Galois problem for a field K--the most basic case being the field of rational numbers--asks what finite groups can arise as Galois groups of extensions of K. In fact, this classical problem admits a vast generalization, which takes into account not only the finite extensions of K but also features of the topology of algebraic varieties defined by polynomial equations with coefficients in K. In this generalization, not only finite groups but also algebraic groups arise as the relevant symmetry groups. I will motivate this general problem, suggest some of its connections to other central phenomena in arithmetic geometry and the Langlands program, and discuss some work on finding the exceptional algebraic groups out in the arithmetic wild.