A Tale of Two Symmetries

Abstract: I will trace the development of ideas from a 1916 paper of Ramanujan, to the formulation of a conjecture of Jean-Pierre Serre in the 1970's, to its resolution in 2009 in my joint work with Jean-Pierre Wintenberger. Serre's modularity conjecture was a stimulus to much work that led to developments which were crucial to the methods Andrew Wiles used in his solution in 1994 of Fermat's Last Theorem. When Wiles announced his results he had said that his methods were ``orthogonal'' to Serre's conjecture. Our proof of Serre's conjecture uses a strategy that relies crucially on Wiles's methods. The mathematics around Serre's conjecture and our solution of it remains a central area in Number Theory. My recently published mathematical memoir Chasing A Conjecture: Inside the Mind of a Mathematician talks about some of these ideas at an impressionistic level. This talk will give a more mathematical introduction to these ideas: it will flesh out the interplay of Galois and Ramanujan symmetries that are the main protagonists of my book.