University of California Berkeley mathematics professor Kenneth Ribet discusses Serre's Modularity Conjecture.
Ribet describes recent work of Khare, Wintenberger, Kisin and others that has led to the proof of a famous conjecture made by Serre in 1972 and then, in a much more precise form, in 1987. The conjecture proposed a precise relation between classical modular forms and certain 2-dimensional mod p representations of the absolute Galois group of the rational field. The proof may be viewed as an elaborate induction on the pair of numerical invariants that Serre associates to a given representation. It employs a number of clever new ideas, together with such inputs as the relative modularity theorems of Taylor-Wiles, Skinner-Wiles and Kisin; potential modularity theorems of Taylor; deformation theory a la Mazur. The talk will focus on the history of the conjecture, numerical examples, and links with the circle of ideas that were used to prove Fermat's Last Theorem in the mid-1990s.
Kenneth Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. His mathematical interests include algebraic number theory and algebraic geometry.
He is credited with paving the way towards Andrew Wiles's proof of Fermat's last theorem. Ribet proved that the epsilon conjecture which was established by Gerhard Frey was indeed true, and thereby proved that Fermat's Last Theorem would follow from the Taniyama-Shimura conjecture.