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.moreless
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