This quantity comprises the accelerated lectures given at a convention on quantity idea and mathematics geometry held at Boston collage. It introduces and explains the numerous rules and methods utilized by Wiles, and to give an explanation for how his outcome could be mixed with Ribets theorem and ideas of Frey and Serre to end up Fermats final Theorem. The publication starts with an outline of the total evidence, through numerous introductory chapters surveying the fundamental thought of elliptic curves, modular services and curves, Galois cohomology, and finite workforce schemes. illustration conception, which lies on the middle of the facts, is handled in a bankruptcy on automorphic representations and the Langlands-Tunnell theorem, and this can be by way of in-depth discussions of Serres conjectures, Galois deformations, common deformation jewelry, Hecke algebras, and entire intersections. The e-book concludes via taking a look either ahead and backward, reflecting at the background of the matter, whereas putting Wiles'theorem right into a extra basic Diophantine context suggesting destiny functions. scholars mathematicians alike will locate this an imperative source.