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This quantity includes the elevated lectures given at a convention on quantity idea and mathematics geometry held at Boston collage. It introduces and explains the various principles 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 turn out Fermats final Theorem. The e-book starts off with an summary of the whole facts, via a number of introductory chapters surveying the elemental thought of elliptic curves, modular features and curves, Galois cohomology, and finite workforce schemes. illustration concept, which lies on the center of the evidence, is handled in a bankruptcy on automorphic representations and the Langlands-Tunnell theorem, and this is often through in-depth discussions of Serres conjectures, Galois deformations, common deformation jewelry, Hecke algebras, and entire intersections. The ebook concludes via taking a look either ahead and backward, reflecting at the heritage of the matter, whereas putting Wiles'theorem right into a extra normal Diophantine context suggesting destiny functions. scholars mathematicians alike will locate this an fundamental source.