From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.
From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key p:oint. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.