Wiles's proof of Fermat's Last Theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem


00:02:18 1 Precursors to Wiles' proof
00:02:28 1.1 Fermat's Last Theorem and progress prior to 1980
00:03:54 1.2 The Taniyama–Shimura–Weil conjecture
00:05:37 1.3 Frey's curve
00:08:18 1.4 Ribet's Theorem
00:09:58 1.5 Situation prior to Wiles' proof
00:11:45 2 Andrew Wiles
00:12:28 3 Announcement and subsequent developments
00:13:01 3.1 Announcement and final proof (1993–1995)
00:17:50 3.2 Subsequent developments
00:19:25 4 Summary of Wiles' proof
00:20:20 5 Mathematical detail of Wiles proof
00:20:31 5.1 Overview
00:25:34 5.2 General approach and strategy
00:29:19 5.3 Structure of Wiles's proof
00:31:07 5.4 Overviews available in the literature
00:32:54 6 Notes



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SUMMARY
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Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993 the proof was found to contain an error. One year later on Monday 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995.Wiles' 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 which were not available to Fermat.
Together, the two papers which contain the proof are 129 pages long, and consumed over seven years of Wiles's research time. John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the [20th] century. Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. For solving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize. When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof".