Solved conjectures of mathematics 

 In reverse chronological order. 

Page maintained by Jean-Claude Evard. Last update: January 1, 2003.

 June 21, 1999: The Shimura-Taniyama conjecture  is proved.
 This conjecture says that every elliptic curve is modular. 
      The proof was achieved in four steps:
      First step: In 1995, Andrew Wiles proved, with a contribution from 
      Richard Taylor, that a large class of elliptic curves, including all semistable 
      elliptic curves, are modular. This was what was needed to deduce Fermat's
      Last Theorem from Ribet's Theorem (See 1995 on this page). 
      This first step was published in the following two publications:
      Andrew Wiles, Modular elliptic curves and Fermat's last theorem,  Ann. 
      of Math. (2) 141 (1995), no. 3, 443--551. Review by Karl Rubin  Link .
      Richard Taylor and AndrewWiles, Ring-theoretic properties of certain 
      Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553--572.
      Review by Karl Rubin  Link .
      Second step: In 1996, Fred Diamond extended the first step to the proof 
      that every elliptic curve over Q with semi-stable reduction at 3 and 5 is 
      modular. This second step was published in the following publication:
      Fred Diamond, On deformation rings and Hecke rings, Ann. of Math.
      (2) 144 (1996), no. 1, 137--166. Review by Richard Taylor:  Link .
      Third step: In 1999, Brian Conrad, Fred Diamond, and Richard Taylor 
      extended the second step to the proof hat every elliptic curve whose 
      conductor is not divisible by 27 is modular. 
      This third step was published in the following publication:
      Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain 
      potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 
      12 (1999), no. 2, 521--567. Review by Ian Kiming  Link .
      Fourth and final step: In 2001, Christophe Breuil, Brian Conrad, Fred 
      Diamond, and Richard Taylor extended the third step to the proof that all 
      elliptic curves are modular. 
      The final step was published in the following e-publication: 
      Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor, On the 
      modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. 
      Soc.14 (2001), no. 4, 843--939 (electronic). Review by Karl Rubin  Link .
Additional information:
1.   Mathematicians who made this event:
      Christophe Breuil  Link  at the University of Paris-Sud at Orsay, France, 
      Brian Conrad  Link  at the University of Michigan at Ann Arbor, 
      Fred Diamond  Link  at Brandeis University, 
      Richard Taylor  Link  at Harvard University.
      Andrew Wiles  Link  at Princeton University.
2.   Brian Conrad, Fred Diamond, and Richard Taylor were Ph.D.
      students of Andrew Wiles, see Genealogy Project:  Link .
3.   Biography  Link  of Andrew Wiles provided 
      by John O'Connor and E. F. Robertson in the School of Mathematics 
      of the University of St. Andrews in Scotland:
4.    Statement of the Taniyama-Shimura  Link  on MathWorld,  
       by Eric Weisstein at Wolfram Research.
5     See Frank Morgan's Math Chat of July 1, 1999:  Link .        
6.     Copy of a related talk presented by Brian Conrad  Link .   
7.     Announcement  Link  of the achievement of the proof, 
        program  Link , and lectures on video  Link , 
        at the conference Modularity of Elliptic Curves and Beyond held at 
        the Mathematical Sciences Research Institute on December 6-10, 1999. 
8.     Announcement  Link  of the achievement of the proof 
        by the Department of Mathematics of Brandeis University.
9.     Announcement  Link  of a colloquium presented at Ohio State University, 
        by Brian Conrad, on Monday, October 11, 1999 . 
10.   Announcement  Link  published in the Notices of the American 
        Mathematical Society of December 1999, 1397--1401, 
        by Henri Darmon  Link  and  Link  at McGill University  Link .
11.   Curving Beyond Fermat  Link , by Ivars Peterson, 
        Science News Online, Vol. 156, No. 21, November 20, 1999.
12.   Curving Beyond Fermat, sources and references  Link ,  by Brian Conrad, 
        Science News Online, Vol. 156, No. 14, p. 221, October 2, 1999 . 
13.   Announcement  Link  of the achievement of the proof 
         by the British Broadcasting Company (BBC), on November 19, 1999. 

 1995: Fermat's Last Theorem is proved,  thanks to the work of 
    Andrew Wiles  Link  at Princeton University, and several other 
    mathematicians, notably, Gerhard Frey  Link  at the Institut für Experimentelle 
    Mathematik in Universität Gesamthochschule in Essen, Germany, Kenneth 
    Ribet  Link  at the University of California at Berkeley, Fields medallist Gerd 
    Faltings  Link  at Max-Planck-Institut für Mathematik, Yukata Taniyama, 
    deceased, Goro Shimura, Professor Emeritus at Princeton University  Link , 
    and Richard Taylor  Link  at Harvard University.
Additional information:
1.   The proof of Fermat's Last Theorem was published in the following paper:
      Andrew Wiles, Modular elliptic curves and Fermat's last theorem,  Ann. 
      of Math. (2) 141 (1995), no. 3, 443--551. Review by Karl Rubin  Link .
2.   The final correction of this proof relies on the following paper:
      Richard Taylor and AndrewWiles, Ring-theoretic properties of certain 
      Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553--572.
      Review by Karl Rubin  Link .
3.   Pages maintained by Eric Weisstein at Wolfram Research:
      History:  Link . 
      Frey curve:  Link . 
      Epsilon conjecture:  Link . 
      Ribet's theorem:  Link .
4.   Ribet's theorem was published in the following paper:
      Kenneth Ribet, On modular representations of 
      ${\text \rm Gal}({\overline Q}/Q)$ arising from modular forms, Invent. . 
      Math 100 (1990), no. 2, 431--476. Review by Glenn Stevens  Link .
5.   History of the subject  Link  and biography of Andrew Wiles  Link  
      by John O'Connor and E. F. Robertson in the School of Mathematics 
      of the University of St. Andrews in Scotland.
6.   The Mathematics of Fermat's Last Theorem  Link   
      by Charles Daney  Link  in the Monterey Peninsula.
 For the first two steps in the direction of the complex Fermat's Last Theorem, 
 see 1993 and 1999. 

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