1 00:00:37,500 --> 00:00:44,499 Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. 2 00:00:46,500 --> 00:00:49,499 One goes into the first room, and it's dark, completely dark 3 00:00:50,000 --> 00:00:55,000 One stumbles around bumping into the furniture, 4 00:00:55,001 --> 00:00:59,201 and gradually, you learn where each piece of furniture is, 5 00:00:59,500 --> 00:01:02,700 and finally, after six months or so, you find the light switch. 6 00:01:03,000 --> 00:01:05,999 You turn it on, and suddenly, it's all illuminated. 7 00:01:06,000 --> 00:01:07,999 You can see exactly where you were. 8 00:01:14,200 --> 00:01:16,199 ANDREW WILES 9 00:01:16,200 --> 00:01:20,199 At the beginning of September, I was sitting here at this desk, 10 00:01:21,700 --> 00:01:25,699 when suddenly, totally unexpectedly, 11 00:01:25,700 --> 00:01:27,699 I had this incredible revelation. 12 00:01:32,200 --> 00:01:35,199 It was the most-- 13 00:01:36,500 --> 00:01:38,499 the most important moment of my working life. 14 00:01:51,500 --> 00:01:54,000 Nothing I ever do again will 15 00:01:55,000 --> 00:01:56,000 I'm sorry. 16 00:02:12,800 --> 00:02:15,300 This the story of one man's obsession 17 00:02:15,301 --> 00:02:18,101 with the world's greatest mathematical problem. 18 00:02:19,000 --> 00:02:20,999 For seven years, professor Andrew Wiles 19 00:02:21,300 --> 00:02:23,299 worked in complete secrecy, 20 00:02:23,300 --> 00:02:25,600 creating the calculation of the century. 21 00:02:26,300 --> 00:02:28,299 It was a calculation which brought him 22 00:02:28,300 --> 00:02:30,299 fame and regret. 23 00:02:33,500 --> 00:02:35,499 So, I came to this. 24 00:02:35,500 --> 00:02:37,499 I was a ten-year-old, 25 00:02:37,500 --> 00:02:40,699 and one day I happened to be looking in my local public library 26 00:02:40,700 --> 00:02:43,699 and I found a book on math 27 00:02:44,700 --> 00:02:47,699 and it told a bit about the history of this problem, 28 00:02:47,700 --> 00:02:51,199 that someone had resolved this problem 300 years ago, 29 00:02:51,800 --> 00:02:54,799 but no one had ever seen the proof. 30 00:02:54,800 --> 00:02:56,799 No one knew if there was a proof. 31 00:02:56,900 --> 00:02:59,899 And people ever since had looked for the proof. 32 00:02:59,900 --> 00:03:01,400 And here was a problem that I,a ten-year-old, 33 00:03:01,401 --> 00:03:03,400 could understand,but 34 00:03:03,599 --> 00:03:08,598 that none of the great mathematicians in the past had been able to resolve. 35 00:03:09,000 --> 00:03:13,300 And from that moment, of course, I just tried to solve it myself. 36 00:03:13,400 --> 00:03:15,399 It was such a challenge, 37 00:03:15,400 --> 00:03:17,399 such a beautiful problem. 38 00:03:19,300 --> 00:03:22,299 This problem was Fermat's last theorem. 39 00:03:25,500 --> 00:03:28,499 Pierre de Fermat was, a 17th century french mathematician 40 00:03:28,500 --> 00:03:32,499 who made some of the greatest breakthroughs in the history of numbers. 41 00:03:36,700 --> 00:03:40,699 His inspiration came from studying the Arithmetica, an Ancient Greek text. 42 00:03:43,000 --> 00:03:44,500 (JOHN CONWAY) Fermat owned a copy of this book, 43 00:03:44,700 --> 00:03:47,799 which is a book about numbers with lots of problems, 44 00:03:47,800 --> 00:03:50,799 which presumably, Fermat tried to solve. 45 00:03:50,800 --> 00:03:52,799 He studied this ; 46 00:03:52,800 --> 00:03:55,300 he wrote notes in the margins. 47 00:03:56,300 --> 00:03:58,600 Fermat's original notes were lost, 48 00:03:59,000 --> 00:04:01,999 but they can still be read in a book published by his son. 49 00:04:02,800 --> 00:04:07,799 It was one of these notes that was Fermat's greatest legacy. 50 00:04:09,000 --> 00:04:12,200 And this is the fantastic observation of master Pierre de Fermat 51 00:04:12,300 --> 00:04:14,299 which caused all the trouble. 52 00:04:14,500 --> 00:04:17,800 "Cubum autem in duos cubos." 53 00:04:18,000 --> 00:04:21,999 This tiny note is the world's hardest mathematical problem. 54 00:04:23,000 --> 00:04:24,999 It's been unsolved for centuries, 55 00:04:25,000 --> 00:04:28,999 yet it begins with an equation so simple that children know it of by heart. 56 00:04:31,100 --> 00:04:36,099 The square of the hypotenuse is equal to the sum of the squares of the other two sides. 57 00:04:38,200 --> 00:04:41,400 Yeah. Well, that's Pythagoras's theorem, isn't it? That's what we all did at school. 58 00:04:42,000 --> 00:04:43,999 So, Pythagoras's theorem, 59 00:04:44,300 --> 00:04:47,299 the clever thing about it is that it tells us 60 00:04:47,500 --> 00:04:49,999 when three numbers are the sides of a right-angle triangle. 61 00:04:50,400 --> 00:04:55,399 That happens just when X squared plus Y squared equals Z squared. 62 00:04:56,800 --> 00:04:59,799 X squared plus Y squared equals Z squared. 63 00:05:01,500 --> 00:05:05,499 And you can ask, "Well, what are the whole number solutions of this equation ?" 64 00:05:06,199 --> 00:05:12,198 You quickly find there's a solution 3 squared plus 4 squared equals 5 squared 65 00:05:13,700 --> 00:05:17,699 Another one is 5 squared plus 12 squared is 13 squared. 66 00:05:18,700 --> 00:05:21,200 And you go on looking, and you find more and more. 67 00:05:21,400 --> 00:05:23,399 So then, a natural question is, 68 00:05:23,400 --> 00:05:27,399 the question Fermat raised : Supposing you change from squares. 69 00:05:28,400 --> 00:05:32,399 Supposing you replace the 2 by 3, 70 00:05:32,400 --> 00:05:33,400 by 4, 71 00:05:33,401 --> 00:05:34,401 by 5, 72 00:05:34,402 --> 00:05:35,402 by 6, 73 00:05:35,403 --> 00:05:37,402 by any whole number "n", 74 00:05:38,800 --> 00:05:43,500 and Fermat said simply that you'll never find any solutions. 75 00:05:43,501 --> 00:05:45,500 However far you look, 76 00:05:45,501 --> 00:05:47,500 you'll never find a solution. 77 00:05:51,000 --> 00:05:53,399 You will never find numbers that fit this equation. 78 00:05:53,700 --> 00:05:55,699 If n is greater than 2, 79 00:05:56,800 --> 00:05:58,500 that's what Fermat said. 80 00:05:58,501 --> 00:06:00,500 What's more, he said he could prove it. 81 00:06:01,500 --> 00:06:06,000 In a moment of brillance, he scribbled the following mysterious note. 82 00:06:07,500 --> 00:06:09,800 Written in Latin, he says he has a truly wonderful proof, 83 00:06:09,801 --> 00:06:12,800 "Demonstrationem mirabilem," of this fact. 84 00:06:12,801 --> 00:06:14,001 And then, the last words are, 85 00:06:14,002 --> 00:06:17,001 "Hanc marginis exigiutas non caperet." 86 00:06:18,200 --> 00:06:21,199 This margin is too small to contain it. 87 00:06:23,500 --> 00:06:28,499 So Fermat said he had a proof, but he never said what it was. 88 00:06:30,700 --> 00:06:33,699 Fermat made lots of marginal notes. 89 00:06:33,700 --> 00:06:35,699 People took them as challenges, 90 00:06:35,700 --> 00:06:39,400 and over the centuries,every single one of them has been disposed of, 91 00:06:39,600 --> 00:06:42,300 and the last one to be disposed of is this one. 92 00:06:42,301 --> 00:06:44,001 That's why it's called the last theorem. 93 00:06:45,600 --> 00:06:49,599 Rediscovering Fermat's proof became the ultimate challenge, 94 00:06:49,600 --> 00:06:53,800 a challenge which would baffle mathematicians for the next 300 years. 95 00:06:56,100 --> 00:06:57,600 Gauss, the greatest mathematician in the world. . . 96 00:06:57,601 --> 00:06:58,601 Oh, yeah. Galois. . . 97 00:07:00,000 --> 00:07:01,000 Kummer, of course. 98 00:07:01,401 --> 00:07:03,601 Well, in the 18th century, Euler didn't prove it. 99 00:07:03,602 --> 00:07:06,601 Well, you know there's only been the one woman, really. 100 00:07:06,602 --> 00:07:07,602 Sophie Germain. 101 00:07:09,399 --> 00:07:12,398 Oh, there are millions. There are lots of people. 102 00:07:13,000 --> 00:07:15,999 But, nobody had any idea where to start. 103 00:07:18,500 --> 00:07:21,499 Well, mathematicians just love a challenge, 104 00:07:21,500 --> 00:07:24,499 and this problem, this particular problem, just looked so simple. 105 00:07:25,400 --> 00:07:28,100 It just looked as if it had to have a solution. 106 00:07:28,101 --> 00:07:30,100 And of course, it's very special 107 00:07:30,101 --> 00:07:32,100 because Fermat said he had a solution. 108 00:07:35,200 --> 00:07:38,199 Mathemeticians had to proof that no numbers fit the equation. 109 00:07:39,400 --> 00:07:43,399 but with the invent of computers couldn't they check each number one by one 110 00:07:44,400 --> 00:07:46,399 and show that none of them fitted ? 111 00:07:50,400 --> 00:07:52,900 Well, how many numbers are there to be dealt with? 112 00:07:52,901 --> 00:07:55,900 You've got to do it for infinitely many numbers. 113 00:07:55,901 --> 00:07:59,900 So, after you've done it for one, how much closer have you got? 114 00:07:59,901 --> 00:08:01,900 Well, there's still infinitely many left. 115 00:08:01,901 --> 00:08:03,900 After you've done it for a thousand numbers 116 00:08:03,901 --> 00:08:05,601 how many, how much closer have you got? 117 00:08:05,602 --> 00:08:07,102 Well, there's still infinitely many left. 118 00:08:07,103 --> 00:08:08,603 After you've done it for a million, 119 00:08:08,604 --> 00:08:10,499 well, there's still infinitely many left. 120 00:08:10,500 --> 00:08:12,200 In fact, you haven't done very many, have you? 121 00:08:15,000 --> 00:08:17,100 A computer can never check every number. 122 00:08:17,300 --> 00:08:21,000 Instead, what's needed is a mathematical proof. 123 00:08:23,000 --> 00:08:26,999 A mathematician is not happy until the proof is complete 124 00:08:27,000 --> 00:08:29,999 and considered complete by the standards of mathematics. 125 00:08:30,000 --> 00:08:32,999 In mathematics, there's the concept of proving something, 126 00:08:33,000 --> 00:08:34,400 of knowing it with absolute certainty. 127 00:08:34,401 --> 00:08:37,400 Which ... Well, it's called "rigorous proof." 128 00:08:37,401 --> 00:08:39,400 That's what mathematics is about. 129 00:08:43,000 --> 00:08:45,999 A proof is a sort of reason, it explains why 130 00:08:46,000 --> 00:08:49,999 no numbers fit the equation without having to check every number. 131 00:08:51,400 --> 00:08:53,399 After centuries of failing to find a proof, 132 00:08:53,400 --> 00:08:56,399 mathematicians began to abandon Fermat. 133 00:08:56,400 --> 00:08:58,399 In favour of more serious maths. 134 00:09:13,000 --> 00:09:15,999 In the '70s, Fermat was no longer in fashion. 135 00:09:16,000 --> 00:09:20,700 At the same time, Andrew Wiles was just beginning his career as a mathematician. 136 00:09:22,200 --> 00:09:27,199 He went to Cambridge as a research student under the supervision of Professor John Coates. 137 00:09:29,000 --> 00:09:31,999 I've been very fortunate to have Andrew as a student, 138 00:09:32,000 --> 00:09:36,999 and even as a research student, he was a wonderful person to work with. 139 00:09:37,000 --> 00:09:38,999 He had very deep ideas then, 140 00:09:39,000 --> 00:09:44,999 and it was always clear he was a mathematician who would do great things. 141 00:09:48,000 --> 00:09:49,200 But not with Fermat. 142 00:09:49,400 --> 00:09:52,399 Everyone thought Fermat's last theorem was impossible, 143 00:09:53,800 --> 00:09:55,799 so Professor Coates encouraged Andrew 144 00:09:55,800 --> 00:09:59,799 to forget his childhood dream and work on more mainstream maths. 145 00:10:01,000 --> 00:10:04,499 When I went to Cambridge, my advisor John Coates 146 00:10:04,500 --> 00:10:07,499 was working on Iwasa theory on elliptic curves 147 00:10:07,500 --> 00:10:09,499 and I started working with him. 148 00:10:15,500 --> 00:10:18,499 elliptic curves were the "in" thing to study. 149 00:10:18,500 --> 00:10:24,499 but perversely, elliptic curves are neither ellipses nor curves. 150 00:10:26,600 --> 00:10:28,800 You may never have heard of elliptic curves 151 00:10:28,900 --> 00:10:30,899 but they're extremely important. 152 00:10:30,900 --> 00:10:32,899 OK. So, what's an elliptic curve? 153 00:10:32,900 --> 00:10:34,899 Elliptic curves. They're not ellipses. 154 00:10:35,500 --> 00:10:37,499 They're cubic curves 155 00:10:37,500 --> 00:10:42,000 whose solution have a shape that looks like a doughnut. 156 00:10:46,399 --> 00:10:51,398 They look so simple, yet the complexity, especially arithmetic complexity, is immense. 157 00:10:55,500 --> 00:10:58,499 Every point on the doughnut is the solution to an equation. 158 00:10:58,650 --> 00:11:04,150 Andrew Wiles now studied these elliptic equations and set aside his dream. 159 00:11:05,700 --> 00:11:09,400 What he didn't realize was that on the other side of the world, 160 00:11:09,401 --> 00:11:14,400 elliptic curves and Fermat's last theorem were becoming inextricably linked. 161 00:11:26,500 --> 00:11:30,499 I entered the University of Tokyo in 1949, 162 00:11:31,800 --> 00:11:35,799 and that was four years after the War, 163 00:11:36,200 --> 00:11:42,199 almost all professors were tired 164 00:11:42,200 --> 00:11:46,199 and the lectures were not inspiring. 165 00:11:49,200 --> 00:11:53,199 Goro Shimura and his fellow students had to rely on each other for inspiration. 166 00:11:53,200 --> 00:11:57,199 In particular, he formed a remarkable partnership 167 00:11:57,200 --> 00:12:00,199 with a young man by the name of Utaka Taniyama. 168 00:12:00,200 --> 00:12:02,699 (Utaka Taniyama) 169 00:12:02,700 --> 00:12:05,699 That was when I became very close to Taniyama. 170 00:12:09,100 --> 00:12:16,099 Taniyama was not a very careful person as a mathematician. 171 00:12:17,200 --> 00:12:19,700 He made a lot of mistakes, 172 00:12:19,701 --> 00:12:24,901 but he made mistakes in a good direction, 173 00:12:27,000 --> 00:12:31,999 and so eventually, he got right answers, 174 00:12:32,000 --> 00:12:35,999 and I tried to imitate him, 175 00:12:36,000 --> 00:12:41,500 but I found out that it is very difficult to make good mistakes. 176 00:12:45,000 --> 00:12:49,999 Together, Taniyama and Shimura worked on the complex mathematics of modular functions. 177 00:12:54,500 --> 00:12:56,700 I really can't explain what a modular function is in one sentence. 178 00:12:56,701 --> 00:12:58,700 I can try and give you a few sentences to explain. 179 00:13:01,500 --> 00:13:03,000 I really can't do it in one sentence. 180 00:13:03,500 --> 00:13:04,700 Oh it's impossible ! 181 00:13:04,701 --> 00:13:07,700 There's a saying attributed to Eichler 182 00:13:08,200 --> 00:13:11,199 that there are five fundamental operations of arithmetic : 183 00:13:11,200 --> 00:13:16,199 addition, subtraction, multiplication, division, and modular forms. 184 00:13:16,999 --> 00:13:21,998 Modular forms are functions 185 00:13:21,999 --> 00:13:25,998 on the complex plane that are inordinately symmetric. 186 00:13:28,300 --> 00:13:31,299 They satisfy so many internal symmetries 187 00:13:31,300 --> 00:13:34,299 that their mere existence seem like accidents. 188 00:13:34,300 --> 00:13:36,299 But they do exist. 189 00:13:38,300 --> 00:13:41,299 This image is merely a shadow of a modular form. 190 00:13:41,300 --> 00:13:45,299 To see one properly, your TV screen would have to be stretched 191 00:13:45,300 --> 00:13:47,500 into something called hyperbolic space. 192 00:13:52,201 --> 00:13:57,701 Bizarre modular forms seem to have nothing whatsoever to do with the humdrum world of elliptic curves. 193 00:14:00,000 --> 00:14:03,999 But what Taniyama and Shimura suggested shocked everyone. 194 00:14:07,000 --> 00:14:14,999 In 1955, there was an international symposium, 195 00:14:15,000 --> 00:14:18,999 and Taniyama posed two or three problems. 196 00:14:21,000 --> 00:14:24,300 The problems posed by Taniyama led to the extraordinary claim 197 00:14:24,501 --> 00:14:29,500 that every elliptic curve was really a modular form in disguise. 198 00:14:32,100 --> 00:14:35,099 It became known as the Taniyama-Shimura conjecture. 199 00:14:39,800 --> 00:14:41,799 What the Taniyama-Shimura conjecture says, 200 00:14:41,800 --> 00:14:44,999 it says that every rational elliptic curve is modular, 201 00:14:45,000 --> 00:14:47,999 and that's so hard to explain. 202 00:14:51,800 --> 00:14:53,799 So, let me explain. 203 00:14:53,800 --> 00:14:58,100 Over here, you have the elliptic world, the elliptic curves, these doughnuts. 204 00:14:58,101 --> 00:15:05,100 And over here, you have the modular world, modular forms with their many, many symmetries. 205 00:15:09,000 --> 00:15:12,500 The Shimura-Taniyama conjecture makes a bridge between these two worlds 206 00:15:12,601 --> 00:15:15,600 These worlds live on different planets. 207 00:15:20,500 --> 00:15:22,700 It's a bridge. It's more than a bridge; 208 00:15:22,701 --> 00:15:24,700 it's really a dictionary, 209 00:15:24,701 --> 00:15:29,700 a dictionary where questions, intuitions, insights, theorems in the one world 210 00:15:30,200 --> 00:15:35,199 get translated to questions, intuitions in the other world. 211 00:15:36,200 --> 00:15:41,199 I think that when Shimura and Taniyama first started talking about the relationship 212 00:15:41,200 --> 00:15:43,199 between elliptic curves and modular forms 213 00:15:43,200 --> 00:15:45,199 people were very incredulous. 214 00:15:45,200 --> 00:15:47,199 I wasn't studying mathematics yet. 215 00:15:47,200 --> 00:15:49,199 by the time I was a graduate student in 1969 or 1970, 216 00:15:49,800 --> 00:15:51,799 people were coming to believe the conjecture. 217 00:15:56,200 --> 00:15:59,199 In fact, Taniyama-Shimura became a foundation 218 00:15:59,200 --> 00:16:02,199 for other theories which all came to depend on it. 219 00:16:03,400 --> 00:16:07,799 But Taniyama-Shimura was only a conjecture, an unproven idea, 220 00:16:09,000 --> 00:16:13,999 and until it could be proved, all the maths which relied on it was under threat. 221 00:16:15,000 --> 00:16:17,999 We built more and more conjectures 222 00:16:18,000 --> 00:16:19,999 stretched further and further into the future, 223 00:16:22,000 --> 00:16:24,999 but they would all be completely ridiculous 224 00:16:25,000 --> 00:16:26,999 if Taniyama-Shimura was not true. 225 00:16:32,700 --> 00:16:36,200 Proving the conjecture became crucial, but tragically, 226 00:16:36,201 --> 00:16:38,701 the man whose idea inspired it 227 00:16:38,702 --> 00:16:41,701 didn't live to see the enormous impact of his work. 228 00:16:43,100 --> 00:16:46,099 In 1958, Taniyama committed suicide. 229 00:16:55,700 --> 00:16:58,699 I was very much puzzled. 230 00:16:58,700 --> 00:17:01,200 Puzzlement may be the best word. 231 00:17:03,000 --> 00:17:04,999 Of course, I was sad but see, 232 00:17:05,000 --> 00:17:06,999 it was so sudden, 233 00:17:07,000 --> 00:17:10,999 and I was unable to make sense out of this. 234 00:17:18,500 --> 00:17:23,499 Taniyama-Shimura went on to become one of the great unproven conjectures. 235 00:17:23,500 --> 00:17:26,499 But what did it have to do with Fermat's last theorem ? 236 00:17:30,000 --> 00:17:31,999 At that time, no one had any idea that 237 00:17:32,000 --> 00:17:35,999 Taniyama-Shimura could have anything to do with Fermat. 238 00:17:37,299 --> 00:17:40,298 Of course, in the '80s, that all changed completely. 239 00:17:53,000 --> 00:17:56,799 Taniyama-Shimura says, "Every elliptic curve is modular," 240 00:17:56,800 --> 00:18:00,799 and Fermat says, "No numbers fit this equation." 241 00:18:01,500 --> 00:18:03,499 What was the connection ? 242 00:18:03,500 --> 00:18:05,700 [music] One way or another 243 00:18:05,701 --> 00:18:06,998 I'm gonna find you 244 00:18:06,999 --> 00:18:09,399 I'm gonna get you get you get you 245 00:18:09,400 --> 00:18:11,400 One way or another 246 00:18:12,300 --> 00:18:14,299 I'm gonna win you. 247 00:18:14,300 --> 00:18:16,299 I'm get you get you get you 248 00:18:16,300 --> 00:18:17,300 One way or another 249 00:18:17,701 --> 00:18:18,701 I'm gonna see you 250 00:18:19,402 --> 00:18:21,602 I wanna wanna meet meet your one day 251 00:18:22,603 --> 00:18:25,098 Well, on the face of it, the Shimura-Taniyama conjecture, 252 00:18:25,099 --> 00:18:29,699 which is about elliptic curves, and Fermat's last theorem have nothing to do with each other, 253 00:18:29,700 --> 00:18:32,699 because there's no connection between Fermat and elliptic curves. 254 00:18:33,700 --> 00:18:37,100 But in 1985, Gerhard Frey had this amazing idea. 255 00:18:39,299 --> 00:18:43,298 Frey, a German mathematician, considered the unthinkable. 256 00:18:43,299 --> 00:18:45,699 What would happen if Fermat was wrong 257 00:18:45,700 --> 00:18:48,699 and there was a solution to this equation after all ? 258 00:18:51,000 --> 00:18:54,999 Frey showed how starting with a fictitious solution to Fermat's last equation, 259 00:18:55,220 --> 00:18:57,620 if, indeed, such a horrible beast existed 260 00:18:57,621 --> 00:19:01,620 he could make an elliptic curve with some very weird properties. 261 00:19:02,400 --> 00:19:05,399 That elliptic curve seems to be not modular. 262 00:19:05,800 --> 00:19:08,799 But Shimura-Taniyama says that every elliptic curve is modular. 263 00:19:10,500 --> 00:19:12,500 So, if there is a solution to this equation, 264 00:19:12,501 --> 00:19:17,001 it creates such a weird elliptic curve it defies Taniyama-Shimura. 265 00:19:19,200 --> 00:19:22,930 So, in other words, if Fermat is false, so is Shimura-Taniyama. 266 00:19:22,931 --> 00:19:27,930 Or, said differently, if Shimura-Taniyama is correct, so is Fermat's last theorem. 267 00:19:29,200 --> 00:19:32,399 Fermat and Taniyama-Shimura were now linked, 268 00:19:32,400 --> 00:19:34,399 apart from just one thing. 269 00:19:35,599 --> 00:19:38,099 The problem is that Frey didn't really prove 270 00:19:38,100 --> 00:19:39,800 that his elliptic curve was not modular. 271 00:19:39,801 --> 00:19:41,800 He gave a plausibility argument, 272 00:19:41,801 --> 00:19:43,800 which he hoped could be filled in by experts, 273 00:19:43,801 --> 00:19:45,800 and then the experts started working on it. 274 00:19:50,000 --> 00:19:53,700 In theory, you could prove Fermat by proving Taniyama, 275 00:19:53,701 --> 00:19:55,700 but only if Frey was right. 276 00:19:55,701 --> 00:19:59,399 Frey's idea became known as the epsilon conjecture, 277 00:19:59,400 --> 00:20:02,399 and everyone tried to check it. 278 00:20:03,000 --> 00:20:07,999 One year later, in San Francisco, there was a breakthrough. 279 00:20:09,500 --> 00:20:11,499 I saw Barry Mazur on the campus, and I said, 280 00:20:11,500 --> 00:20:13,000 "Let's go for a cup of coffee." 281 00:20:13,001 --> 00:20:15,000 And we sat down for cappuccinos at this cafe, 282 00:20:17,700 --> 00:20:19,699 and I looked at Barry and I said, 283 00:20:19,900 --> 00:20:21,600 You know, I'm trying to generalize what I've done 284 00:20:21,601 --> 00:20:24,600 so that we can prove the full strength of Serre's epsilon conjecture. 285 00:20:24,601 --> 00:20:26,501 And Barry looked at me and said, 286 00:20:26,502 --> 00:20:28,202 But you've done it already. 287 00:20:28,203 --> 00:20:31,202 All you have to do is add on some extra gamma zero of m structure 288 00:20:31,203 --> 00:20:33,202 and run through your argument, and it still works, 289 00:20:33,203 --> 00:20:34,903 and that gives everything you need. 290 00:20:34,904 --> 00:20:37,903 And this had never occurred to me, as simple as it sounds. 291 00:20:37,904 --> 00:20:39,903 I looked at Barry, I looked at my cappucino, 292 00:20:39,904 --> 00:20:41,104 I looked back at Barry, and I said, 293 00:20:41,204 --> 00:20:42,404 My God. You're absolutely right. 294 00:20:42,800 --> 00:20:44,000 Ken's idea was brilliant. 295 00:20:53,700 --> 00:20:58,699 I was at a friend's house sipping iced tea early in the evening, 296 00:20:59,400 --> 00:21:02,399 and he just mentioned casually in the middle of a conversation, 297 00:21:02,400 --> 00:21:06,399 "By the way, did you hear that Ken has proved the epsilon conjecture?" 298 00:21:07,099 --> 00:21:09,098 And I was just electrified. 299 00:21:09,700 --> 00:21:14,699 I knew that moment the course of my life was changing, 300 00:21:14,700 --> 00:21:21,699 because this meant that to prove Fermat's last theorem, 301 00:21:21,700 --> 00:21:24,699 I just had to prove Taniyama-Shimura conjecture. 302 00:21:24,700 --> 00:21:28,699 From that moment, that was what I was working on. 303 00:21:28,700 --> 00:21:33,699 I just knew I would go home and work on the Taniyama-Shimura conjecture. 304 00:21:41,300 --> 00:21:44,299 Andrew abandoned all his other research. 305 00:21:44,799 --> 00:21:47,099 He cut himself off from the rest of the world, 306 00:21:47,100 --> 00:21:53,099 and for the next seven years, he concentrated solely on his childhood passion. 307 00:21:59,500 --> 00:22:01,000 I never use a computer. 308 00:22:01,001 --> 00:22:05,000 I sometimes might scribble. I do doodles. 309 00:22:06,400 --> 00:22:10,399 I start trying to find patterns, really, 310 00:22:12,000 --> 00:22:13,600 so I'm doing calculations 311 00:22:13,601 --> 00:22:15,600 which try to explain some little piece of mathematics, 312 00:22:16,599 --> 00:22:20,998 and I'm trying to fit it in with some previous broad 313 00:22:20,999 --> 00:22:26,998 conceptual understanding of some branch of mathematics. 314 00:22:29,300 --> 00:22:31,800 Sometimes, that'll involve going and looking up in a book 315 00:22:31,801 --> 00:22:33,800 to see how it's done there. 316 00:22:33,801 --> 00:22:36,101 Sometimes, it's a question of modifying things a bit, 317 00:22:36,291 --> 00:22:39,691 sometimes, doing a little extra calculation. 318 00:22:39,692 --> 00:22:43,992 And sometimes, you realize that nothing that's ever been done before is any use at all, 319 00:22:43,993 --> 00:22:46,992 and you just have to find something completely new. 320 00:22:46,993 --> 00:22:50,992 And it's a mystery where it comes from. 321 00:22:52,000 --> 00:22:53,100 I must confess, 322 00:22:53,101 --> 00:22:58,100 I did not think that the Shimura-Taniyama conjecture was accessible to proof at present. 323 00:22:58,101 --> 00:23:00,601 I thought I probably wouldn't see a proof in my lifetime. 324 00:23:00,602 --> 00:23:03,601 I was one of the vast majority of people who believed 325 00:23:03,602 --> 00:23:06,902 that the Shimura-Taniyama conjecture was just completely inaccessible, 326 00:23:06,903 --> 00:23:08,403 and I didn't bother to prove it 327 00:23:08,683 --> 00:23:10,183 even think about trying to prove it 328 00:23:10,184 --> 00:23:13,183 Andrew Wiles is probably one of the few people on earth 329 00:23:13,184 --> 00:23:15,183 who had the audacity to dream that 330 00:23:15,184 --> 00:23:17,183 you could actually go and prove this conjecture. 331 00:23:17,484 --> 00:23:20,483 In this case, certainly the first several years, 332 00:23:20,484 --> 00:23:22,483 I had no fear of competition. 333 00:23:23,000 --> 00:23:26,999 I simply didn't think I or anyone else had any real idea how to do it. 334 00:23:32,000 --> 00:23:35,999 Andrew was embarking on one of the most complex calculations in history. 335 00:23:37,500 --> 00:23:39,499 For the first two years, he did nothing 336 00:23:39,500 --> 00:23:42,499 but immerse himself in the problem, 337 00:23:42,500 --> 00:23:45,499 trying to find a strategy which might work. 338 00:23:49,400 --> 00:23:51,399 So, it was now 339 00:23:51,600 --> 00:23:54,599 known that Taniyama-Shimura implied Fermat's last theorem. 340 00:23:56,600 --> 00:23:58,599 What does Taniyama-Shimura say ? 341 00:23:58,600 --> 00:24:00,599 It says that all elliptic curves should be modular. 342 00:24:03,000 --> 00:24:04,500 Well, this was an old problem, 343 00:24:04,501 --> 00:24:06,001 been around for twenty years, 344 00:24:06,002 --> 00:24:09,001 and lots of people had tried to solve it. 345 00:24:10,000 --> 00:24:13,999 Now, one way of looking at it is that you have all elliptic curves, 346 00:24:14,000 --> 00:24:15,599 and then you have the modular elliptic curves, 347 00:24:15,600 --> 00:24:17,599 and you want to prove that there are the same number of each. 348 00:24:17,800 --> 00:24:20,799 Now, of course, you're talking about infinite sets, 349 00:24:20,800 --> 00:24:22,399 so you can't just count them, per se, 350 00:24:22,400 --> 00:24:24,050 but you can divide them into packets, 351 00:24:24,100 --> 00:24:26,799 and you can try to count each packet and see how things go. 352 00:24:26,800 --> 00:24:30,399 And this proves to be a very attractive idea for about thirty seconds, 353 00:24:30,400 --> 00:24:32,399 but you can't really get much further than that. 354 00:24:32,400 --> 00:24:35,399 And the big question on the subject was how you could possibly count, 355 00:24:35,400 --> 00:24:38,600 and in effect, Wiles introduced the correct technique. 356 00:24:45,700 --> 00:24:48,699 Andrew's trick was to transform the elliptic curves 357 00:24:48,700 --> 00:24:50,699 into something called Galois representations. 358 00:24:50,700 --> 00:24:53,699 which would make counting easier. 359 00:24:54,500 --> 00:24:56,099 Now it was a question of comparing 360 00:24:56,100 --> 00:24:59,099 modular forms with Galois representations 361 00:24:59,100 --> 00:25:01,099 not elliptic curves. 362 00:25:05,000 --> 00:25:06,999 Now, you might ask, and it's an obvious question, 363 00:25:07,000 --> 00:25:09,999 why can't you do this with elliptic curves and modular forms ? 364 00:25:10,000 --> 00:25:11,999 Why couldn't you count elliptic curves, 365 00:25:12,000 --> 00:25:14,999 count modular forms, show they're the same number ? 366 00:25:15,000 --> 00:25:18,999 Well, the answer is, people tried and they never found a way of counting them, 367 00:25:20,000 --> 00:25:21,999 and this was why this is they key breakthrough, 368 00:25:22,000 --> 00:25:25,999 that I had found the way to count not the original problem, 369 00:25:26,000 --> 00:25:27,999 but the modified problem. 370 00:25:28,000 --> 00:25:31,999 I'd found a way to count modular forms in Galois representations. 371 00:25:33,599 --> 00:25:35,898 This was only the first step, 372 00:25:35,899 --> 00:25:39,898 and already, it had taken three years of Andrew's life. 373 00:25:43,000 --> 00:25:45,999 My wife's only known me while I've been working on Fermat. 374 00:25:46,500 --> 00:25:49,499 I told her a few days after we got married. 375 00:25:51,800 --> 00:25:56,300 I decided that I really only had time for my problem and my family, 376 00:25:58,099 --> 00:26:01,299 So, I'd found this wonderful counting mechanism, 377 00:26:01,300 --> 00:26:04,800 and I started thinking about this concrete problem in terms of Iwasawa theory. 378 00:26:05,101 --> 00:26:10,100 Iwasawa theory was the subject I'd studied as a graduate student, 379 00:26:10,101 --> 00:26:16,100 and, in fact, with my advisor, John Coates, I'd used it to analyze elliptic curves. 380 00:26:19,010 --> 00:26:24,009 Iwasawa theory was supposed to help create something called a Class Number Formula. 381 00:26:24,700 --> 00:26:30,699 But several months passed and the Class Number Formula remained out of reach. 382 00:26:32,000 --> 00:26:33,999 So, at the end of the summer of '91, 383 00:26:34,000 --> 00:26:35,999 I was at a conference, 384 00:26:36,000 --> 00:26:40,999 and John Coates told me about a wonderful new paper of Matthias Flach, a student of his, 385 00:26:41,000 --> 00:26:43,499 in which he had tackled the class number formula, 386 00:26:43,500 --> 00:26:46,499 in fact, exactly the class number formula I needed. 387 00:26:47,200 --> 00:26:53,199 So, Flach, using ideas of Kolyvagin, 388 00:26:53,800 --> 00:26:57,300 had made a very significant first step 389 00:26:57,301 --> 00:27:00,300 in actually producing the class number formula. 390 00:27:01,801 --> 00:27:04,800 So, at that point, I thought, 'This is just what I need. 391 00:27:04,801 --> 00:27:06,800 This is tailor-made for the problem. 392 00:27:08,599 --> 00:27:12,598 I put aside of the completely the old approach I'd been trying, 393 00:27:12,599 --> 00:27:16,598 and I devoted myself day and night to extending his result. 394 00:27:31,800 --> 00:27:36,799 Andrew was almost there, but this breakthrough was risky and complicated. 395 00:27:37,000 --> 00:27:41,999 After six years of secrecy, he needed to confide in someone. 396 00:27:43,000 --> 00:27:46,500 January of 1993, Andrew came up to me one day at tea, 397 00:27:46,501 --> 00:27:48,101 asked me if I could come up to his office ; 398 00:27:48,402 --> 00:27:49,901 there was something he wanted to talk to me about. 399 00:27:50,002 --> 00:27:52,001 I had no idea what this could be. 400 00:27:52,402 --> 00:27:54,401 I went up to his office. 401 00:27:54,402 --> 00:27:55,902 He closed the door. 402 00:27:56,000 --> 00:27:59,999 He said he thought he would be able to prove Taniyama-Shimura. 403 00:28:00,000 --> 00:28:01,999 I was just amazed. This was fantastic. 404 00:28:03,099 --> 00:28:10,098 It involved a kind of mathematics that Nick Katz is an expert in. 405 00:28:10,099 --> 00:28:14,098 I think another reason he asked me was 406 00:28:14,099 --> 00:28:18,098 that he was sure I would not tell other people, 407 00:28:18,400 --> 00:28:20,399 I would keep my mouth shut. 408 00:28:20,400 --> 00:28:21,399 Which I did. 409 00:28:24,200 --> 00:28:27,199 Andrew Wiles and Nick Katz had been spending 410 00:28:27,200 --> 00:28:30,199 rather a lot of time huddled over a coffee table 411 00:28:30,200 --> 00:28:32,199 at the far end of the common room 412 00:28:32,500 --> 00:28:34,499 working on some problem or other. 413 00:28:34,500 --> 00:28:36,499 We never knew what it was. 414 00:28:38,299 --> 00:28:40,298 In order not to arouse any more suspicion, 415 00:28:40,299 --> 00:28:45,298 Andrew decided to check his proof by disguising it in a course of lectures, 416 00:28:45,299 --> 00:28:47,298 which Nick Katz could then attend. 417 00:28:47,599 --> 00:28:49,598 Well, I explained at the beginning of the course 418 00:28:49,599 --> 00:28:54,098 that Flach had written this beautiful paper 419 00:28:54,099 --> 00:28:58,098 and I wanted to try to extend it to prove the full class number formula. 420 00:28:58,099 --> 00:29:01,098 The only thing I didn't explain was that proving the class number formula 421 00:29:01,099 --> 00:29:04,098 was most of the way to Fermat's last theorem. 422 00:29:09,000 --> 00:29:10,200 So, this course was announced. 423 00:29:10,201 --> 00:29:12,200 It said "Calculations on Elliptic Curves," which could mean anything. 424 00:29:14,201 --> 00:29:16,200 It didn't mention Fermat, it didn't mention Taniyama-Shimura. 425 00:29:16,201 --> 00:29:19,200 There was no way in the world anyone could have guessed that it was about that, 426 00:29:19,201 --> 00:29:20,400 if you didn't already know. 427 00:29:25,000 --> 00:29:26,000 None of the graduate students knew, 428 00:29:26,001 --> 00:29:27,500 and in a few weeks, they just drifted off, 429 00:29:27,501 --> 00:29:31,500 because it's impossible to follow stuff if you don't know what it's for, pretty much. 430 00:29:31,501 --> 00:29:35,100 It's pretty hard even if you do know what it's for. 431 00:29:35,101 --> 00:29:39,100 But after a few weeks, I was the only guy in the audience. 432 00:29:45,000 --> 00:29:47,300 The lectures revealed no errors, 433 00:29:47,301 --> 00:29:51,300 and still, none of his colleagues suspected why Andrew was being so secretive. 434 00:29:54,299 --> 00:29:57,298 Maybe he's run out of ideas. That's why he's quiet. 435 00:29:57,299 --> 00:29:58,698 You never know why they're quiet. 436 00:30:02,000 --> 00:30:04,999 The proof was still missing a vital ingredient, 437 00:30:05,000 --> 00:30:07,999 but Andrew now felt confident. 438 00:30:08,000 --> 00:30:10,999 It was time to tell one more person. 439 00:30:12,700 --> 00:30:14,699 So, I called up Peter and asked him 440 00:30:14,700 --> 00:30:17,699 if I could come 'round and talk to him about something. 441 00:30:23,500 --> 00:30:26,099 I got a phone call from Andrew 442 00:30:26,100 --> 00:30:30,099 saying that he had something very important he wanted to chat to me about. 443 00:30:31,000 --> 00:30:32,999 And sure enough, he had some very exciting news. 444 00:30:34,900 --> 00:30:36,899 I said, "I think you'd better sit down for this." 445 00:30:37,400 --> 00:30:38,599 He sat down. 446 00:30:38,600 --> 00:30:42,599 I said, "I think I'm about to prove Fermat's last theorem." 447 00:30:43,500 --> 00:30:46,499 I was flabbergasted, excited, disturbed. 448 00:30:47,200 --> 00:30:50,699 I mean, I remember that night finding it quite difficult to sleep. 449 00:31:02,510 --> 00:31:04,509 But, there was still a problem. 450 00:31:07,599 --> 00:31:11,198 Late in the spring of '93, I was in this very awkward position 451 00:31:11,199 --> 00:31:14,198 that I thought I'd got most of the curves being modular, 452 00:31:14,199 --> 00:31:18,798 so that was nearly enough to be content to have Fermat's last theorem, 453 00:31:19,000 --> 00:31:24,999 but there were these few families of elliptic curves that had escaped the net. 454 00:31:26,400 --> 00:31:32,399 I was sitting here at my desk in May 93, 455 00:31:32,400 --> 00:31:34,399 still wondering about this problem, 456 00:31:34,400 --> 00:31:39,399 and I was casually glancing at a paper of Barry Mazur's, 457 00:31:39,400 --> 00:31:42,399 and there was just one sentence which 458 00:31:42,400 --> 00:31:46,999 made a reference to actually what's a 19th century construction, 459 00:31:47,000 --> 00:31:51,999 and I just instantly realized that there was a trick that I could use, 460 00:31:52,000 --> 00:31:55,999 that I could switch from the families of elliptic curves I'd been using. 461 00:31:56,599 --> 00:31:58,598 I'd been studying them using the prime three. 462 00:31:58,599 --> 00:32:01,598 I could switch and study them using the prime five. 463 00:32:01,599 --> 00:32:05,598 It looked more complicated,but I could switch from 464 00:32:05,599 --> 00:32:10,598 these awkward curves that I couldn't prove were modular to a different set of curves, 465 00:32:10,599 --> 00:32:12,598 which I'd already proved were modular, 466 00:32:12,599 --> 00:32:15,598 and use that information to just go that one last step. 467 00:32:16,599 --> 00:32:21,198 And, I just kept working out the details, 468 00:32:21,199 --> 00:32:25,198 and time went by, and I forgot to go down to lunch, 469 00:32:25,199 --> 00:32:27,198 and it got to about tea-time, 470 00:32:27,199 --> 00:32:31,198 and I went down and Nada was very surprised that I arrived so late, 471 00:32:31,199 --> 00:32:33,598 and then she -- 472 00:32:33,599 --> 00:32:39,198 I told her that I believed that 473 00:32:39,199 --> 00:32:41,198 I'd solved Fermat's last theorem. 474 00:32:49,700 --> 00:32:53,299 I was convinced that I had Fermat in my hands, 475 00:32:53,300 --> 00:32:58,299 and there was a conference in Cambridge organized by my advisor, John Coates. 476 00:33:00,499 --> 00:33:03,098 I thought that would be a wonderful place. 477 00:33:03,399 --> 00:33:06,398 It's my old hometown, and I'd been a graduate student there. 478 00:33:06,799 --> 00:33:11,798 It would be a wonderful place to talk about it if I could get it in good shape. 479 00:33:18,799 --> 00:33:23,798 The name of the lectures that he announced was simply "Elliptic Curves and Modular Forms." 480 00:33:24,299 --> 00:33:26,298 There was no mention of Fermat's last theorem. 481 00:33:30,000 --> 00:33:33,999 Well, I was at this conference on L functions and elliptic curves, 482 00:33:34,000 --> 00:33:36,999 and it was kind of a standard conference and all of the people were there. 483 00:33:37,000 --> 00:33:38,999 Didn't seem to be anything out of the ordinary, 484 00:33:39,000 --> 00:33:40,999 until people started telling me that 485 00:33:41,200 --> 00:33:45,199 they'd been hearing weird rumors about Andrew Wiles's proposed series of lectures. 486 00:33:47,500 --> 00:33:49,099 I started talking to people 487 00:33:49,220 --> 00:33:52,219 and I got more and more precise information. I have no idea how it was spread. 488 00:33:52,900 --> 00:33:55,100 Not from me. Not from me. 489 00:33:56,300 --> 00:34:00,299 Whenever any piece of mathematical news had been in the air, 490 00:34:00,300 --> 00:34:02,899 Peter would say, "Oh, that's nothing. 491 00:34:03,000 --> 00:34:04,999 Wait until you hear the big news. 492 00:34:05,000 --> 00:34:06,000 There's something big going to break." 493 00:34:06,300 --> 00:34:07,799 Maybe some hints, yeah. 494 00:34:09,699 --> 00:34:12,698 People would ask me, leading up to my lectures, 495 00:34:12,899 --> 00:34:15,898 what exactly I was going to say. 496 00:34:15,899 --> 00:34:17,898 And I said, "Well, come to my lecture and see." 497 00:34:22,699 --> 00:34:24,698 It's a very charged atmosphere. 498 00:34:24,800 --> 00:34:28,799 A lot of the major figures of arithmetical, algebraic geometry were there. 499 00:34:30,000 --> 00:34:31,999 Richard Taylor and John Coates. Barry Mazur. 500 00:34:32,700 --> 00:34:35,699 well, I'd never seen a lecture series in mathematics like that before. 501 00:34:36,900 --> 00:34:40,899 What was unique about those lectures were the glorious ideas, 502 00:34:42,199 --> 00:34:43,798 how many new ideas were presented, 503 00:34:44,599 --> 00:34:47,098 and the constancy of its dramatic build-up. 504 00:34:47,099 --> 00:34:49,098 T' was suspenseful until the end. 505 00:34:50,699 --> 00:34:52,698 There was this marvelous moment 506 00:34:52,699 --> 00:34:56,698 when we were coming close to a proof of Fermat's last theorem. 507 00:34:56,999 --> 00:35:01,598 The tension had built up, and there was only one possible punch line. 508 00:35:02,300 --> 00:35:07,799 So, after I'd explained the 3/5 switch on the blackboard, 509 00:35:08,000 --> 00:35:12,999 I then just wrote up a statement of Fermat's last theorem 510 00:35:13,199 --> 00:35:17,198 said I'd proved it, said, "I think I'll stop there." 511 00:35:25,000 --> 00:35:26,799 The next day, what was totally unexpected was 512 00:35:26,800 --> 00:35:31,799 that we were deluged by inquiries from newspapers, 513 00:35:31,800 --> 00:35:34,799 journalists from all around the world. 514 00:35:46,000 --> 00:35:48,999 It was a wonderful feeling after seven years 515 00:35:49,000 --> 00:35:51,999 to have really solved my problem. 516 00:35:52,000 --> 00:35:54,999 I'd finally done it. 517 00:35:57,199 --> 00:36:00,698 Only later did it come out 518 00:36:00,699 --> 00:36:03,198 that there was a problem at the end. 519 00:36:09,700 --> 00:36:11,299 Now, it was time for it to be refereed, 520 00:36:11,300 --> 00:36:15,299 which is to say,for people appointed by the journal 521 00:36:15,300 --> 00:36:18,299 to go through and make sure that the thing was really correct. 522 00:36:24,300 --> 00:36:26,299 So, for two months, July and August, 523 00:36:26,800 --> 00:36:32,799 I literally did nothing but go through this manuscript line by line, 524 00:36:33,500 --> 00:36:36,499 and what this meant concretely was that essentially every day 525 00:36:37,900 --> 00:36:41,399 sometimes twice a day, I would e-mail Andrew with a question : 526 00:36:41,400 --> 00:36:46,098 "I don't understand what you say on this page, on this line. It seems to be wrong," 527 00:36:46,099 --> 00:36:47,598 or "I just don't understand." 528 00:36:48,400 --> 00:36:52,399 So, Nick was sending me e-mails, and at the end of the summer, 529 00:36:52,400 --> 00:36:54,399 he sent one that seemed innocent at first, 530 00:36:54,400 --> 00:36:56,399 and I tried to resolve it. 531 00:36:56,599 --> 00:36:58,198 It's a little bit complicated, 532 00:36:58,199 --> 00:37:02,198 so he sends me a fax, but the fax doesn't seem to answer the question, 533 00:37:02,199 --> 00:37:03,498 so I e-mail him back, 534 00:37:04,000 --> 00:37:05,000 and I get another fax, 535 00:37:05,001 --> 00:37:07,000 which I'm still not satisfied with. 536 00:37:07,529 --> 00:37:11,528 And this, in fact, turned into the error 537 00:37:11,529 --> 00:37:14,528 that turned out to be a fundamental error, 538 00:37:14,529 --> 00:37:18,528 and that we had completely missed when he was lecturing in the spring. 539 00:37:19,099 --> 00:37:21,098 That's where the problem was, 540 00:37:22,099 --> 00:37:26,098 in the method of Flach and Kolyvagin that I'd extended. 541 00:37:27,099 --> 00:37:30,098 So, once I realized that at the end of September, 542 00:37:30,099 --> 00:37:35,098 that there was really a problem 543 00:37:35,099 --> 00:37:38,098 with the way I'd made the construction, 544 00:37:38,099 --> 00:37:41,999 I spent the fall trying to 545 00:37:42,000 --> 00:37:44,999 think what kind of modifications could be made to the construction. 546 00:37:45,000 --> 00:37:48,999 There are lots of simple and rather natural modifications 547 00:37:49,000 --> 00:37:50,999 that any one of which might work. 548 00:37:51,000 --> 00:37:52,999 And 549 00:37:53,000 --> 00:37:55,999 every time he would try and fix it in one corner, 550 00:37:56,000 --> 00:37:57,999 it would sort of -- 551 00:37:58,000 --> 00:38:00,999 Some other difficulty would add up in another corner. 552 00:38:02,000 --> 00:38:04,999 It was like he was trying to put a carpet in a room 553 00:38:05,000 --> 00:38:07,999 where the carpet had more size than the room, 554 00:38:08,000 --> 00:38:09,999 but he could put it in in any corner, 555 00:38:10,000 --> 00:38:12,999 and then when he ran to the other corners, it would pop up in this corner. 556 00:38:13,000 --> 00:38:15,999 And whether you could not put the carpet in the room 557 00:38:16,000 --> 00:38:17,999 was not something that he was able to decide. 558 00:38:22,200 --> 00:38:25,199 I think he externally appeared normal, but 559 00:38:25,200 --> 00:38:30,799 at this point he was keeping a secret from the world 560 00:38:32,000 --> 00:38:36,999 and I think he must have been, in fact, pretty uncomfortable about it. 561 00:38:37,099 --> 00:38:40,698 Well, you know, we were behaving a little bit like Kremlinologists. 562 00:38:40,699 --> 00:38:45,698 Nobody actually liked to come out and ask him how he's getting on with the proof. 563 00:38:45,699 --> 00:38:49,698 So, somebody would say, "I saw Andrew this morning." 564 00:38:50,699 --> 00:38:54,698 "Did he smile?" "Well, yes. But he didn't look too happy." 565 00:38:57,500 --> 00:39:00,499 The first seven years I'd worked on this problem, 566 00:39:00,500 --> 00:39:02,499 I loved every minute of it. 567 00:39:02,500 --> 00:39:03,799 However hard it had been, 568 00:39:04,000 --> 00:39:06,399 there'd been setbacks often, 569 00:39:06,400 --> 00:39:10,399 there'd been things that had seemed insurmountable, 570 00:39:10,400 --> 00:39:17,399 but it was a kind of private and very personal battle 571 00:39:17,400 --> 00:39:19,399 I was engaged in. 572 00:39:23,000 --> 00:39:26,999 And then, after there was a problem with it, 573 00:39:28,500 --> 00:39:31,499 doing mathematics in that kind of 574 00:39:31,900 --> 00:39:35,900 rather over-exposed way is certainly not my style, 575 00:39:36,499 --> 00:39:38,498 and I have no wish to repeat it. 576 00:40:05,000 --> 00:40:07,999 In desperation, Andrew called for help 577 00:40:08,000 --> 00:40:10,999 he invited his former student, Richard Taylor, 578 00:40:11,000 --> 00:40:12,999 to work with him in Princeton. 579 00:40:13,000 --> 00:40:16,499 But no solution came, after a year of failure, 580 00:40:16,500 --> 00:40:18,499 he was ready to abandon his flawed proof. 581 00:40:20,299 --> 00:40:23,298 In September, 582 00:40:23,299 --> 00:40:27,298 I decided to go back and look one more time 583 00:40:27,299 --> 00:40:33,298 at the original structure of Flach and Kolyvagin 584 00:40:33,299 --> 00:40:38,798 to try and pinpoint exactly why it wasn't working, 585 00:40:38,799 --> 00:40:40,798 try and formulate it precisely. 586 00:40:40,799 --> 00:40:42,798 One can never really do that in mathematics, 587 00:40:42,799 --> 00:40:45,798 but I just wanted to set my mind to rest 588 00:40:45,799 --> 00:40:48,798 that it really couldn't be made to work. 589 00:40:57,000 --> 00:40:59,798 And I was sitting here at this desk. 590 00:40:59,799 --> 00:41:03,298 It was a Monday morning, September 19th, 591 00:41:03,799 --> 00:41:07,798 and I was trying, convincing myself that it didn't work, 592 00:41:07,799 --> 00:41:10,798 just seeing exactly what the problem was, 593 00:41:11,399 --> 00:41:15,298 when suddenly, totally unexpectedly, 594 00:41:15,299 --> 00:41:18,298 I had this incredible revelation. 595 00:41:18,299 --> 00:41:22,698 I realized what was holding me up 596 00:41:22,699 --> 00:41:26,198 was exactly what would resolve the problem 597 00:41:26,199 --> 00:41:30,399 I had had in my Iwasawa theory attempt three years earlier, 598 00:41:31,000 --> 00:41:37,999 was -- It was the most -- 599 00:41:39,000 --> 00:41:41,999 the most important moment of my working life. 600 00:41:59,599 --> 00:42:01,598 It was so indescribably beautiful ; 601 00:42:02,099 --> 00:42:04,098 it was so simple and so elegant, 602 00:42:05,499 --> 00:42:11,498 and I just stared in disbelief for twenty minutes. 603 00:42:11,799 --> 00:42:14,798 Then, during the day, I walked around the department. 604 00:42:14,799 --> 00:42:16,798 I'd keep coming back to my desk 605 00:42:16,799 --> 00:42:18,298 and looking to see if it was still there. 606 00:42:18,299 --> 00:42:19,399 It was still there. 607 00:42:22,599 --> 00:42:25,598 Almost what seemed to be stopping the method of Flach and Kolyvagin 608 00:42:25,599 --> 00:42:29,598 was exactly what would make horizontal Iwasawa theory. 609 00:42:29,599 --> 00:42:32,598 My original approach to the problem from three years before 610 00:42:32,599 --> 00:42:34,598 would make exactly that work 611 00:42:34,999 --> 00:42:40,998 So, out of the ashes seemed to rise the true answer to the problem. 612 00:42:47,200 --> 00:42:52,199 So, the first night, I went back and slept on it. 613 00:42:52,899 --> 00:42:54,898 I checked through it again the next morning, 614 00:42:54,899 --> 00:42:57,398 and by eleven o'clock, 615 00:42:57,399 --> 00:42:59,398 I was satisfied 616 00:43:00,399 --> 00:43:02,398 and I went down and told my wife, 617 00:43:02,399 --> 00:43:05,398 "I've got it. I think I've got it. I've found it." 618 00:43:06,400 --> 00:43:08,399 And it was so unexpected, 619 00:43:08,400 --> 00:43:11,399 I think she thought I was talking about a children's toy or something 620 00:43:11,900 --> 00:43:13,399 and said, "Got what?" 621 00:43:13,900 --> 00:43:17,899 And I said, "I've fixed my proof. I've got it." 622 00:43:30,400 --> 00:43:34,399 I think it will always stand as one of the high achievements of number theory. 623 00:43:34,900 --> 00:43:36,199 It was magnificent. 624 00:43:36,900 --> 00:43:38,899 It's not every day that you hear the proof of the century. 625 00:43:39,599 --> 00:43:44,598 Well, my first reaction was, "I told you so." 626 00:43:51,299 --> 00:43:54,298 The Taniyama-Shimura conjecture is no longer a conjecture, 627 00:43:55,400 --> 00:43:58,399 and as a result, Fermat's last theorem has been proved. 628 00:44:00,099 --> 00:44:03,098 But is Andrew's proof the same as Fermat's? 629 00:44:05,000 --> 00:44:07,999 Fermat couldn't possibly have had this proof. 630 00:44:09,199 --> 00:44:11,298 It's a 20th century proof. 631 00:44:11,400 --> 00:44:14,399 There's no way this could have been done before the 20th century. 632 00:44:15,400 --> 00:44:19,399 I'm relieved that this result is now settled. 633 00:44:19,900 --> 00:44:21,199 But I'm sad in some ways, 634 00:44:21,400 --> 00:44:26,399 because Fermat's last theorem has been responsible for so much. 635 00:44:26,400 --> 00:44:28,399 What will we find to take its place? 636 00:44:32,100 --> 00:44:35,099 There's no other problem that will mean the same to me. 637 00:44:36,099 --> 00:44:40,098 I had this very rare privilege of being able to pursue 638 00:44:42,099 --> 00:44:46,098 in my adult life what had been my childhood dream. 639 00:44:46,099 --> 00:44:50,098 I know it's a rare privilege, 640 00:44:50,099 --> 00:44:52,098 but if one can do this, 641 00:44:52,099 --> 00:44:56,098 it's more rewarding than anything I could imagine. 642 00:45:08,099 --> 00:45:12,098 One of the great things about this work is it embraces the ideas of so many mathematicians. 643 00:45:12,199 --> 00:45:13,999 I've made a partial list. 644 00:45:15,400 --> 00:45:20,399 Klein, Fricke, Hurwitz, Hecke, Dirichlet, Dedekind. . . 645 00:45:21,000 --> 00:45:22,300 The proof by Langlands and Tunnell. . . 646 00:45:23,000 --> 00:45:25,598 Deligne, Rapoport, Katz. . . 647 00:45:25,599 --> 00:45:29,598 Mazur's idea of using the deformation theory of Galois representations. . . 648 00:45:30,000 --> 00:45:33,198 Igusa, Eichler, Shimura, Taniyama. . . 649 00:45:33,699 --> 00:45:34,798 Frey's reduction. . . 650 00:45:35,000 --> 00:45:36,200 The list goes on and on. 651 00:45:36,299 --> 00:45:42,298 Bloch, Kato, Selmer, Frey, Fermat.