The Abel Prize Award Ceremony May 24, 2016 The University Aula, Oslo Procession accompanied by the Abel Fanfare (Klaus Sandvik) Performed by musicians from The Staff Band of the Norwegian Armed Forces H.R.H. Crown Prince Haakon enters the University Aula Come Again John Dowland (1563 1626) Opening speech by Professor Ole M. Sejersted President of The Norwegian Academy of Science and Letters From Winterreise: Gute Nacht Franz Schubert (1797 1828) The Abel Committee s citation by Professor John Rognes Chair of the Abel Committee H.R.H. Crown Prince Haakon presents the Abel Prize to Sir Andrew Wiles Acceptance speech by the Abel Laureate Youkali Kurt Weill (1900 1950) / Roger Fernay (1905 1983) H.R.H. Crown Prince Haakon leaves the University Aula Procession leaves The Prize Ceremony will be followed by a reception at Midtgolvet in Det Norske Teatret. During the reception, the Laureate will be interviewed by Nadia Hasnaoui. More info on page 11.
Professor Ole M. Sejersted President of The Norwegian Academy of Science and Letters Your Royal Highness, Minister, Excellencies, Dear Abel Prize Laureate Sir Andrew Wiles, Dear friends of mathematics and basic science, Ladies and Gentlemen On behalf of The Norwegian Academy of Science and Letters, I am very pleased and honored to welcome you to this year's Abel Prize Award Cermony. The Abel Prize was established in 2002, 200 years after Niels Henrik Abel's birth. The Abel Prize is administered by The Norwegian Academy of Science and Letters on behalf of the Norwegian Ministry of Education and Research and is awarded for outstanding work in the field of mathematics. The main purpose of the Abel Prize is to recognize pioneering scientific achievements in mathematics. Another great Norwegian mathematician, Sophus Lie, proposed the establishment of an Abel Prize more than a century ago, but as Fridtjof Nansen put it: "The Abel Prize promised by blessed King Oscar disappeared into thin air along with the union". It was only in 2001 that the Norwegian Government announced the establishment of an Abel Fund, the purpose of which was to award a prize in Abel's honor. In addition to recognizing the contribution made by the prize winner, the Abel Prize is intended to help enhance the status of mathematics in society and encourage children and young people to become interested in mathematics. The Abel Prize highlights the fundamental importance of basic research in finding solutions to the major challenges faced by society. In mathematics, we often find clear examples that advanced solutions that were initially the result of a desire to solve a theoretical problem have subsequently been of unintentional and unexpectedly great practical importance. This year's prize winner, Sir Andrew Wiles, receives the prize "for his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory". Mathematicians have tried to prove Fermat's Last Theorem for 350 years, without success, indicating that mathematicians regard this as one of the great mathematical puzzles. The Abel Committee says that Sir Andrew's work has heralded a new era in number theory. To me, this indicates that the work on the theorem required the development of an entirely new mathematical foundation, the significance of which goes far beyond the actual proving of the theorem. Sir Andrew and Niels Henrik Abel had one thing in common they both worked on elliptic functions. The Norwegian Academy of Science and Letters is pleased and honored to have responsibility for awarding of the Abel Prize. The Academy's main purpose is to support the advancement of science and scholarship, and in its strategy it attaches importance to contributing to good research policy, disseminating knowledge as a basis for decisions and strengthening the best research. The award ceremony and the various activities during the Abel week, especially those aimed at young people, thus provide the Academy with an excellent opportunity to showcase its work and goals. To achieve the intention behind the Abel Prize, the Academy has appointed an Abel Board, which has been charged with stimulating an interest in mathematics among children and young people by supporting projects in kindergartens, schools and science centres, etc. The initiatives supported include the Norwegian Mathematical Olympiad for upper secondary pupils and the UngeAbel contest for teams of pupils in year 9. This year's young winners are here with us in the audience today. You are important ambassadors for mathematics. Fostering an interest in mathematics in young people also requires good teachers. Niels Henrik Abel's teacher, Bernt Michael Holmboe, was pivotal to Abel's career. To emphasize the important role teachers play, a teacher is honored each year with the Holmboe Memorial Prize through the Norwegian Mathematical Council. This year's Holmboe Prize winner is Ingunn Valbekmo from Byåsen School in Trondheim. I would like to thank Petroleum Geo-Services (PGS) for their support in Norway and internationally to stimulate interest in mathematics, especially among children and young people, in developing countries, specifically Ghana. This is a prime example of long-term public private collaboration. Sir Andrew Wiles and all the previous Abel Prize laureates are important role models for young researchers. The Heidelberg Laureate Forum is an annual event where former winners of prizes in mathematics and computer science meet talented young researchers from around the world for mutual inspiration. I would like to thank the Heidelberg Laureate Forum Foundation for having taken the initiative to arrange these annual forums. Today is the fourteenth time the Abel Prize is awarded. Over the years the Abel Prize has achieved a very high status in the mathematical research community all over the world. The Abel Committee, with members nominated by the International Mathematical Union and the European Mathematical Society, deserves much of the honor for the recognition that the Prize has attained internationally. I would like to take this opportunity to thank the Abel Committee, headed by Professor John Rognes, for their excellent work. The Norwegian Academy of Science and Letters would also like to thank the Norwegian mathematics community in general and everyone who helps underscore the importance of mathematics in society. The knowledge society is dependent on the kind of outstanding basic research that the Abel Prize seeks to recognize. The Award Ceremony provides an excellent opportunity to draw attention to this. Last but not least, the Abel Board works tirelessly to showcase mathematics and the importance of mathematics. I would especially like to thank the chair of the Abel Board, Professor Kristian Ranestad, for his dedication and hard work. Distinguished guests and most honored prize winner, Sir Andrew Wiles: This is a great day, not only for you, but for the field of mathematics, for all of us who are here today, and for society at large. Once again, I welcome you to this year's award ceremony. 4 5
Professor John Rognes Chair of the Abel Committee Your Royal Highness, Minister, Your Excellencies, honoured Laureate, dear colleagues and guests! How did Fermat's last theorem come to be important? The story starts well over three thousand years ago. A Babylonian clay tablet lists integer solutions to the equation x 2 + y 2 = z 2. The first row reads 119 2 + 120 2 = 169 2. Over a thousand years later this equation became known as the Pythagorean theorem about the sides of a right triangle. Pythagoras also discovered a connection between music and mathematics. The ratio of the frequencies of two notes that sound good together is a fraction with small numerator and denominator. For an octave, the ratio is two to one. For a major third, the ratio is five to four. That three major thirds sound like one octave corresponds to 5 3 being almost twice 4 3, that is, 125 is close to 128. This coincidence is special enough that human hearing recognises it, perhaps already from birth. Multiplying by eight we find that 2 10 = 1024 is close to 1000, which we recognise in the confusion about what a kilobyte really is. That such coincidences exist, but that they are rare enough to be audible to us, expresses a subtle relationship between addition and multiplication. Fermat's equation x n + y n = z n probably arose out of pure curiosity. What happens if you replace squares by n-th powers, for n>2? More recently, this equation has become a touchstone for our understanding of the subtle relations between addition and multiplication. Looking at examples, it appears to be rare that the sum of two numbers with small prime factors only has small prime factors. In particular, it seems that the sum of two n-th powers is never itself an n-th power. Early attempts to prove this inspired new mathematics connected to number systems where z n y n can be factorised. But Fermat's equation first became really important when it was tied to elliptic curves and modular forms. According to Martin Eichler, modular forms constitute the fifth fundamental operation of arithmetic, together with subtraction, division, and the two I have already mentioned. Modular forms are exceptionally symmetric objects, and we can therefore prove that there are very few of them. The marvel of Andrew Wiles' modularity theorem is that it connects the empirical observations of rare numerical coincidences to a provable mathematical fact. Since we know that there are no modular forms of a specific type, we now also know that there are no solutions to Fermat's equation. How did Wiles come to solve this problem? He read about the equation at an early age. When he heard, in 1986, that Ken Ribet had connected the modularity conjecture to Fermat's last theorem, Wiles realised that he had an exceptional opportunity to try to prove these two results. His earlier work on elliptic curves and Iwasawa theory could give him exactly the expertise needed to attack the problem. He was already a tenured professor at Princeton University, and was free to judge, by himself, how he should best use his research time. The project he decided to grapple with was unusually daring. Hardly anyone else thought that the modularity conjecture could be proven. Therefore the mathematical world was overwhelmed when he seven years later announced that he had found the elusive proof. But this was not the end of the story. By the end of the year, it had become clear that the most recent part of the proof was incomplete. A calculation was not fully justified. The joy turned into dread. How did he manage to keep working, after the mistake was found? Wiles knew that the conceptual part of his proof already gave deep new insights, but to prove the modularity conjecture and Fermat's last theorem he needed the calculation. Only when he could understand why the calculation failed was he able to see the way ahead. A method he had previously abandoned would nonetheless lead to the goal. The persistence shown by Wiles in this year, and his remarkable capacity for clear-headed thinking under pressure, are feats comparable to his initial seven-year effort. The discussions Wiles had with Richard Taylor in this period surely also provided crucial support. The French have an expression for the smart answer you only think of when leaving a party, after saying goodbye to the hosts, and being on your way down the stairs. That answer is called l'esprit d'escalier. In English I have heard the expression garden gate thought, which suggests that you have already left the house and are about to exit the garden when the idea strikes. More common is carriage wit, which must mean that you are sitting in a vehicle, well on your way home, when the right reply comes to you. Either way, it appears that the Frenchman finds his answer a little sooner than the Englishman. In our case, over 350 years sooner. But, and this is to the undisputed honour and glory of this year's Abel laureate, while Pierre de Fermat seems to have jumped to his conclusion, Andrew Wiles was, in the end, completely right! 6 7
Sir Andrew Wiles University of Oxford Abel Laureate 2016 Photo: John Cairns Photo: John Cairns Your Royal Highness, Minister, Excellencies, Ladies and gentlemen As a ten year old eager to explore mathematics I rummaged in the popular mathematics section of my local public library and found a copy of a book called 'The Last Problem' by E.T. Bell. I did not even have to open the book. On the bright yellow front cover it told the story of the 1907 Wolfskehl prize offered for the solution of a famous mathematical problem. The problem itself was on the back cover. I was hooked. It was a wonderful find for me. Apparently inside mathematics there was hidden treasure! A little over three hundred years previously a Frenchman by the name of Pierre de Fermat had solved a beautiful sounding problem, but he had buried the proof and now there was a prize for finding it! For Fermat had written in his copy of a Greek text on arithmetic: ''It is impossible on the other hand to divide a cube into two cubes, a fourth power into two fourth powers or likewise any power higher than the fourth into two like powers. For this truly I have a wonderful proof but this margin is too small to contain it.'' In my teenage years I tried to master the kind of mathematics that Fermat had known in the belief that I could recapture his lost proof. I scoured Fermat's writings for clues. I learned what it was to do research. As I read more I learned that the subject of modern number theory was also born with Fermat. It had grown alongside attempts to solve this problem. In the 19th century the results of Kummer related to it became the backbone of algebraic number theory. Kummer made great progress on it but he could not resolve it. However people following those methods since Kummer's time had achieved very, very little in understanding the problem. The methods were simply not strong enough. By now I had become a professional mathematician and part of the great communal enterprise of mathematics that is as old as recorded history. Looking with a professional eye at these early attempts I awoke to the realization that Fermat had probably been mistaken. And in this awakening I came also to the realization that working on the Fermat problem would be irresponsible. Then in 1985, starting with a novel idea, Gerhard Frey suggested a completely new approach to the problem, which was confirmed by Ken Ribet a year later. Hearing this was electrifying for me. Fermat had re-entered mainstream mathematics. Suddenly it became possible to resume my quest. And this time it was going to be the quest of a lifetime. Fermat did not leave any clues because he did not have a solution, but nature itself leaves clues. I just had to find them. There was never going to be a one line proof. Nor do proofs come just because one has been born with mathematical perfect pitch. There is no such thing. One has to spend years mastering the problem so that it becomes second nature. Then, and only then, after years of preparation is one's intuition so strong that the answer can come in a flash. These Eureka moments are what a mathematician lives for, the burst of creativity that is all the more precious for the years of hard work that go into them. The moment in the morning of September 1994 when I resolved my last problem is a moment I will never forget. The first steps are critical. For one has to set off in a direction, there are many to choose from, but if the first steps are wrong then you can never make progress. Fortunately there are clues, correctly read, which can tell you that you are going in the right direction. The most essential companion is faith because you have to believe that there is a solution, and not just a solution in the abstract but a solution that is accessible within your own lifetime. I could not have begun this journey without the help and generosity of my parents and teachers; my high school teacher who gave me a copy of a famous number theory text, Dominic Welsh who guided my undergraduate studies and John Coates who guided my graduate studies. Of course I depended too on the huge combined effort of the many mathematicians over the centuries who have built up modern mathematics, as well as the smaller number that I was lucky enough to meet and learn from on the way. It is a pleasure to express my deep gratitude to The Norwegian Academy of Science and Letters and the Abel committee for awarding me this prize. It is an unparalleled honour. And finally I thank my wife and daughters who helped me in my long struggle with this problem and who I am very happy to have here with me today to help celebrate this award. 8 9
Music for a while Reception and interview Nadia Hasnaoui in conversation with the Abel laureate The Prize Award Ceremony will be followed by a reception at Det Norske Teatret, Kristian IVs g. 8. During the reception, Sir Andrew Wiles will be interviewed by NRK TV-host Nadia Hasnaoui. Light refreshments will be served. The event is open for all guests attending the Award Ceremony. Photo: Åsa Maria Mikkelsen Tora Augestad was born in Bergen, and educated at the Norwegian Academy of Music in Oslo and the Royal College of Music in Stockholm. She holds a Master's degree in cabaret singing. Augestad moved to Berlin in 2007, and has since been working with some of Europe's leading ensembles for contemporary music. The trademark of Tora Augestad and Music for a while is the group's personal, organic and creative approach to the material, regardless of its origins. Whether the point of departure is Kurt Weill's familiar cabaret melodies or songs and pieces from the classical repertoire, two elements consistently stand out: vocalist Tora Augestad's vocal control and unique communication skills, and the band's elegant, seamless and genre-crossing treatment of the material. Music for a while tear down boundaries and crumbling conventions, bringing the music even closer to its origins. In addition to singer Tora Augestad, the ensemble features a dream team of Norwegian jazz musicians: Stian Carstensen, accordion Mathias Eick, trumpet Pål Hausken, drums & percussion 10 11