My Search for Ramanujan

by Ken Ono

  Instead of dwelling on the pain of Missoula, he had redirected my thoughts. I looked inside myself. He was right. I had become a different person. I was enjoying mathematics for its own sake. Under Gordon’s tutelage, the walks on the Santa Monica boardwalk, the poetry, the Chopin nocturnes, and the mathematics had transformed me. I saw the world differently, and I had become a mathematician. And I was now able to see beauty everywhere around me.

  But why the Mickey Mouse quotation, I wondered. Gordon explained that his first encounter with Mickey Mouse was in his role as the sorcerer’s apprentice in the film Fantasia. When his master the sorcerer had gone to bed, leaving his apprentice to carry heavy buckets of water to fill a cauldron, Mickey put on the magician’s hat and transformed a broomstick into a legion of water bearers. But when the cauldron was full, Mickey was unable to find the magic formula to tell the broomsticks to stop. A flood ensued, and Mickey was punished. He had transgressed, had been a naughty mouse. But he had dared to attack a difficult problem, and although he got himself into trouble, he went on to conquer the world as one of the best-loved of all creatures, a mouse who brings a smile to every face on the planet at the mere sight of his trademark ears.

  Gordon said something like this:Ken, be like naughty Mickey Mouse. You are already a magician creating lovely mathematics. Mickey had an inauspicious beginning as the apprentice. He tried to do something before he was ready. But he ultimately triumphed and became a worldwide symbol of joy and magic. I predict that you, too, will emerge as someone who was meant to follow in your father’s footsteps. Your energy and youthful enthusiasm are palpable. I already see it. Now, if you can dream it, you can do it.

  I couldn’t believe what I was hearing. His words vanquished the depression that I had brought back with me from Missoula. Right at the moment that the script called for criticism—Ken-chan, you screw up—Gordon offered me the praise that I had so desperately sought as a tiger boy. But it was more than that. He was in effect praising me for screwing up: Ken, you messed up, but that’s because you reached beyond your grasp. Now go out and conquer the world. Gordon’s words became my battle cry. I had a new voice in my head: “Ken, be like naughty Mickey.”

  Mathematicians who knew me in the early 1990s will now understand why I attended conferences wearing a stylish baseball cap emblazoned with the image of Mickey Mouse. I didn’t wear the cap as a souvenir of Disneyland or because I wanted to be a Mouseketeer. I wore it to remind me of Gordon’s inspirational words. That cap was my talisman.

  That meeting with Gordon turned out to be one of the most uplifting moments of my life, one whose memory brought me to tears a few years later as I attempted to recount it at a conference I had organized with George Andrews in honor of Gordon’s sixty-fifth birthday. I wanted to tell the world how important Gordon was to me.

  It was time to get back to work. We had three weeks to prepare for the Rademacher Centenary Conference at Penn State. Some of the most important figures in algebraic number theory would be there: George Andrews (Penn State), Bruce Berndt (University of Illinois), Dorian Goldfeld (Columbia University), Harold Stark (UCSD), among many others. I had learned from the mistakes I had made in Missoula, and together we prepared a talk for this high-powered audience.

  We huddled over the coffee table in Gordon’s reading room, scribbling an outline, which we then shaped into a well thought out presentation that I later printed on overhead transparencies. When we were done, the room was a sea of paper, as if a powerful box fan had been turned on full blast in front of an unsuspecting stack of recycling paper.

  Since Ramanujan scholars Andrews and Berndt would be in the audience, we decided that I should speak about congruences for modular forms, the theory that Deligne, Serre, and Swinnerton-Dyer had developed to explain some of Ramanujan’s classic formulas. I had made some small contributions to the field, and using Ramanujan’s well-known results as motivation, I hoped to keep the attention of the audience long enough to explain my findings. And if I succeeded, then perhaps I would impress some experts with my results.

  I began my presentation with Ramanujan, using his renown as a hook to draw my audience in. Not that I needed a hook. In contrast to my lectures in Missoula, this time I was speaking to an audience of experts in the field, a mathematician’s version of “preaching to the choir.” My lecture was well received, and I was even invited to submit a paper to the conference proceedings. Several famous number theorists approached me after my lecture, and they congratulated me on the nice work I had done. Although my results didn’t astound anyone, I had produced respectable work that experts felt was worthwhile. As the German mathematician Leopold Kronecker wrote in a letter to his colleague Georg Cantor, quoting an old saying, “When kings are building, there is work for carters.” He went on to say that a mathematical researcher has to be both king and carter, but at this stage of my career, and especially after the Missoula disaster, I was delighted to earn praise for my yeoman work. After all, even Mickey Mouse had gotten his start as a water carrier. Those number theorists had been kind, and they seemed genuinely to want to encourage me as a junior member of their guild.

  In Lutherville after the Rademacher conference (left to right: Doug Bowman, Ken Ono, Takasan)

  With the success of the Rademacher conference buoying me up, I could look back over the past eight years with a certain satisfaction. I had come a long way, and as I traced my career from high school to college to graduate school, I saw a common thread, and that thread was Ramanujan. When I was a high-school student, Ramanujan, because he had been a hero to my father, had been the Ariadne’s thread that allowed me to escape the labyrinth in which I felt hopelessly trapped. Ramanujan’s story of achieving success by following his passion even if it meant twice flunking out of college had inspired my father and then me as well. Thanks to Ramanujan, for reasons that I still did not fully understand, my parents had let me run away from my former life. Then when I was drifting as an unmotivated college student, Ramanujan had given me hope, inspiring me, just as my clock was running out, to apply myself to my studies, despite my fear that I would discover that I wasn’t good enough to be a mathematician. I did well enough at UChicago to be rescued by Paul Sally, who went out of his way to help me get into a graduate program.

  Then as a graduate student, I had followed Ramanujan’s lead again. But now I was following Ramanujan’s actual mathematics. I presented my work in the inspirational context of Ramanujan’s compelling story, and my talk was well received. Each time I looked at a positive event in my life, there was Ramanujan.

  I was making progress toward completion of my dissertation, and I had an advisor who was fully committed to me. But that was not enough to subdue the voices in my head. My parents continued to doubt my prospects, and I still yearned for their acceptance and approval. My first published papers and my lecture at the Rademacher conference weren’t sufficient to earn their praise. For them, there was only one acceptable outcome—which actually wasn’t my goal—a position at the Institute for Advanced Study or at the very least a professorship at a top university.

  Even if it had been my goal, the harsh realities of the outside world continued to confirm my feeling of inadequacy. The academic job market was brutal, and I witnessed close friends suffer the indignity of a failed job search. There were a few dozen graduate students at the Rademacher meeting at Penn State, and many of them had presented results that were much more impressive than mine. There would be postdoctoral fellowships for perhaps three or four of them, and tenure-track positions, at schools I had never heard of, for a few others. The odds were not in my favor. There certainly wouldn’t be a job for me at Montana, and I was afraid that there might not be a job for me anywhere.

  Nevertheless, I had complete faith in Gordon, and I continued to work on my thesis, creating mathematics for its own sake. At least now the negative voices in my head had some competition: “Ken, be like Mickey. If you can dream it, you can do it.”

  I defended my dissertation
in March 1993. I had worked out theorems that I was passionate about. And they were mine. Like a sorcerer, I had conjured up a proof that previously had not existed. And though I was still a sorcerer’s apprentice, I didn’t have to fear that my theorems would get me in trouble. The only danger was that they might not be good enough. But I had done my best. Following my advisor as a role model, I had become a mathematician who pursued the beauty of mathematics for its own sake, without worrying too much about whether anybody would care about my work. But now it was time to worry. If I didn’t want to starve like Ramanujan, I was going to have to get a job. And moreover, without a job, I would be even more at the mercy of my voices. I wasn’t confident that my results would pique the interest of many mathematicians, and I feared that I wouldn’t get any job offers from research universities. At the time I defended my thesis, I had no job offers, and the annual hiring cycle would soon be over.

  My failure to get any offers gave the voices in my head new power:Ken-chan, your thesis not very good. You say you do math for own sake, but results you get mediocre. You should have been studying all along instead of riding bike and partying. The UChicago junior professor and Montana professor both speaking truth. You wasting everybody’s time.

  Then ten weeks after I defended my thesis, in June 1993, by which time all the postdoctoral fellowships and tenure-track positions had been filled, I received an unexpected email from Andrew Granville, a British mathematician at the University of Georgia. He was inviting me to apply for a one-year visiting position; the math department at UGA, he wrote, needed one more instructor to cover its fall courses. That email had come out of left field. I hadn’t even applied to UGA. But for whatever reason, I was being invited to apply for a job. If I was interested, I should send my CV immediately to his colleague Carl Pomerance, a UGA number theorist well known for his work on prime numbers.

  I responded immediately to Granville’s request, but I was not optimistic that my application would result in an offer. My two hundred job applications had not generated a single interview, so why would the UGA solicitation yield anything different? Perhaps the same email had been sent to thirty other new PhDs. And suppose I was lucky enough to get the offer. What would be the point of moving across the country—without Erika, who had another year of school—for a one-year job? And what would a one-year stopgap do to improve my future prospects? It seemed much more likely that the job would only delay the inevitable realization that there was no place for me in academia.

  Then on the morning of June 23, 1993, my third wedding anniversary, I heard the most incredible news, news that would change my life. Wrapped in only my bath towel, I sat down at my desk in our Santa Monica bungalow to read my email before taking a shower. I was surprised to find dozens of messages. In the early 1990s, before the era of web browsers and the need for spam filters, I received fewer than half a dozen emails a day. Surprisingly, all of the messages had the same subject line.

  The story is told that the writers, critics, and actors who gathered daily in the 1920s at the Algonquin Hotel for lunch once held a contest to see who could come up with the most shocking newspaper headline. Dorothy Parker, famous for her acid wit, won with the two words “Pope Elopes.” To a mathematician, the subject line of all my emails was more shocking even than that: “Wiles proves FLT.”

  A few hours earlier, Andrew Wiles, of Princeton University, had announced a proof of Fermat’s last theorem at a conference at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. His announcement took place in an auditorium a short walk from the same halls that Ramanujan had roamed eighty years earlier.

  Fermat’s last theorem was surely the most famous open problem in all of mathematics. It all began around 1637, when the French jurist and amateur mathematician Pierre Fermat wrote in the margin of his copy of Diophantus of Alexandria’s Arithmetica that he had discovered a “truly marvelous proof” of a certain assertion, “which this margin is too small to contain.” The assertion is easily stated. It involves only the counting numbers 1, 2, 3, and so on. What Fermat claimed was that for every integer n > 2, there are no nonzero integers a, b, and c for which . Of course, for n = 2, there are such numbers a, b, c, such as and . Such examples have been known at least since the time of Pythagoras, who lived more than two millennia before Fermat. In fact, there are infinitely many such so-called Pythagorean triples, all given as side lengths of right triangles. What Fermat was claiming was that for every larger value of n, there are no such triples.

  That marginal note came to the attention of the mathematical community after Fermat’s death in 1665, when his son published a new edition of the Arithmetica complete with Fermat’s marginal notes. For the next 350 years, mathematicians failed to reproduce Fermat’s marvelous proof, and the consensus is that he did not have such a proof. Indeed, although Fermat produced a proof for the special case , the fact that he never mentioned his “marvelous proof” again suggests that he realized that his method for the special case could not be generalized.

  Over the centuries, mathematicians chipped away at what came to be known as Fermat’s last theorem—so called because it was the last of his claims still lacking a proof. In the process, a great deal of powerful and elegant mathematics was created, comprising whole new branches of the subject. The fame of the Fermat problem grew and grew, until Fermat’s last theorem could be found even in the Guinness Book of World Records, where it was listed as one of the “most difficult mathematical problems.”

  Wiles had based his work on Berkeley mathematician Ken Ribet’s earlier discovery that Fermat’s claim would follow from a verification of the modularity conjecture, a bizarre claim about modular forms and elliptic curves that was first stated by Yutaka Taniyama at the 1955 Tokyo–Nikko conference, which had occasioned the pivotal event in my father’s life, being discovered by André Weil. You could say that in the 350 years since Fermat, mathematics had gone a very long distance out of its way to attack the Fermat problem, and Ribet had suggested how it might be possible to come back a short distance correctly. Amazingly, Wiles’s proof also made use of Galois representations, the subject I had been studying in my thesis. I couldn’t believe it.

  The proof of Fermat’s last theorem became a major news item. A story appeared on the front page of the New York Times, and Wiles was named one of People magazine’s twenty-five most intriguing people of 1993, alongside the likes of Oprah Winfrey and Bill and Hillary Clinton. I had no idea that these events would soon change my life. After all, I had nothing to do with the proof of Fermat’s conjecture.

  A few days later, I received an offer from UGA. I suppose that the excitement generated by Wiles’s proof of Fermat’s last theorem gave a mathematician who studied modular forms, even a lowly one like me, a certain cachet. Emboldened by the Fermat hoopla, I was poised to take the risk and accept the one-year UGA offer. Without any other options, I had a simple choice: accept the UGA offer or leave academia. When I asked Gordon what I should do, he said without hesitation, “Go to Georgia.” But I could not think of accepting the offer without a serious discussion with Erika, who would have to stay behind at UCLA to finish her second bachelor’s degree. Erika agreed with Gordon, and I accepted the offer that afternoon.

  Now in addition to Ramanujan in my corner, I had Fermat. Everyone was suddenly interested in the three topics that made up the title of Wiles’s famous talk at Cambridge: “Modular forms, elliptic curves, and Galois representations.” Those were all topics that I had studied in my dissertation, and I began to feel that I had expertise that would be of interest to more than just a few of my mathematician colleagues.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_29

  29. My Hardy

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy &
History of Science, Boston University, Boston, MA, USA

  Athens, Georgia (1993–1994)

  I moved to Athens, Georgia, in August 1993. It was only a one-year position, and after that, the future was as uncertain as ever. But I had done it. I had completed my PhD. I was now Dr. Ken Ono, mathematician. To get this far, I had not traveled an easy path, but inspired by Ramanujan and guided by caring mentors such as my brother Santa, Paul Sally, and Basil Gordon, I had achieved an important milestone. I had hoped that my accomplishment would have elicited the recognition and praise from my parents that I had long sought and now felt I merited. After all, earning a doctorate was part of the formula that my parents had laid out for me long ago. It was a major accomplishment, and I had come through. But I received no acknowledgment from them. I didn’t bother to go to my graduation ceremony, assuming that they wouldn’t want to attend. My father had advised a number of PhD students, and I would have thought that he would appreciate what it meant to end one’s apprenticeship and set forth into the world as a certified member of the profession. I suppose he viewed obtaining a doctorate as just one more chore that was expected of you, like brushing your teeth or walking the dog.

  I was now twenty-four years old, and I had quite given up hope of ever earning the praise that I had desperately sought earlier in my life. I had developed defense mechanisms to protect my psyche. Basically, I simply buried my former life. My years growing up in Lutherville were consigned to a black hole. I never told anyone about the period of my life before UChicago. I rarely called home, and I rarely communicated with my brothers. Instead, I had Erika nurturing me at home, and I had Gordon nurturing me in my research. At Woodbury University, where I had taught during my last years of graduate school, I was nurtured by Zelda Gilbert, a psychology professor.