My Search for Ramanujan

by Ken Ono

  Soon after I passed my qualifying exams, I was awarded a master’s degree. This meant that I was due a small raise as a teaching assistant. But I had a different idea. I applied for a full-time teaching position at Woodbury University, a small school in Burbank. Amazingly, they offered me a job teaching three college algebra courses a semester at a salary of $25,000. That was a huge raise over my teaching assistantship. Moreover, I could teach the early morning classes, the ones nobody else wanted to teach. By taking the job, I would earn a much higher salary, obtain valuable teaching experience, and still be able to work toward my doctorate by attending classes and seminars in the afternoon at UCLA. Some of my professors and classmates questioned my decision, arguing that it would take time away from my studies. As it turned out, it was one of the best decisions I ever made, for at Woodbury, I learned that I love to teach.

  Part IV

  Finding My Way

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_26

  26. My Teacher

  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

  I finally passed my remaining qualifying exams, and on paper, I had earned the right to advance to candidacy in the UCLA doctoral program. This may have been true on paper, but in reality, I had no idea what I was supposed to do next. I was poised to drift through the program or else drift right out of it. I needed direction.

  In the spring of 1991, I took an algebraic number theory course from Professor Basil Gordon. Gordon loved the material, and the students in the class could sense his deep devotion to the subject. While Herstein’s lectures in his number theory class at UChicago were intimidating tirades, Gordon’s lectures were inspirational sermons. It was like being at a poetry reading. For him, a theorem was not just some odd mathematical fact. It was a work of art whose aesthetic qualities could be described, as could its place in the ongoing intellectual dialogue of mathematics and the questions it raised for further research. Gordon would sometimes compare a theorem to a famous work of art or classic poem. It was not unusual for him to juxtapose the majesty of a theorem of Gauss with the breathtaking beauty of a Michelangelo sculpture. I soon understood that Gordon’s relationship with mathematics was unusual. He viewed himself as an artist whose medium happened to be mathematics. It was clear that he thought about mathematics in a way that was very different from my view, which had always involved performance on exams and the memorization of formulas and proofs. I wanted to know more, and I didn’t have to wait long to get my chance.

  It was several weeks into the course, during a lecture about ideal class groups, a subject developed by Gauss a century and a half earlier. Gordon was just finishing the proof a theorem about prime-order torsion elements in these groups using a method introduced decades earlier by MIT mathematician Nesmith C. Ankeny and Penn State professor Sarvadaman Chowla. As I listened, it began to dawn on me that there was a much more conceptual proof that made use of elliptic curves. After Gordon completed the proof, I raised my hand and offered my alternative proof, which made use of geometric ideas of Mordell and Weil. Gordon’s response was to ask my classmates to applaud my proof, and he invited me to his office after class.

  I nervously made my way to his office, worried that he would scold me for my presumptuousness. Had he been mocking me when he asked the class to applaud?

  Gordon’s office seemed strangely out of place at UCLA. It could have been the office of an Oxford don or one from the Hogwarts School in the Harry Potter novels. The walls were lined with beautiful barrister bookcases, their contents beckoning from behind hinged glass doors. An enormous ornate Persian carpet graced the floor, concealing the weathered 1960s-era floor tiles. The desk overflowed with papers and letters, nearly enveloping a set of antique gold pens.

  I was in the presence of a gentleman, someone whom I could imagine sitting by the fireplace with G.H. Hardy in the Reading Room at Trinity College enjoying a cup of tea. Gordon and Hardy? Come to think of it, Hardy had also compared mathematics to art, music, and poetry; perhaps I was onto something. Gordon’s manner evoked images of a different time and place, perhaps a nineteenth-century English manor. Our discussion was brief. Although the proof I had offered in class was not a new result, he had been impressed with my insight. He had been following my career at UCLA, and he told me that he would be honored to be my doctoral advisor. He was thinking about retirement, and he wanted me as his final PhD student. Although I was surprised and puzzled by his offer, I accepted on the spot. That meeting with Gordon marked my birth as a mathematician.

  Basil Gordon was indeed a gentleman and a scholar, a polymath who was a direct descendant of the Gordon family of British distillers, producers of Gordon’s gin. He was the step-grandson of the famous American general George Barnett, who served as the major general commandant of the Marine Corps during World War I. I was pleased to learn that we had both grown up in Baltimore. He had attended Baltimore Polytechnic Institute and received his master’s degree in mathematics from Johns Hopkins University in 1953. He earned his doctorate in mathematics and physics from Caltech in 1956 working under the mathematician Tom Apostol and the iconic physicist Richard Feynman. Gordon was drafted into the U.S. Army, where he worked with rocket scientist Wernher von Braun. He was part of the team that worked out the path of the satellite Explorer I so precisely that it remained in orbit for a full dozen years after its launch in 1958. Gordon joined the UCLA faculty in 1959.

  Gordon and I developed an unusual routine. Instead of weekly sessions in his office at UCLA, we met at his home in Santa Monica. Gordon lived in an elegant house on Palisades Avenue, a quiet street a few blocks from Santa Monica Beach, about a mile from our bungalow on 16th Street. That was convenient for me, and since our meetings would last for hours, working with him at his home on Saturdays meant that we could work without interruption. That home was more museum than residence. There were hundreds, perhaps thousands, of books, a grand piano, and antique crystal. Original works of art adorned the walls.

  With Basil Gordon in 1992 (photo by Keith Kendig)

  We rarely began our meetings by diving straightaway into the mathematics. Instead, Gordon might begin by playing a Chopin nocturne on the piano. Sometimes, he would recite poetry from memory. Gordon had a photographic memory and could effortlessly quote reams of literature. I recall him intoning the first few pages of Melville’s Moby Dick. He’d close his eyes, entering a trancelike state, and then begin:Call me Ishmael. Some years ago—never mind how long precisely—having little or no money in my purse, and nothing particular to interest me on shore, I thought I would sail about a little and see the watery part of the world.

  To leave the safe familiarity of the shore and sail off into unknown territory, that is what it is like to do mathematics. Gordon was constantly reminding me that our mathematical research, as difficult and as confusing as it can be, is an art form, an exploration, an adventure, something to be appreciated, something to be lived. How could we possibly prove a good theorem if we viewed mathematics as a chore? We weren’t hanging sheetrock, we were creating a masterpiece, cultivated over weeks, months, even years of deep thought and imagination. And so it was music and poetry that set the tone before we began scribbling figures and equations on our yellow pads.

  I learned a great deal from Gordon on those Saturday afternoons. We would spend hours huddled in his den, struggling with difficult concepts, trying to break an impasse and find a way to bridge a logical gap in an argument. From time to time, on rare occasions made the sweeter for their rarity, we were rewarded with a breakthrough, an elegant argument, a watertight proof. Those moments of revelation were so awesomely gratifying that we quickly forgot the doubt and despair that can creep into the soul when one has lost one’s way.

  The
n we might go out for lunch at the corner Italian bistro, followed by a long walk to the beach. We must have looked an odd couple—an Asian-American young man in his early twenties with a mullet haircut and neon clothes strolling slowly with a sixty-year-old gentleman, nose painted in zinc oxide, dressed in khaki pants, polo shirt, and low-cut white canvas sneakers. Perhaps not so odd; this was, after all, Southern California. But anyone catching snippets of our conversation would have been baffled by our passionate outpourings about continued fractions, modular forms, and Galois representations. Of course we were passionate. We were talking about great mathematical works of art created by the likes of Gauss, Euler, Galois, Serre, Shimura, Taniyama, Weil, and, of course, Ramanujan.

  From Basil Gordon I learned what it means to “do mathematics.” When I was a child, I understood as a child, and I thought that math was only about manipulating numbers and “solving for x.” In college, I thought mathematics was about memorizing theorems and proofs, mastering techniques for carrying out difficult calculations, and solving textbook problems.

  Gordon taught me that “doing mathematics” begins with a state of mind that allows you to travel to a place deep inside the subconscious to open body, mind, and spirit to the contemplation of a mathematical idea. Doing mathematics is a mental voyage in which clarity of thought and openness to insight make it possible to see the deeper beauty of a mathematical structure, to enter a world where triumph over a problem depends less on conscious effort than on confidence, creativity, determination, and intellectual rigor.

  Although I had never practiced such meditative techniques with mathematics, it sounded just like what I had been doing for years on my bicycle, when, for example, on my solo training rides, I was racing the likes of Eddy Merckx up the imagined slopes of Mont Ventoux. The state of complete concentration that I achieved helped me find the strength to pedal a higher gear, propelling me a bit faster than I thought I could manage. Gordon taught me that a similar practice could be instrumental in discovering and proving difficult theorems that might otherwise be off limits.

  When I do math now, I enter a trancelike state in which I travel to a special place to visit my old friends, objects called tau, p(n), and f(q). If you looked at me while I was working at home, you might think that I had dozed off. I also do much of my best work on long solo runs and rides on my mountain bike. Those activities make me more alert and mentally productive, both while I am exerting myself physically and in the hours afterward.

  I would later find my spiritual self. Like many scientists, I would come to believe that the unimaginable complexity and symmetry exhibited by numbers can be nothing less than evidence of their divine origin. The deep trancelike states and the mental freedom that I achieve in running and cycling are spiritual. They give me an openness of mind and soul that allows mathematical secrets to be revealed. Perhaps it is something like what Ramanujan experienced when he received visions from the goddess Namagiri.

  Gordon encouraged me to study the theory of modular forms, a subject that has its origins in much older works of Euler and Jacobi. He steered me in the direction of the more modern treatment of the subject as developed by Pierre Deligne, the Belgian mathematician who proved a conjecture of Weil that in turn solved a problem of Ramanujan. He also had me study papers by the French mathematician Jean-Pierre Serre, who together with Weil inspired my father and the other young Japanese mathematicians at the 1955 Tokyo–Nikko conference.

  I learned that the work of Deligne and Serre on a subject called modular Galois representations was somehow rooted in the work of Ramanujan. I had not thought about Ramanujan for two years, and so it came as a surprise to learn that his work played a role in the development of cutting-edge mathematics. Serre wrote a lovely article in 1973 in which he explained how the English mathematician Swinnerton-Dyer had recast some old results of Ramanujan, who had obtained them by means of the masterful manipulation of power series, into the geometric framework of infinite Galois theory that Deligne was developing.

  Ramanujan’s formulas, known as congruences for tau, exemplified and somehow anticipated the deep conjectures of Serre that Deligne was proving at the time, work that contributed to the Fields Medal that Deligne was awarded in 1978. (Serre was awarded the Fields Medal in 1954. Serre and Deligne are two of only four mathematicians to have been awarded the Fields Medal, the Wolf Prize, and the Abel Prize.) Deligne’s research proved Ramanujan’s congruences as well as the far-reaching Weil conjectures. Ramanujan’s seemingly old-fashioned formulas seemed deeply and strangely intertwined with the ultramodern theories developed by those great mathematicians.

  My dissertation research was forging another connection to Ramanujan. It would not be long before that connection would be transformed into a personal search for Ramanujan the mathematician. But I still had to go a very long distance out of my way before that would happen.

  During my second year at UCLA, Robert Kanigel published his book The Man Who Knew Infinity: A Life of the Genius Ramanujan. I bought the book as soon as it came out and almost devoured it in a single sitting. It offered vivid descriptions of exotic sites in south India, and it filled in many details about Ramanujan’s life, about which I had actually known quite little. Ramanujan suddenly became for me more than a source of magical formulas. He became a mystical figure whose life seemed to contradict every stereotype I had of mathematicians. Kanigel’s description of Ramanujan’s life made it seem like something more out of the Arabian Nights than the history of twentieth-century mathematics. It was a most improbable tale.

  After finishing the book, I felt that I would like one day to make a trip to India to pay homage to Ramanujan. Later, that desire would grow to the point that I would feel compelled to make a pilgrimage to search for Ramanujan himself, the mathematician and the man. It would become my calling.

  Kanigel’s book included expert commentary from mathematicians George Andrews, of Penn State, and Bruce Berndt, of the University of Illinois. Berndt was devoting his career to working out Ramanujan’s unproven claims, systematically working through his writings, making sense of the Indian genius’s assertions and supplying proofs wherever they were lacking. This task would take him decades to complete, and he did not work alone. He enlisted the help of many mathematicians, mostly doctoral students and newly minted PhDs. Berndt was a mathematical guru, who helped many young mathematicians at early stages of their careers. Although I didn’t know it at the time, those two men, Andrews and Berndt, would soon play important roles in my own life.

  George Andrews in 2008 (photo courtesy of the American Mathematical Society)

  Bruce Berndt with Ramanujan’s slate in the 1980s (photo courtesy of Bruce Berndt)

  For the next two years, I worked on my dissertation and taught classes at Woodbury. I filled my need for physical activity as a slow member of the Santa Monica Track Club, which boasted Olympic gold-medalist Carl Lewis among its lightning fast members. Erika and I enjoyed life as a young married couple in picturesque Santa Monica. It should have been a halcyon time, but in the depths of my psyche lurked still the parental voices that ate away at my self-esteem like the eagle gnawing eternally at Prometheus’s liver. I survived thanks to Erika’s and Gordon’s nurturing. Without either of them, I would have surely dropped out of UCLA, perhaps creating a second black hole of memories.

  I didn’t expect that my thesis would be of interest to many mathematicians, but that didn’t bother me. A thesis is supposed to be the first step in the life of a professional mathematician, not the final magnum opus. Gordon had taught me to love mathematics for its own sake, and that discovery sustained me. My goal was to finish the dissertation, and Gordon assured me that I was well on my way to success. Strangely, I believed him. We were having a wonderful time proving theorems, and proving them for their own sake.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_27

  27. Hitting Bottom

/>   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

  Montana (1992)

  Basil Gordon had awakened me. I had been dangerously adrift at UCLA, and his mentoring coaxed me back into the life I was meant to lead. Like Ramanujan, I had developed an addiction for mathematics. I was in love with mathematical beauty. I had a passion for doing mathematics. Yet I had no idea how things might turn out for me professionally. Desire does not always lead to fulfillment. Would my theorems be good enough for anyone to care about? Might I have a thesis in me but not much else?

  Although I was pleased with Gordon’s assessment of my progress, I didn’t entertain any thoughts of a top-flight research career. The voices in my head told me that I had no chance. So I set my sights lower. My wish was to land a position at the University of Montana in Missoula, Erika’s hometown. To secure a teaching position at UM would have been a dream come true. We would have bought a house near campus and set down roots in that lovely college town, with Erika’s family nearby.

  In the spring of 1992, I learned about a conference in Missoula, one that I thought might offer the opportunity of a lifetime. The Pacific Northwest Sectional Meeting of the Mathematical Association of America (MAA) would be held at the University of Montana in June. I was ecstatic to learn that the distinguished plenary lecturer at the meeting would be Bruce Berndt, the celebrated Ramanujan expert about whom I had just read so much in The Man Who Knew Infinity. He was also masterfully training many young mathematicians. If I could meet him, then maybe he could help me, just as André Weil had helped my father in 1955 at the Tokyo–Nikko conference.