My Search for Ramanujan

by Ken Ono

  I had survived by blocking out the past, and I could look forward to the future with at least some optimism. I had not won one of the precious tenure-track jobs or postdoctoral fellowships that would have set me on the path to a research career, but the one-year position that I had been offered was beginning to look like a terrific opportunity. Like Ramanujan, I had been discovered by an English analytic number theorist, my own personal G.H. Hardy. I like to think of the email that I had received from Andrew Granville telling me of the possibility of a job at UGA as something like the letter Ramanujan received from Hardy in response to his “I beg to introduce myself” letter. Excited by Hardy’s reply, Ramanujan had proclaimed, “I have found a friend in you, who views my labours sympathetically.” Those words echoed in my mind almost exactly eighty years after they were written. And my Hardy—and come to think of it, my Weil as well—would also prove to be my next mentor, nurturer, and friend.

  Andrew Granville, a Cambridge-educated number theorist, was a budding star at UGA, and it was he who had arranged the visiting assistant professorship for me. It seems that I had impressed him at a conference earlier in the year, and he wanted to learn about modular forms because of their role in the ongoing work on Fermat’s last theorem. Without Andrew’s job offer, my career as a mathematician might have been over before it had begun, and now he was offering me something more, the chance to work with him on mathematical research. It was a wonderful opportunity, and I vowed to make the most of it.

  With Andrew Granville in 2013

  Athens, a charming college town sixty-five miles northeast of Atlanta, was well known for its music scene, as the home of the rock band REM and the hip music venue the “40 Watt” club. Although Erika was unable to join me in Athens for the year, she helped me make the move. We had never driven across the United States, so we decided to take advantage of my move from the West Coast to the Southeast to see a bit of what lies between. We packed the car with a few of my belongings, mounted my fancy Team Miyata bicycle on the roof of our little blue hatchback, and set off.

  I wish I could write that the drive was uneventful, but that would be an untruth. The drive was eventful, too eventful. First, we suffered our way across the desert in our little go-kart without air conditioning. Then, somewhere in Arizona, my bike slipped its tether and flew off the roof of the car. I gasped as I watched the reflection of my expensive racing cycle bounce down the highway in the rearview mirror. Thankfully, there were no other cars around. Amazingly, my bike survived its attempted escape.

  Near Tupelo, Mississippi, our car began to lose power. It refused to go faster than forty miles an hour. We limped into town and found a service station. The mechanic quickly diagnosed the problem—a malfunctioning catalytic converter. He didn’t have the parts; nobody in Tupelo had parts for a Hyundai. The Korean automaker had no presence in the southeast at the time. The mechanic said that he knew a lawnmower repairman who might be able to help us out. A lawnmower repairman? Was he kidding? Our little car might be a go-kart, but it was no lawnmower. We decided to take our chances on the highway. Poking along at forty miles per hour, we made it as far as Birmingham, Alabama, which became the Hyundai’s final resting place. We were 220 miles short of our destination.

  I called Andrew from Birmingham and explained that we were having serious car trouble and would likely arrive a day later than we had anticipated. Without a second thought, he offered to make the long drive from Athens to fetch us. I was stunned by his gracious offer, and although we didn’t accept it, instead buying an eight-year-old red Honda sports car to complete the journey, I knew then that I would be in good hands at UGA. That road trip was quite an experience, and from it, we learned at least one important lesson: that Honda we bought may have been old, but it had air-conditioning!

  Although I was in Athens for only one year, it was an enormously significant time as the last stage of my apprenticeship. Andrew guided my transition from graduate student to young professor, showing what it meant to be a professional mathematician, a colleague who not only does mathematics and teaches classes, but shares in the responsibility of determining the future of the profession and the subject.

  Although Andrew was only in his early thirties, he was way ahead of me professionally and served as a perfect role model. He was producing huge volumes of first-rate theorems, covering an unusually broad array of topics: binomial coefficients, combinatorics, graph theory, prime numbers, to name but a few. His work on prime numbers had earned him an invitation to speak at the 1994 International Congress of Mathematicians, one of the most coveted honors awarded to research mathematicians. He was brilliantly advising several PhD students, and I became in effect his postdoctoral advisee. Unlike that unfortunate Hyundai, Andrew was firing on all cylinders. He is the role model that I have done my best to emulate throughout my career.

  We spent many hours doing math, gossiping about mathematicians, and enjoying pints of Bass ale at the Globe, the local Bohemian pub popular among the young faculty.

  Like Sally and Gordon, Andrew believed in me, and that knowledge was a comfort instead of a source of anxiety. Andrew treated me like a colleague. To my surprise, he gave me early drafts of his papers and asked me to critique his work and his writing. How had I earned the right to question and critique him? He asked me to be a role model to his graduate students. What had I done to merit that? I had been a graduate student myself just a few months earlier. Was I worthy of being addressed as “professor”? Of course not, said the voices in my head; to them, I was still an impostor.

  Andrew’s unexpected show of respect for my opinions and judgment was eye-opening. It told me that he genuinely believed in my mathematical ability and competence. In a way, that quiet demonstration of respect was the ultimate expression of praise and approval. Although it could never fill the void that only parental praise and approval could satisfy, his confidence in me gave me strength. It was just the boost I needed.

  Concerning the right to ask questions, Andrew taught me that it is indeed a right. It isn’t a privilege granted only to the select few. It is a practice, he said, that should be second nature to all scientists. Questioning leads to deeper and more meticulous research and to more clearly presented results. It reveals new avenues of inquiry. It is the engine of progress. The ability to ask a question is not something you acquire with seniority. Instead, it is a skill that you must work on at every stage of your development.

  That realization gave rise to new voices in my head, voices that asked lots of questions. My self-confidence as a mathematician was born out of those new voices. They told me that I was adequate enough to formulate meaningful questions, and they began to take up the cudgels against the old voices that hammered away at my self-esteem.

  Andrew and I wrote one joint paper that year in Athens. It was a semi-important paper in representation theory. We completed a program that aimed to classify the “defect-zero p-blocks for finite simple groups,” a project born out of an old problem raised by the Harvard mathematician Richard Brauer in the 1930s. It is interesting to note that Brauer, like Weil, happened to be one of the distinguished American delegates at the 1955 Tokyo–Nikko conference. Our result won me invitations to speak at several first-rate universities.

  The most important ingredient in our proof was Deligne’s theorem, the work I had studied with Gordon that confirmed a deep conjecture of Ramanujan. I was thrilled to put Ramanujan to good use. His mysterious calculations had inspired the deep mathematics that we needed to crack an unsolved problem in an area of mathematics that didn’t even exist during his lifetime. That is the magic of Ramanujan. His formulas are prophecies that have anticipated important discoveries and have guided generations of mathematicians that followed him.

  This was the second time that I had been rewarded for following Ramanujan’s mathematics. The first had been at the Rademacher conference, where I had presented my results in the context of Ramanujan’s formulas from almost a century earlier. Now with Andrew, I had made use
of a theorem that solved one of Ramanujan’s claims, and we had settled an open conjecture. I was beginning to get the idea that Ramanujan was more than an inspiration in my life. Perhaps I was meant to follow Ramanujan’s mathematics, too. He had been the source of everything good in my young career.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_30

  30. Hitting My Stride

  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

  Urbana and Princeton (1994–1997)

  Hardy had judged Ramanujan worthy and had invited him into the community of professional mathematicians. Eighty years later, Andrew Granville extended such an invitation to me, and like the earlier pair, we collaborated on research. When we had completed our one joint project, Andrew encouraged me to take aim at some well-known unsolved problems. My year at UGA was an important transition, one that I like to think of as somewhat analogous to Ramanujan’s first days in Cambridge. We both had a lot to learn about the world of professional mathematics, and we both had mentors to guide our way.

  Andrew helped me spread my wings in search of independence. He told me that I was ready to fly solo and recommended that I study Gauss’s class numbers and Euler’s partition numbers. Andrew believed that my knowledge and ability could lead me to exciting new results on these two types of numbers. I am pleased to say that I lived up to his faith in me. In a project that would take five years to complete, I obtained important results on Gauss’s class numbers in joint work with Winfried Kohnen, of Heidelberg University.

  In addition to my research, I was teaching several sections of calculus. I had enjoyed teaching when I was in graduate school, and at UGA, I continued to find teaching rewarding. One thing that struck me right away was that unlike my students in Burbank, who always addressed me as “Ken,” here I was “Professor,” or occasionally the more informal “Sir.” I felt that such formality created a barrier between me and my students, but there was nothing I could do about Southern decorousness, and I must admit that being addressed with such an honorific for the first time in my life reinforced the positive voices that were beginning to do battle against the negative ones. I felt that in teaching, I was giving back something of what I had received from the many wonderful teachers, going back to my earliest years, who had given me so much. I told my students how much it meant to me to be their teacher, and even now, twenty years later, I still hear from some of them from time to time. I have thereby learned that my students take many different paths in life when then leave my class. I heard from one student who is now a physician. Another competed in the 1996 Atlanta Olympics representing Greece. One has become an accomplished botanist. Teaching has become an integral part of who I am.

  Now my first semester at UGA was coming to an end, and it was time to think about the next academic year. There had been talk of an extension of my UGA position for a second year, but nothing was certain, and so I was girding myself for another grueling job search. In mid-December, I attended a conference in Asilomar, California, in the hope that my short contributed talk would increase my visibility as I entered the job market. Even if only a dozen or so mathematicians heard my presentation, being noticed by the right person could lead to an offer.

  Erika joined me for the conference. We drove up together from Los Angeles, arriving early. Erika understood how important the conference was for my career—I couldn’t afford to have a repeat of Missoula—and so she suggested that we take a walk on Asilomar Beach, on the picturesque Monterey Peninsula, a few miles north of the fabled Pebble Beach Golf Course. I needed to clear my mind and mentally prepare myself for the meeting. The cool, stiff ocean breeze refreshed us as we strolled on the boardwalks that crisscross the natural dunes. Erika reminded me that I had done good work, and as a result, good things were bound to happen. Part of me knew that I indeed had begun to make my mark, but hearing it from Erika, I began to believe that I really had something to offer and that I would be able to put that across in my talk. As Erika encouraged me, I heard echoes of Gordon: “Be like Mickey. If you can dream it, you can do it.”

  Wearing my trademark Mickey Mouse hat, I was standing in line at the conference registration desk waiting to get my information packet when one of the organizers pulled me aside. He told me that the steering committee had decided to offer a presentation that evening, after dinner, on the proof of Fermat’s last theorem. Everyone wanted to know about the proof, and in particular, they all wanted to know about modular forms and elliptic curves. Someone had recommended me as the speaker. Would I give a talk on Wiles and Fermat?

  Would I? I couldn’t believe my good fortune. Although I had had absolutely nothing to do with the proof, I was being given the opportunity to address my fellow mathematicians because my small corner of mathematics had become world news overnight. I took a hot shower, all the while saying, “Oh my God, Oh my God, Oh my God.” Then I got to work and wrote the talk. I had learned from my debacle at Missoula to gauge my audience. They didn’t want the details of Wiles’s proof. They wanted the big picture. I wrote out the main points by hand on overhead transparencies. Everyone was there. My talk was well received, and the next day, my contributed talk, on the rather abstruse topic of Shimura sums related to quadratic imaginary fields, was packed. I began to believe that I might actually have some sort of career as a mathematician.

  I devoted the spring of 1994 to the other numbers that Andrew had called to my attention: Euler’s partition numbers. Ramanujan was the first mathematician to obtain deep results about them. Although I didn’t know it at the time, my decision to work on these numbers marked the beginning of my search for Ramanujan the mathematician. Although I had benefited from following Ramanujan’s mathematics twice before, my decision to study partitions was a dive headfirst into some of the deepest waters of Ramanujan’s mysteries. Although I had other projects and plans, my commitment to Ramanujan had been set in motion.

  The partition numbers seem to arise from a child’s counting game involving only adding and counting. It is simple to explain what these numbers are about. The equalities 3 = 2 + 1 = 1 + 1 + 1 illustrate that there are three ways of “partitioning” the number 3. Next, we can observe that 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1, which shows the five ways of partitioning the number 4. Repeating this process of adding and counting for an arbitrary number n defines the partition function p(n). Thus our two examples are denoted by p(3) = 3 and p(4) = 5. The partition numbers grow at an astonishing rate. One can calculate p(10) = 42, p(20) = 627, p(30) = 5604, p(100) = 190,569,292, and p(1000) = 24,061,467,864,032,622,473,692,149,727,991.

  Ramanujan proved very surprising divisibility properties for these numbers. One of his mysterious identities involves the sequence of all partition numbers of the form p(5n + 4). The first few numbers in the sequence are p(4) = 5, p(9) = 30, p(14) = 135, p(19) = 490, p(24) = 1575. Each of these numbers is a multiple of 5, and what Ramanujan’s identity showed is that p(5n + 4) is a multiple of 5 for every value of n. He also proved analogous theorems for 7 and 11, namely that for every n, p(7n + 5) is a multiple of 7, and p(11n + 6) is a multiple of 11. These three statements, for the three primes 5, 7, and 11, are now known as Ramanujan’s partition congruences.

  It is natural to ask whether the primes 5, 7, and 11 are somehow special. For the prime 2, for instance, is there a similar progression that yields only even partition numbers? That is, do there exist a whole number A and a whole number B such that p(An + B) is always even for every value of n? Or is there an analogous progression of partition numbers all of which are divisible by some larger prime number, such as 13 or 677 or 7753?

  Ramanujan touched on these questio
ns with enigmatic words in a paper he published in 1919, writing, “It appears that there are no equally simple properties … involving primes other than these three.” For decades, mathematicians were unsure what he meant. Did he know about or suspect properties for other primes, but ones that were very difficult to describe? Or was Ramanujan conjecturing that there are no such properties at all for other primes?

  Andrew suggested that I study this mystery for the prime 2. In the 1960s, the Canadian-Indian mathematician M.V. Subbarao made Ramanujan’s enigmatic claim precise in this case. He conjectured that there is no sequence like those discovered by Ramanujan, that is, no sequence of the form p(An + B), in which all of the partition numbers are even. Although my work fell short of proving Subbarao’s conjecture, I proved in the spring of 1994 a theorem that established most of it. The Austrian mathematician Cristian-Silviu Radu would complete my proof fifteen years later. My theorem attracted some attention, and it earned me further invitations to lecture.

  The academic year was now drawing to a close. Erika had graduated from UCLA with a bachelor’s degree in nursing, and leaving California behind, she joined me in Athens for the summer. It was wonderful to be together again after a year living apart. Back then, the phone company was advertising its long-distance service with the slogan “reach out and touch someone.” But long-distance phone calls, as many as we had, couldn’t replace our need for each other. I can attest that reaching out and touching are better from up close. I have no idea how Ramanujan managed without Janaki with him England. He must have felt very lonely and isolated.