by Ken Ono
Erika and I enjoyed Andrew’s company and that of the other friends I had made. It was a summer full of hot and muggy bike rides in the hills around Athens, and evenings at the Globe enjoying pints of Bass ale. The summer of 1994 was a very sweet time for us.
Although I had hoped to stay at UGA for the 1994–1995 academic year, budget constraints made that impossible, and I accepted an offer of a visiting assistant professorship at the University of Illinois. It was very difficult leaving Andrew and Georgia after only a year. Through Andrew’s wonderful mentoring, I had been transformed into a professional mathematician. But I felt like a bird being kicked out of the nest, and I had doubts whether I could remain aloft and continue to produce results in a new environment without him nearby.
Erika and I moved to Urbana, Illinois, in August 1994. I was delighted to work alongside such fine mathematicians as Nigel Boston, who had arranged the position for me; my friend Doug Bowman; and Bruce Berndt, the Ramanujan scholar whom I had spectacularly failed to impress two years earlier in Missoula. Berndt was devoting his career to resolving all of Ramanujan’s claims, and with my newly found passion for Ramanujan’s mathematics, I was excited to work with him and his graduate students. In fact, I had gradually formulated a plan to work long-term with Berndt on Ramanujan’s mathematics.
Erika and I would spend only one year in Urbana. Erika found a job in the oncology ward of the local hospital, while I spent the year doing mathematics, teaching, and making some difficult academic choices. Although my work on Subbarao’s conjecture and my paper with Andrew on Brauer’s problem had earned me a measure of notoriety, I had reached a scientific crossroads, a pivotal moment. Shortly after arriving in Urbana, I realized that I had a choice to make.
As college students, future mathematicians are exposed to a wide variety of mathematical subjects. As graduate students, they specialize in some of those subjects in their coursework, and then in their dissertations, they make an original contribution generally in a single specialized area. Early in a career, it is common to pursue questions that are closely related to the work that was done in the dissertation. But at some point, that particular well runs dry or one finds oneself left with questions that are too difficult. Or else one’s interest simply turns elsewhere. And of course, some research careers just peter out.
At Urbana, I encountered my own personal crossroads. I could work with Berndt and his students, as I had originally planned, chipping away at the unresolved claims that abound in Ramanujan’s notebooks. For if Ramanujan was a mathematical king, he had certainly left behind plenty of work for carters. Indeed, Ramanujan’s writings seemed to offer a nearly endless supply of unproven claims and identities to work on. But there was another path I might follow. Following Kronecker’s advice, I could choose to take a crack at being my own ruler and assume responsibility for figuring out in what field of mathematics I wished to pitch my tent. Berndt and I had not made any explicit plans for research together, and so I felt no obligation one way or the other. It was like choosing between a promotion to a secure position of responsibility in a large firm or quitting and forming my own startup.
Ramanujan’s mathematics, as it is written in his notebooks and letters, presents a major challenge to contemporary mathematicians, who are trained to build frameworks of theory. It is from those new theories that formulas, expressions, and relationships flow. Some mathematicians build theories, while others, the problem solvers, become expert technicians who masterfully apply those theories. There is a similar dichotomy in physics between the theoreticians and the experimentalists. The challenge represented by Ramanujan is that there was no “theory of Ramanujan” to apply to make sense out of his writings. And without such a theory, mathematicians are left to supply proofs to an enormous collection of disparate formulas and claims.
Berndt has chosen to work directly with Ramanujan’s writings, seeking meaning in the complicated formulas and expressions one by one. The challenge has been formidable, and together with his students, he had been successfully resolving those beautiful enigmas.
But Ramanujan had become much more to me over the years than the source of enigmatic formulas, and so my view of him and his mathematics began to seek a wider horizon. While studying Ramanujan’s claims for their own sake would have been an interesting and important challenge, I had come to suspect that many of those claims could be seen as enticing hints at theories that were just begging to be discovered and developed. I believed that Ramanujan’s discoveries had come to him as fragments of a vision of something higher, and so discovering those theories would ultimately give rise to new mathematics. I viewed Ramanujan’s claims as gifts for intrepid mathematicians of the future, elusive gifts that concealed their true reward, an offering to any mathematician able and willing to plumb their deeper meaning.
I first began to suspect that there were such unexplored depths in Ramanujan’s mathematics when I was working with Gordon at UCLA. Ramanujan’s strange formulas had offered hints of the important theories that Deligne had developed. Indeed, Deligne was awarded the Fields Medal for proving overarching conjectures in arithmetic geometry put forth by Serre and Weil. Those theories that arose from Deligne’s work have created a framework that has defined a massive part of mathematics for the last forty years. For example, without Deligne’s work, we still wouldn’t have a proof of Fermat’s last theorem. The conjectures Deligne proved were inspired by some of Ramanujan’s strange claims for his tau function, claims that seemed insignificant when he first recorded them a century ago.
Instead of working with Berndt, I began, with Nigel Boston’s help, to investigate areas of mathematics that although new to me, were known to involve modular forms, the objects I had studied in my dissertation. Instead of following the path of least resistance, I chose to broaden my fields of expertise with the idea that I was investing in my mathematical future. I worked to assemble the knowledge and tools that I believed would be essential for my mathematical purpose: to become one of the heirs to the rich legacy that Ramanujan had left to mathematicians of the future, a legacy available to anyone with the talent, dedication, and insight to seek out the theories implicit in Ramanujan’s bequest to the future: the claims he recorded without proof in his letters and notebooks.
I had a strong sense that Ramanujan’s claim about partitions in his 1919 paper in which he proved the stunning divisibility patterns for the primes 5, 7, and 11 was such a gift. He wrote, “It appears that there are no equally simple properties … involving primes other than these three.” He didn’t elaborate on what he meant. Did Ramanujan know of other properties that were more complex? First of all, if there had been other simple properties, Ramanujan’s genius would have found them. And if he had discovered more complex properties, he surely would have written them down. But this was Ramanujan, and there was another possibility: that he sensed the presence of other properties but couldn’t see precisely what they were. I read and reread that paper many times, and I almost came to believe that those words were meant for me. In reading between the lines, I became convinced that he had been aware of other, less simple, properties. He was speaking to me, and he was beckoning to me to find them.
I now had new voices in my head, and they were the words Ramanujan had left for me and mathematicians like me. They were the clues he had left behind for us. Ramanujan was telling me that what he had seen in his visions was a fragment of something larger. My mathematical search for Ramanujan now became a search for an encompassing theory.
That year in Urbana, I wrote further papers in arithmetic geometry and representation theory, subjects that at first glance have nothing to do with Ramanujan’s mathematics. I was beginning to develop my view of the implications of Ramanujan’s mathematics, which I would later write about in a book I called The Web of Modularity. The functions I had studied in my dissertation, the so-called modular forms, seemed to appear in so many different areas, forming a web of interconnected subjects, that they must have a deep mathematical signi
ficance. Thus it was that during my year in Urbana, I developed a personal relationship to Ramanujan’s mathematics, and the work I did in recognizing the many implications and roles for those functions, constructing the web of modularity, has driven and sustained my career—a mathematical search for Ramanujan.
In November, out of the blue, I received a letter from the Institute for Advanced Study in Princeton. Nothing about the envelope suggested that the letter inside would be important. It was an ordinary white office envelope with Institute for Advanced Study as the return address. It could have been an announcement of the seminar schedule for all I knew. But it was much more than that. I was like the boy Charlie from Roald Dahl’s children’s book Charlie and the Chocolate Factory when he tore the wrapper off his Wonka bar to discover the last of five golden tickets, offering a tour of the magical factory and a lifetime supply of chocolate. Instead of all the chocolate I could ever want, I had been given an even sweeter prize. The letter was from the renowned number theorist and Fields medalist Enrico Bombieri. It consisted of a single paragraph offering me a two-year membership at the Institute.
Andrew Granville had recommended me to Bombieri, praising my goal of searching for Ramanujan’s number theory, and Bombieri had apparently found my goal worthy of support. I had only one duty—to pursue my search for Ramanujan. The letter was a dream come true. I would have the privilege of working at an institution made famous by the likes of Einstein, Dyson, Gödel, Oppenheimer, and Weil, among many other luminaries.
And of course, the Institute had played a special role in my family history. André Weil, who had discovered my father in Japan forty years earlier, was a longtime faculty member. Over the years, he had arranged several visiting positions for my father, including the 1968–1969 academic year, the year I was born. My mother would take long walks on the Institute campus, pushing me in a carriage. Erika and I would have our first child, Aspen, at the Institute in 1996, and history would almost repeat itself: instead of long walks pushing a carriage, Erika and I would glide around the grounds on roller blades with Aspen strapped safely in a pink baby jogger.
Thanks to the strong support of my mentors Sally, Gordon, and Granville, I had somehow reached a level beyond my wildest dreams. I had dropped out of high school ten years earlier, and four years later, my complex analysis professor at UChicago had tried to talk me out of pursuing a doctorate. Now I was pursuing my own research program at the Institute for Advanced Study. And woven through all that history was my faithful guide Ramanujan.
My decision to search for Ramanujan the mathematician would mean going a bit more distance out of my way. I wanted to increase my knowledge, and that would slow down my publication rate, something that anyone trying to land a permanent job must take into account. The Institute’s offer was therefore a godsend. I could concentrate on my mathematics in an environment free of other distractions and responsibilities; I had access to world-class libraries at the Institute and Princeton University, and I would be able to learn from some of the world’s most brilliant and talented mathematicians. I vowed to make the most of this special opportunity.
Erika and I moved to Princeton in August 1995, my third consecutive August move. We lived in a two-bedroom Bauhaus apartment at 69 Einstein Drive. Erika found work as a nurse in Trenton, and I did my number theory research.
My first task was to complete my web of modularity. After that initial investment, I would then turn to my search for Ramanujan’s mathematics.
The 1995–1996 academic year at the Institute was devoted to an examination of the proof of Fermat’s last theorem. Although it turned out that the original proof by Wiles had a flaw, he was able to correct it in a supplementary paper written with his former graduate student Richard Taylor. Due to the importance of their work, the Institute had invited many of the world’s leading number theorists to spend the year in a collaborative environment in which they could push number theory even further. It was an awesome year, one that would contribute to my growth as a mathematician and help my career in many ways. With so many experts to talk to, I was able to complete the bulk of the work for my web of modularity in good order.
We made many friends that year, mostly other young mathematicians. Two of our closest friends were Princeton graduate students Kannan “Sound” Soundararajan and Chris Skinner. Sound would later become a professor at Stanford, and Chris would become a professor at Princeton. We were fans of the TV show X-files, and we made frequent trips to nearby Iselin, New Jersey, for Indian food. Chowpatty was our favorite restaurant, and that is where I developed a taste for south Indian vegetarian dishes like pav bhaji and masala dosa.
My enthusiasm for cycling rubbed off on Chris and Sound. I helped them shop for mountain bikes, and we rode often on the trails in the area, even after one of us flipped over the handlebars on a narrow and rocky descent and landed in the emergency room. When our daughter Aspen was born in June 1996, Chris and Sound became her first “uncles.”
I wrote papers with Chris and Sound during my Princeton years. Chris and I would ultimately write three papers on mathematics related to the Birch–Swinnerton-Dyer conjecture, one of the notorious “Millennium Problems” whose solution would bring a million-dollar prize. This was part of filling in my web, the groundwork I felt was necessary before I could begin my search for Ramanujan’s mathematics in earnest.
When I wasn’t thinking about that problem, Ramanujan’s words were on my mind, as if he were somehow speaking to me. In addition to his comments about the absence of further “simple properties” for the partition numbers, I was deeply interested in a 1916 paper on quadratic forms in which similar puzzling words appeared, this time about the absence of a “simple law.” Sound and I became enamored with the problem implied by Ramanujan’s words. We had to figure out what he meant.
In some ways, Sound is a modern-day Ramanujan. Born to Brahmin parents in Chennai (Madras), he discovered mathematics as a young boy. He was a prodigy, and he came to the West seeking to make a name for himself. As a high-school student in 1989, when I was racing my bike against people like Greg Lemond and Eric Heiden for Pepsi-Miyata, he attended the Research Science Institute at MIT, arguably the most renowned summer science research program for high-school students. There he began to hone his skills in analytic number theory. In 1991, he won a silver medal representing India at the International Mathematical Olympiad in Sweden. He attended college at the University of Michigan, where he wrote an honors thesis that earned him the prestigious Morgan Prize for undergraduate research in mathematics.
Sound and I wanted to figure out what Ramanujan had meant by the absence of a “simple law” for his quadratic form. Quadratic forms are objects that mathematicians have studied for centuries. One of the most famous theorems about them is due to the eighteenth-century mathematician Joseph Lagrange. He proved that every positive integer—no exceptions!—can be expressed as the sum of four perfect squares. It’s like a magic trick: Pick a number, any number. How about 374? Then I can pull four integers out of my hat such that their squares add up to 374. For example, . I could also have written (as you can see, there is nothing necessarily unique about such representations). Lagrange proved that there is nothing special about 374. You can find a similar representation for every positive integer.
Now in solving a mathematical problem, it is crucial to ask the question in a way that leads to a solution. So instead of asking whether every positive integer can be written as a sum of four squares, Lagrange considered the quadratic form and proved that by plugging in all possible integer combinations for a, b, c, d into that quadratic form, you obtain all of the numbers 1, 2, 3, 4, … .
In the 1916 paper that had intrigued Sound and me, Ramanujan was considering the quadratic form , in relation to which he wrote, “the odd numbers that are not of the form , viz., 3, 7, 23, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391 … do not seem to obey a simple law.” Whereas Lagrange’s quadratic form could represent every positive integer, t
here are numbers that cannot be written in the form (the reader can easily verify that the numbers in Ramanujan’s list above cannot be thus represented).
There are plenty of odd numbers that can be obtained by this quadratic form, such as , where we have chosen x = 1, y = 4, and z = 2. There seemed to Ramanujan to be no simple law that would explain his list and show how it continues. What did he mean that there doesn’t appear to be a “simple law”?
In 1990, Bill Duke and Rainer Schulze-Pillot proved a fantastic theorem that implied that all odd numbers from some point on must be represented in this way. That meant that Ramanujan’s list petered out eventually, since the number of odd integers that cannot be represented by the quadratic form in question is finite. You could say, then, that Ramanujan was right: there was no simple law. Indeed, you could say that there was no law at all! It was just a finite list of the relatively few (however many it might be, a finite number is small compared to infinity) integers that happened not to have such a representation. On the other hand, for us, the law that we had to find was this: what is the last number on the list? And was finding it going to be simple, or was it going to be hard?
Sound and I ran a computer program, and we found that Ramanujan’s list could be extended by the odd numbers 679 and 2719. But after that, the well ran dry. For every larger odd number we tried—and we tried all the way up to the super-huge number 1,000,000,000,000,000—we found that it could be expressed by the quadratic form using some choice of x, y, and z. We concluded that we had found the “from some point on” from Duke and Schulze-Pillot’s theorem. That was the “simple law.” It must be true that every odd number larger than 2719 can be expressed by Ramanujan’s quadratic form, and we set out to prove it. Although we firmly believed that we were right, we couldn’t come up with a proof, no matter how hard we tried. When mathematicians are unable to prove something that they believe to be true, they sometimes are able to give a proof on the assumption that some unproven conjecture is true. In our case, we were able to prove that 2719 is the last number in the sequence on the assumption of the truth of the generalized Riemann hypothesis.
