My Search for Ramanujan

by Ken Ono

  I had been pondering those bewildering expressions for an hour when a glorious insight came to me as if handed down from on high. It was one of those inexplicable moments of clarity that come in a flash when one is in a deep meditative state. It was one of those moments when the clear outline of a solution emerges from the fog.

  I excitedly jumped to my feet, banging my forehead so hard on the sloping ceiling that I sent the squirrels in the attic scurrying. I was stunned by the blow, but my mind was clear: I understood what Ramanujan meant by his expressions, and I knew how to compute them. Ramanujan had found a way to relate the partition numbers to special functions I already knew well. It was time to do some calculations. I went to my computer and wrote a little program to evaluate one of the formulas. The computer spit out the terms one by one, and the results showed that the partition numbers were in perfect harmony with the values of the special function. Ramanujan was right, and I had figured out his secret.

  I ran out of my office wanting to hoot and holler in sheer joy. But I thought better of that and ran to the bathroom to splash some cold water on my face to make sure I wasn’t dreaming. I was so excited that I drenched my shirt, which I then took off. At that moment, in walked Dale Brownawell, a fellow number theorist, to find me shirtless, wet, and sporting a huge bump on my forehead. Understandably shocked by the sight of me, he said, “What in the world happened to you?” I didn’t want to have to explain my “eureka” moment on the sofa, since there was still work to be done before I would have anything to say to anyone, and so I simply said that I had hit my head on the coat hook in the bathroom stall. I was not very convincing and probably gave Dale cause for unnecessary speculation.

  I spent the next weekend pondering Ramanujan’s strange formulas, now with the firm belief that they had to be correct, “because if they were not true, no one would have had the imagination to invent them,” as Hardy said after receiving his first letter from Ramanujan. After trying a variety of tricks from the theory of modular forms, I was finally able to devise a procedure that produced Ramanujan’s formulas and thereby proved his claims. From those puzzling claims, which Ramanujan had offered without proof in his strange unpublished manuscript, I assembled a theory, which I was immediately able to put to good use.

  By combining my new results with work of Deligne and Serre and results of Shimura, one of the stars among the young Japanese mathematicians at the 1955 Tokyo–Nikko conference, I was finally able to make sense of Ramanujan’s enigmatic quotation. I proved that the partition numbers have patterns for virtually every prime. For primes larger than 11, however, those patterns turn out to be monstrosities. For instance, the number of ways of partitioning 4,063,467,631n + 30,064,597, no matter what you choose for n, is always a multiple of 31. I proved that there are infinitely many such patterns for every prime other than 2 and 3.

  Ramanujan was right about there being no simple properties for primes larger than 11, and I am certain that his unpublished formulas were the first steps in his effort to prove the complex properties that he knew must be out there. Moreover, I firmly believe that he expected that he would eventually find the necessary tools for proving them. He never succeeded. That I was able to succeed where he had failed is because I had at my disposal the powerful mathematical machinery assembled fifty years after Ramanujan’s death by Deligne, Serre, and Shimura. When I proved my theorem, I was standing on the shoulders of giants.

  My paper proving this theorem appeared in the Annals of Mathematics, the same journal that had published the proof of Fermat’s last theorem just a few years earlier. My result made world news, and I was rewarded for figuring out Ramanujan’s enigma with fellowships from the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation. I was also one of sixty young scientists and engineers to be honored by President Bill Clinton with a Presidential Early Career Award. My parents attended the ceremony at the White House, and I was proud to have them there.

  That evening, after the White House ceremony, my father bestowed on me the magical letter that Ramanujan’s widow had sent him in 1984, the one that had come to symbolize my path from high-school dropout to professional mathematician. He said,Ken-chan. Ramanujan’s widow sent it to me for the little part I played in honoring his memory. Your work, that is the genuine gift. There is no better way to honor Ramanujan, and my son, you were the one to do it. I was merely meant to be a temporary keeper of the letter, and now I pass it on to you, its rightful owner. I am so proud of you.

  I finally heard the words I had so desperately craved my entire life. That evening, I cried tears of joy under a steaming hot shower, realizing that I had finally achieved my impossible dream. At that moment, many of the voices in my head vanished. I have never heard from them again. Although I still have some voices, they are of the sort that everyone has. The voices that had once nearly driven me mad disappeared that day a few blocks from the White House. As fate would have it, I was thirty-two years old, the age at which Ramanujan, the “gift from Kumbakonam,” passed away, leaving behind writings that have been speaking to me.

  In front of the White House before my Presidential Early Career Award ceremony

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_33

  33. The Idea of Ramanujan

  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

  How have I been lucky enough to get to where I am today?

  The story of Ramanujan has inspired generations of mathematicians. It inspired Weil. It inspired my father as an uncertain mathematician in postwar Japan, and in turn it inspired me as a troubled teenager in the 1980s.

  My search for Ramanujan shall go on. I shall continue to search for Ramanujan the mathematician. Perhaps more importantly, I shall continue to promote the “idea of Ramanujan”—that greatness is often found in unusual and unpromising circumstances, and it must be recognized and nurtured. Indeed, had it not been for the goodwill of Ramanujan’s friends and parents, and people like G.H. Hardy, André Weil, Paul Sally, Basil Gordon, Andrew Granville, Bruce Berndt, George Andrews, and countless others, I wouldn’t have had anything to write about. Ramanujan, my father, and I—we three would have never happened.

  I will continue mentoring bright math students, the future Ramanujans, through my annual summer undergraduate research programs, my training of PhD students, and my mentoring of postdoctoral students. What is interesting about many of the students that I have mentored, like the prodigies Jayce Getz, Daniel Kane, Eric Larson, Hannah Larson, Alison Miller, Maria Monks, Evan O’Dorney, Aaron Pixton, among others too many to mention, is that they have often come from nontraditional or unpromising circumstances. What unites them is that they have been drawn by the beauty of mathematics. I owe it to them and my mentors, and I owe it to Ramanujan, to do my part. Searching for Ramanujan is my calling; it is my life’s purpose.

  How was I lucky enough to get to where I am today? That is the question that opened this book. The preceding pages prove that the answer is multidimensional, and the many steps in the proof amazingly trace my search for Ramanujan. I benefited from the tough love of my parents. They fostered qualities in me that have been essential to my success. I am ambitious and competitive. I am restless, anxious to take on the next challenge.

  But I also emerged as a fragile adolescent with low self-esteem. I needed nurturing from many people along the way to reach my happy place, and for some inexplicable reason, Ramanujan was there with me every step of the way.

  I recently gave a TEDx talk that I called “How to live mathematically, but not by the numbers.” As the product of tough-loving Japanese-American parents, and as a professor at Emory who has present-day tiger children in class, I felt the strong need to offer advice.

  Let me of
fer this advice here.

  How will you choose to live your life?

  That is the question I posed in my TEDx talk. It is the most important math problem that most people are ever asked to solve. It is a question that researchers have studied, and some have even offered mathematical formulas as solutions.

  To keep it simple, I maintain that many people today are living mathematically: they “live by the numbers.” Many high-school students, college students, and their parents will know exactly what I mean. Kids today are frantically living their lives in pursuit of numbers such as high grade-point averages, strong ACT and SAT test scores, a large number of “Advanced Placement” courses and tests, admission to a top-ranked college, and so on. Many parents, and society as a whole, emphasize the importance of these numbers with the idea that strong numbers lead to success, and success leads to happy and fulfilled lives.

  Yet such a frantic high-stakes pursuit has its risks.

  But despite those risks, does the frantic pursuit of credentials work? How do we measure happiness and fulfillment? It is natural to ask those who are facing death. Well-known studies have determined the top deathbed regrets, and their findings are thought-provoking. These are the top five deathbed regrets from Bronnie Ware’s book The Top Five Regrets of the Dying: 1.I wish I had had the courage to live a life that was true to myself, not the life others expected of me.

  2.I wish I hadn’t worked so hard.

  3.I wish I had had the courage to express my feelings.

  4.I wish I had stayed in touch with my friends.

  5.I wish I had let myself be happier.

  These regrets speak to me. It turns out that many of the choices I made in my life have warded off the possibility of such regrets.

  I struggled to live my own life. I found strength in my decision to quit the violin, and I found freedom in my bicycle racing. I strongly believe that both of those choices helped me find the courage to live a life that was true to myself. My choices were made in the pursuit of happiness, and it was my ability to express my feelings with conviction that allowed me to make those choices.

  I was a goofball in college. I was an unmotivated student who was worried about living up to parental expectations as a math major at UChicago. Although I ended up becoming a mathematician, I did not work on math full throttle, living the life that my parents expected of me. As a result, I did not come to resent mathematics, and I was able to remain open to the idea of pursuing mathematics for its own sake when Gordon taught me how to appreciate its intrinsic beauty. Of course, thanks to that choice, I ended up enjoying many rich collegiate experiences that I would otherwise have missed out on. They were choices made in the pursuit of happiness.

  Among the top five deathbed regrets, in case anyone is keeping score, the only one I haven’t checked off is number 4: I wish I had stayed in touch with my friends. Writing this book has been a cathartic experience. I have fondly recalled some joyous times while struggling to find the words and courage to reveal my darkest moments. Many of these darkest moments were new to Erika, who has been closer to me than anyone. It is my hope that with this book, I will be able to reconnect with some of my friends from the black hole of my former life.

  Friends, I apologize for abandoning you. It is my sincere hope that you will understand from this book why I had to leave you behind for a time. I went a very long distance out of my way to block out the period of my life in which you were present, and I am ready now to come back the short distance that separates us.

  Let me end with the advice that I offered in my TEDx talk. I espoused the idea of “living mathematically, but not by the numbers” as an approach to life that highlights the skills that are important for solving math problems, but applied to everyday life. There are four qualities that I think we should try to enhance in ourselves: creativity, flexibility, confidence and determination, and rigor. Let me explain.1.Creative people find ways to handle life’s challenges. I believe that curious and creative people tend to be happier people. Their open minds allow them to appreciate the world’s infinite complexity.

  2.Flexible people are able to approach problems from different points of view. Flexible people are equipped to meet and work with people from different backgrounds and cultures.

  3.Confident and determined people have the courage and strength to manipulate difficult ideas. They have the confidence to attack challenges even when a solution is not obvious.

  4.Rigorous people pay attention to details, and they are more likely to find opportunities in unexpected places.

  Live mathematically, but not by the numbers.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_34

  34. My Spirituality

  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 am often asked whether I believe that Ramanujan’s findings truly came to him as visions from a goddess. I didn’t believe this for most of my life. Perhaps if I were a Hindu, I would have had an easier time subscribing to such a view. Instead, I wish to offer my opinion on a simpler question: was Ramanujan’s mathematics divine in origin?

  I have read most of Ramanujan’s papers multiple times. I have read virtually everything ever written about him, and I have read and reread his letters and notebooks many times. I have tried to develop some sense of what he was like, and what motivated his thinking. The deeper I dig, the more in awe I am of Ramanujan. Because of my growing sense of wonder, I have thought quite a bit about the source of Ramanujan’s ideas.

  Earlier, I was convinced that his claim of visions from a goddess was poppycock. But now I have changed my mind. His claims and formulas, as intimidating as they are at first glance, are awe-inspiring in their beauty and rightness. The more I read Ramanujan’s work, the greater are the depths that are revealed. How was it possible for an untrained youth ignorant of modern mathematics to produce those wonderful formulas? Reading Ramanujan’s writings has become a spiritual experience for me. I sense in his revelations to me a divine source of revelation to him.

  I now firmly believe that Ramanujan’s ideas are divine in origin, though I am much less sure just what I mean when I use the word “divine.” I am not saying that Ramanujan had a direct line to God, whatever that might mean. Instead, I share Carl Sagan’s view that “science invariably elicits a sense of reverence and awe … The cumulative worldwide build-up of knowledge over time converts science into something … that is surely spiritual.” From this point of view, I believe that all science is spiritual. Francis Collins, director of the National Institutes of Health and the former leader of the project that mapped the human genome, is a strong advocate of this view, and he is well known for having said that “The God of the Bible is also the God of the genome. He can be worshiped in the cathedral or the laboratory.”

  So I leave aside the question of the nature of God. Our knowledge of the divine can be obscure at best. As Saint Paul tells us, “now we see through a glass, darkly.” But like Carl Sagan, like Francis Collins, like Alfred Tennyson in his “Higher Pantheism” that was quoted earlier in this book, I see evidence of the divine everywhere. And while some may find it in the cosmos, or the genome, or the seas, the hills, and the plains, I have seen it most vividly in the work of Ramanujan. It doesn’t matter whether Ramanujan believed in the literal existence of the goddess Namagiri or whether he saw in her merely the form that divine inspiration took in his sleeping mind. A version of Namagiri that Western readers may find more accessible can be found in Coleridge’s poem about Kubla Khan and his famous pleasure dome, in which the lyric voice records a vision, not of a goddess but of a damsel with a dulcimer, and tells us that a creative person’s task is to convert such visions into poetry, or art, or mathematics:Could I revive within me

 
Her symphony and song,

  To such a deep delight ’twould win me,

  That with music loud and long,

  I would build that dome in air,

  That sunny dome! those caves of ice!

  And such a person is indeed divinely blessed:For he on honey-dew hath fed,

  And drunk the milk of Paradise.

  My search for Ramanujan has transformed me. Raised with no religion, I found the strong need to come to terms with the sense of awe elicited by Ramanujan’s formulas, and more generally the beautiful infinite complexity of the universe. Thanks to this awakening, I now wish to experience the world, to enjoy its beauty and its people. My curiosity pushes me, and it has left me open to discovering my spirituality.

  On my thirty-fifth birthday, March 20, 2003, the United States, in a coalition with British, Australian, Polish, and Danish military forces, invaded Iraq with the goal of toppling the regime of Saddam Hussein. Despite the precision of modern military technology, there were many innocent victims at the outset, Iraqi men, women, and children accidentally killed in the line of fire. Within days of the invasion, a church in Madison, Wisconsin, home to the University of Wisconsin, where I was a professor, began planting white flags on their lawn to honor those children. It was a church that we had driven by countless times without taking notice. With the flags, however, the church got our attention, and as the conflict dragged on, the lawn of the church was transformed into a sea of white flags, a poignant expression of compassion and humanity in memory of those innocent victims. That garden of flags, a symbol of the congregation’s prayer for a quick end to the hostilities, provided a daily reminder of our good fortune in a world in which so many others were victims of conflict.