On a Saturday morning in 1984, when Ken Ono was in high school, he opened his family's mailbox in Baltimore and found an envelope as thin as rice paper covered in brilliantly colored stamps. It was addressed to his father, a reserved Japanese mathematician. When Ono handed over the mail, the elder Ono looked up from the yellow legal pad on which he was always scribbling equations and set down his ballpoint pen. Gently, he pried open the seal and unfolded the letter inside.
“Dear Sir,” it began. “I understand … that you have contributed for the sculpture in memory of my late husband…. I am happy over this event.” It was signed “S. Janaki Ammal,” whom the red-inked letterhead identified as the widow of the “(Late) Srinivasa Ramanujan (Mathematical Genius).”
That was the first time the younger Ono had heard of the legendary Ramanujan. A self-taught mathematical prodigy from India, he made cryptic claims around a century ago that “seemed scarcely possible to believe,” his British collaborator Godfrey Harold (“G. H.”) Hardy once wrote. Yet his work has inspired entirely new fields of mathematics and hinted at theories that, in several cases, won their inventors the Fields Medal—mathematics' equivalent of the Nobel Prize.
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As Ono studied to become a mathematician—he is now a professor of number theory at Emory University—he never had reason to pay Ramanujan much mind. As far as he knew, the “Mathematical Genius” had not left behind new insights into Ono's particular specialty in number theory, modular forms—abstract two-dimensional objects revered for their remarkable symmetry.
Ramanujan resurfaced in Ono's life in a big way in 1998, when he was 29. While assembling an anthology of the prodigy's work, mathematician Bruce C. Berndt of the University of Illinois at Urbana-Champaign had come on a largely neglected manuscript. Because the paper dealt with modular forms, Berndt e-mailed Ono a digital scan, thinking he might be able to decipher some strange claims.
Two thirds of the way through the text, Ono stopped. In neat schoolboy script, Ramanujan had penned six bold mathematical statements that seemed utterly bizarre to Ono, even though they touched on his area of expertise.
Ono was dumbfounded. He was certain the statements were false. “I looked at them and I said, ‘No way. This is crap.’”
His first instinct was to try to prove Ramanujan wrong.
Part and Parcel
It is a mystery how Ramanujan thought up much of the mathematics he wrote down. He educated himself using an outdated English tutoring book, and in his mid-20s, while working as a government clerk, he began broadcasting his ideas in letters to mathematicians in England. He received one reply. It came from Hardy, then an up-and-coming professor, who invited Ramanujan to come work with him in Cambridge. After just three years abroad, Ramanujan fell ill during the food shortages of World War I. Emaciated and feverish, he returned to India and died in 1920. He was 32 years old.
In addition to 37 published papers, Ramanujan left behind a small library of letters, partially completed manuscripts and three leather-bound notebooks. Examining them, Hardy and others found that he had rediscovered classic theorems—rules about how numbers behave—that were first recorded by mathematicians at the tops of their fields. And Ramanujan noticed more patterns that no one else saw. A trained mathematician would know to back up each finding with a proof, a sequence of logical arguments that would convince her or his colleagues of its truth. But Ramanujan did not bother. He filled page after page with long lists of theorems and calculations that he worked out in his head or on a chalk slate, rarely pausing to explain how he arrived at them. The three notebooks alone contain more than 3,000 such conclusions about the nature of numbers, which mathematicians have worked hard to prove or disprove since Ramanujan's death.
Berndt began digging through the Ramanujan archive in the 1970s. He was still at it more than two decades later, when he got to the manuscript with the six arresting statements—the ones Ono was determined to prove wrong. They drew a parallel between modular forms and so-called partition numbers, which are a sequence of integers (that is, whole numbers) that represent all the ways you can add up smaller integers to get the one you started with. Partition numbers come from the partition function, which, like any function, describes a relation between two things: it takes a given input x and spits out the corresponding output f(x). The partition function, p(n), counts the combinations of positive integers that sum to a given integer n. For example, p(4) is 5: 1 + 1 + 1 + 1, 1 + 1 + 2, 2 + 2, 1 + 3 and 4.
The partition function and the numbers that it generates might seem straightforward, but for centuries theoreticians have struggled to find patterns among these numbers so that they can predict them, calculate them, or relate them to other functions and theorems. Ramanujan made one of the first real breakthroughs. He and Hardy together devised a method to quickly approximate partition numbers. To check the accuracy of their approximations, they solicited a retired British artilleryman and calculation wizard named Percy Alexander MacMahon (aka Major MacMahon) to work out the first 200 partition numbers by hand. As it turned out, Ramanujan and Hardy's approximations were impressively precise. Even more important, studying MacMahon's list led Ramanujan to one of his most famous observations. MacMahon had arranged the values of p(n), starting with n = 0, in five columns. Ramanujan noticed that every entry in the last column—that is, every fifth partition number starting with p(4)—is divisible by 5, and he proved that this pattern continues forever. It was a stunning revelation. Remember, partitions are about adding numbers. No one imagined that they could have properties involving division.
Ramanujan saw there were even more patterns like this. He proved, for example, that every seventh partition number starting with p(5) is divisible by 7. Similarly, every 11th partition number starting with p(6) is divisible by 11. Mysteriously, the “Ramanujan congruences” stop there. “It appears that there are no equally simple properties for any moduli involving primes other than these,” Ramanujan wrote in a 1919 paper, referring to the prime numbers 5, 7 and 11.
After he died, mathematicians wondered if partitions might have some not so simple properties, and they tried to find them. Yet by the late 1990s they had not dug up more than a handful of additional congruences involving seemingly random primes and powers of primes, including 29, 17
3 and 236. They began to suspect that such patterns were unpredictable—and very, very rare.
After grappling with those six bygone statements in Ramanujan's manuscript, however, Ono was shocked to realize that those suspicions might be very, very wrong. Mathematicians had long believed partition numbers were related only to a small subset of modular forms. To Ono's bewilderment, Ramanujan's six statements linked the two fields in a profound way that no one had anticipated.
Because Ramanujan did not record proofs, Ono could not directly identify errors in the prodigy's thought process. So he decided to plug some numbers into the formulas Ramanujan included in the statements, hoping these examples might reveal some flaws. Yet the formulas worked every time. “Holy cow!” Ono said to himself. He realized that Ramanujan had to be right “because you couldn't possibly be creative enough to make something like that up and have it be true 100 times unless you knew why that formula was true, always.” Then he closed his eyes and thought hard about what Ramanujan understood that no one else had.
Ono knew that modular forms “are littered with congruences”—those same patterns of divisibility that Ramanujan had found a few instances of among the partition numbers. As Ono contemplated the six statements, it occurred to him that if he thought of the partition function as a modular form in disguise, he could show that they were true.
Another thought immediately followed: he realized, laughing out loud, that with a few adjustments, the theories he had developed about modular forms could be powerful tools not just for verifying Ramanujan's genius but also for unearthing deeper secrets about the partition function. “It was something like getting a fancy new telescope,” Ono reminisces. “Once you have it, if you start scanning space—where in this space the stars are the partition numbers—you'll see there are lots and lots of galaxies.”
In this way, Ono was able to prove that partition congruences are not rare at all. Mathematicians had assumed there were few beyond 5, 7 and 11. But in fact, as Ono discovered, there are infinitely many.
Ono's peers hailed the finding as groundbreaking. He was not satisfied, however. Even though he could prove that partition congruences are everywhere, he could not tell you where to find them. If you lined the partition numbers in order, you might want to know how often a congruence would turn up. If you saw one, could you predict when you would see the next? Ono did not have a clue.
When a problem stumps Ono, he refuses to obsessively chew on it in his mind until it is as inelastic as old gum. Instead he files it away inside his head alongside other unsolved problems until it resurfaces. The problem of how to predict partition congruences lay dormant for five years, until postdoctoral fellow Zachary A. Kent arrived at Emory in the spring of 2010. It just popped up one day in conversation, and soon they were talking about it all the time—in their offices, over coffee and on a long walk in the woods north of Atlanta.
Little by little, they built in their minds a labyrinthine superstructure into which the partition numbers could be neatly arranged. They discovered this organization using a theoretical device, which mathematicians call an operator. The particular operator they chose takes any prime number (13, say), selects powers of that prime (13
2, 133, and so on), and divides them into the partition numbers. Incredibly, the numbers it spits out obey a fractal structure—they repeat in near-identical patterns at different scales, like the branches of a snowflake. This outcome shows that the partition numbers are not just a random sequence of numbers with incidental symmetries sprinkled among them willy-nilly. Rather these numbers have a “beautiful inner structure,” Ono says, that makes them predictable and much more fascinating to study.
It took several months for Ono, Kent and their collaborator Amanda Folsom of Yale University to work out all of the kinks in their new theory. But at last, they were able to prove that partition congruences appear in a calculable manner. They exist for every prime and every prime power. Beyond 11, though, the patterns get much more complex, which is probably why Ramanujan never worked them out.
Ono and his collaborators presented their findings at a specially convened symposium at Emory in 2011. Afterward, messages of congratulations flooded Ono's in-box. “It's a dramatic and surprising discovery,” says George E. Andrews, an expert on partitions at Pennsylvania State University. “I don't think even Ramanujan could have dreamt it.”
Beautiful Answers
Investigating Ramanujan's insights has led Ono to other revelations that may one day be useful in fields outside mathematics. By melding Ramanujan's prescience with modern mathematics, Ono and his colleagues have devised powerful computational tools. Beyond advancing understanding in pure mathematics, these tools could lead to better ways of encrypting computer data and studying black holes.
Working with Jan Bruinier of the Technical University of Darmstadt in Germany, Ono constructed a formula for computing large partition numbers quickly and exactly—the holy grail Ramanujan never obtained. Ono calls this calculator “the Oracle.” In addition to crunching partitions, he says, it could be used to study certain kinds of elliptic curves—geometric objects that look something like the surface of a doughnut.
Cryptographers use elliptic curves to create algorithms for encrypting computer data. The success of these schemes hinges on their ability to generate mathematical puzzles that are impossible to solve in a timely manner. A common algorithm called RSA, for example, rests on the difficulty of factoring the product of two very large prime numbers. Newer methods use points on an elliptic curve, whose relations are even harder to discern. If the Oracle or related discoveries can shed light on other, more elusive connections, cryptographers could potentially use this knowledge to devise stronger encryption systems.
Ono's work has also unveiled one of the greatest mysteries of Ramanujan's mathematical legacy. Three months before he died, Ramanujan, bedridden by fever and pain, dashed off one last letter to Hardy in England. “I am extremely sorry for not writing you a single letter up to now,” he wrote. “I discovered very interesting functions recently which I call ‘Mock’ theta functions…. They enter into mathematics as beautifully as the ordinary theta functions.”
Theta functions are essentially modular forms. Ramanujan surmised that it is possible to describe new functions—the mock theta functions—that look nothing like modular forms and yet behave similarly at special inputs called singularities. Nearing these points, the outputs of a function balloon to infinity. Consider, for example, the function f(x) = 1/x, which has a singularity at x = 0. As an input x gets closer and closer to 0, the output f(x) grows infinitely large. Modular forms have an infinite number of such singularities. Ramanujan intuited that for every one of these functions, there is a mock theta function that not only shares the same singularities but also produces outputs at these points that climb toward infinity at almost exactly the same rates.
It was not until 2002 that a Dutch mathematician, Sander Zwegers, formally defined mock theta functions, using ideas shaped decades after Ramanujan's death. Yet still mathematicians could not explain Ramanujan's assertion that these functions mimic modular forms at their singularities.
The machinery behind Ono and Bruinier's Oracle finally solved the puzzle. With Folsom and Robert Rhoades of Stanford University, Ono used it to derive formulas for calculating the outputs of mock theta functions as they approach singularities. And indeed, they found that Ramanujan's conjecture was correct: these outputs were remarkably like the outputs near corresponding singularities in modular forms. In one case, for instance, the mathematicians found that the difference between them gets very close to 4, a surprisingly neat and almost negligible divergence in this universe of infinite numbers.
Physicists have recently begun using mock theta functions to study a property of black holes called entropy—a measure of how close a system is to achieving a perfect state of energy balance. Some scientists believe that formulas akin to Ono's may allow them to probe such phenomena with finer precision.
Ono cautions that we should not make too much of potential applications for his work. Like many theoreticians, he believes that practical purposes are not what make such discoveries great. Great discoveries, he argues, are great the way a painting or sonata is great. “Ken's theorems aren't going to supply us with an infinite amount of green energy or cure cancer or anything like that,” Andrews agrees. Mathematical discoveries often assume important roles in science and technology only after they sit around for a few decades. It is difficult, if not impossible, to predict what those roles will be.
Ono can still recall the giddy pleasure of seeing Ramanujan's congruences written out for the first time, his father's steady hand scripting the unfamiliar symbols on his yellow legal pad. “Why just three?” he remembers asking. “Nobody knows,” his father told him.
As he recounts this story, Ono is sitting in his family dining room in Georgia. On the wall behind him is a framed photograph of the bronze bust of Ramanujan that was commissioned for his widow with $25 donations from Ono's father and hundreds of other mathematicians and scientists around the world. “I never in my wildest dreams imagined I'd one day get to say, ‘You know what, Dad? Those congruences aren't the only ones—not by a long shot.’”