# It Was a Mystery More Than 80 Years Old – an Equation That Had Perplexed the World’s Best Mathematicians.; In His Short Life, Srinivasa Ramanujan Posed This and Other Mock Theta Functions.; Now, a Couple of UW Number Theorists; Have Revealed the Secrets;

By MARK JOHNSON

Two number theorists from the University of Wisconsin-Madison were flying above the clouds on their way to a conference in the summer of 2005, and halfway between Detroit and Manchester, N.H., a more than 80-year-old mathematical mystery unraveled.

Strange, complex formulas known as mock theta functions began to yield their secrets, opening a world of possibilities: new ways to make the Internet more secure, calculate the energy in black holes and understand how particles interact with each other.

At the front of the aircraft, Ken Ono was reading an article by George Andrews in which the Pennsylvania State University professor listed six great problems for mathematicians to solve in the new millennium. The last two problems referred to the baffling functions first described in 1920 by Indian mathematician Srinivasa Ramanujan on his deathbed.

As with much of the work in his short life – Ramanujan died at 32 – he set down the mock theta functions without showing how he knew they were true, without even showing the trail of thought that led to his discoveries. For decades, mathematicians have regarded the functions as tantalizing clues – but to what?

Now, as Ono read the article, he realized that he knew how to solve one of the problems. Some of the formulas needed were similar to those he and colleague Kathrin Bringmann had been using for a theory they were developing.

He rushed down the aisle to Bringmann’s seat.

“Kathrin,” Ono said, “you have to read this. They look exactly like our functions.”

So far, that high-altitude flash of insight has led to three papers by Ono, 38, and Bringmann, 29. In the first, they solved Andrews’ fifth problem for the millennium. In the remaining two, they solved Problem 6, “more or less,” Ono said.

Last week, Andrews called Ono and Bringmann’s work “just absolutely outstanding. Brilliant. I really didn’t expect their achievements to occur in my lifetime.”

Andrews knew how intractable these problems were. He had first confronted Problem 5 in his PhD thesis in the 1960s.

In 1987, renowned physicist Freeman Dyson had called the mock theta functions “a challenge for the future.” Now 83 and retired from the Institute for Advanced Study at Princeton University, where Einstein once worked, Dyson hailed the discoveries by Ono and Bringmann, though he confessed their complexity is daunting.

“I read their papers, but I only understood about 20 percent,” Dyson said, adding that what he did understand “is quite beautiful.”

Ono and Bringmann’s third paper was published this month in the Proceedings of the National Academy of Sciences.

Modern applications

Few mathematicians have been pondering the functions, said Bruce Berndt, a Ramanujan expert and professor of mathematics at the University of Illinois. “They’re so mysterious. It was just an impossible task. Where to begin?”

Number theory, in general, has played a significant role in the development of the Internet; computers apply it when they communicate with one another through e-mail and when they gather information in fractions of a second without becoming overburdened.

The mysterious functions Ono and Bringmann tackled are similar to classical theta functions, useful in solving equations that measure the distribution of heat over time – important in power plants – and in a branch of atomic physics known as statistical mechanics. They have also been applied to math problems that date back to the Greeks. Using theta functions, mathematicians can determine, for example, that there are four ways of expressing the number 1,105 as the sum of two squares without having to find all of the ways by trial and error.

In a 1920 letter to his British mentor and collaborator G.H. Hardy, Ramanujan said he had discovered a new class of theta functions and set down 17 examples, the mock theta functions. More than 50 years later, a half-dozen or so additional examples surfaced when Andrews, then a visiting professor at UW, stumbled upon 140 pages of previously undiscovered Ramanujan work – the so-called “Lost Notebook” – in the library of Trinity College at the University of Cambridge. Until now, so little was known about these functions that mathematicians have yet to explore all of their uses.

“It would not surprise me to see significant applications in physics,” Andrews said.

The functions have been used in number theory, probability theory and certain branches of theoretical chemistry. In combination with other ideas, Ono said, they may help scientists compute the overall energy in a black hole or test and improve Internet security – advances that go beyond current applications of number theory.

For Ono, a math professor at UW, the drive to solve the riddle of mock theta functions harkened back to a moment in adolescence when he wanted to become a professional cyclist, when he found the notion of being a mathematician “quite queer.”

He was 15 and living in Baltimore. His father, Takashi Ono, a Japanese immigrant, was a number theorist at Johns Hopkins University.

One day, an envelope arrived for the elder Ono. There were Indian stamps on the front. Inside was a letter typed on rice paper.

Ono watched his father read the single page and could tell he was moved. The letter was from Ramanujan’s widow, thanking him for helping to pay for a bust honoring her late husband.

At dinner that night, Ono’s father told the story of Ramanujan. As a mathematician’s son, Ono knew how such stories were supposed to go. Great mathematicians went to college, then graduate school. They read books, books and more books.

Life of a genius

Almost all of Ramanujan’s initial training came from a single book. Moreover, mathematics became such an obsession that he ignored other subjects. He flunked out of one college, then a second.

In his biography of Ramanujan, “The Man Who Knew Infinity,” Robert Kanigel described the young Indian sitting on his porch, scribbling away on a large slate on his lap and using his elbow to wipe away the rare mistake. Once satisfied with a result, he recorded it in his notebook. Eventually, the frustrated clerk took a bold step and wrote to three English mathematicians, sharing some of his results. The first two dismissed him.

The third, Hardy, reviewed his work and saw traces of genius. Hardy persuaded Ramanujan to join him in England, where he tutored the Indian and collaborated with him. In the space of five years, Ramanujan published more than 30 papers and left a trail of ideas for future mathematicians to investigate.

While in England, he contracted tuberculosis. Ramanujan returned to India in 1919 and died the next year, though not before leaving a final problem – the mock theta functions.

Although the story made a lasting impression on the son, Ken Ono entered the University of Chicago planning to be a premed major. His medical career ended with the first exam in organic chemistry. In 1989, he received a bachelor’s in pure mathematics and four years later earned a PhD from University of California, Los Angeles. In July 1999, he was hired to teach at UW, and in 2004, he met Bringmann.

Born in Mnster, Germany, Bringmann had always enjoyed problem- solving. Although she excelled in math, the subject offered few job prospects in her homeland.

She first encountered the work of Ramanujan about five years ago while writing a thesis. Like Ono, she was struck by the Indian mathematician’s unconventional story and his originality in a subject that schools sometimes reduced to mere recipes. She met Ono because her PhD adviser had collaborated with him, forwarded an abstract of her thesis and inquired about postdoctoral jobs.

Developing a theory

In Bringmann, Ono found someone with a gift for the hard, “dirty calculations” he neither enjoyed nor excelled at. Ono brought an ability to see the big picture, to envision how a complex theory could be laid out clearly in mathematical papers.

For a year leading up to the New Hampshire flight, they had been working on a theory of Maass forms, a sophisticated relative of trigonometry functions. They had not expected to plunge into the mystery of mock theta functions.

The functions, in Ono’s words, were like a shadow. Since Ramanujan discovered them more than 80 years ago, there had been one major advance. In 2002, Dutch mathematician Sander Zwegers had made a connection between Maass forms and mock theta functions. He also had found a missing part, something called a period integral, which when added to Ramanujan’s functions made complete sense.

“It comes out of nowhere,” Ono said of the period integral. “The magic is that Zwegers saw how to use this piece.”

But this advance still left mathematicians well short of a theory for the functions. After the conference in New Hampshire, Ono and Bringmann went straight to work following up on their insight from the plane, giving the shadow meaning.

The work was demanding and never confined itself to their offices in Van Vleck Hall. Bringmann pondered the problems on her twice-a- day runs. Ono called her by cell phone in the midst of 30-mile bike rides.

A single paper required calculations that ran on for 20 pages or more. A single mistake could kill their theory.

Within a month, they had written the first of their papers explaining mock theta functions.

“What can you deduce about a person from their shadow?” Ono said, returning to his analogy. “Our theory is like the actual biology. Here’s the person.”

Still, Ramanujan took one aspect of the mystery to his grave. How did this untrained mathematician, facing death, make a discovery that eluded the most brilliant minds for so long?

“Think of it,” Ono said. “2007 and we’re throwing all of our knowledge that we’ve learned, all of the knowledge accumulated over decades, to figure out what this man was talking about in 1920.”

Last spring, Takashi Ono presented his son with a gift.

The framed letter from Ramanujan’s widow.

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