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Theorists Crack Long-standing Math Mystery

Wisconsin State Journal :: LOCAL :: D4

Sunday, March 4, 2007
Ron Seely

The problem had to do with the nature and functions of a set of strangely recurring numbers called "mock theta functions."

Ken Ono, Manasse Professor of Letters and Science, and Kathrin Bringmann, a German mathematician and postdoctoral researcher at UW-Madison, used relatively new mathematical techniques to solve the long-standing problem.

The feat is being hailed by other mathematicians as a major breakthrough and has made possible the use of the newly deciphered mock theta functions to tackle difficult problems in other fields ranging from physics and chemistry to several other branches of math.

In 2000, George Andrews, a leading number theorist at Pennsylvania State University, called mock theta functions one of the most difficult math puzzles of the new millennium. He was shocked at the accomplishment of Ono and Bringmann, which was explained in a series of three papers in the Proceedings of the National Academy of Sciences.

"This is something I really didn't expect anybody to do," Andrews said of the work by Ono and Bringmann. "It is an outstanding piece of work, a breathtakingly wonderful achievement."

Mock theta functions are among a more thoroughly understood class of mathematical expressions known as the "theta" functions that have been in use for hundreds of years. These functions are basically a series of numbers that have proven useful in various mathematical calculations.

While mock theta functions are composed of a series of infinite numbers similar to the other, better-known theta functions, they are much more powerful. They have arisen in surprising places, in calculations in fields that include physics, chemistry and even in cancer research.

Ramanujan, nearing death at the young age of 32, seemed to be on the verge of explaining mock theta functions. In fact, much of that explanation was apparently in a letter he wrote just prior to his death.

Ono and Bringmann used that letter in their own work. But a large part of that final, telling letter was lost and modern-day mathematicians were left with just 17 examples of the functions and their uses. Missing was the crucial information about how to derive them and how to more broadly apply them to solve other problems.

Ono and Bringmann came to their own realizations, perhaps reprising many of Ramanujan's lost observations. They arrived at their conclusions by recognizing important connections between mock theta functions and another more modern family of mathematical formulations known as Harmonic Maas Forms.

It wasn't the first time such connections were recognized; a Dutch mathematician named Sander Zwegers had also recognized the relationship between the two theories.

Ono, while on a flight to New Hampshire, was reading an article on mock theta functions when he realized the way in which mock theta functions and the Harmonic Maas Theory meshed.

"It was an uncanny set of coincidences," Ono said.

Ono's completion of Ramanujan's lost work continues a fascinating relationship between Ono and the late Indian genius. Ono, along with many students he has mentored, has long pondered the proofs and problems left by Ramanujan.

In 2005, he visited the Indian home of the mathematician he so respects. In writing about that visit, he had this to say about the man who had so helped to shape his career.

"For me, there is a poetic resolution to the question of whether one can rationally explain the legend of Ramanujan: this true story is one of magic. Ramanujan was an untrained mathematician, toiling largely in isolation, whose work was born entirely out of imagination. He was a pioneer and a self-taught anticipator of great mathematics, and this is indeed magical. After all, great mathematics is magic, something we can understand but whose inspiration we cannot comprehend. Ramanujan was a gift to the world of mathematics."

Fierce hurricanes

Atmospheric scientists at UW-Madison have provided convincing evidence that there is indeed a connection between a warming Atlantic Ocean and the increasing ferocity of hurricanes such as Katrina.

But to arrive at their conclusions, the scientists had to resolve some perplexing problems with data that are being used to probe the relationship between warming oceans and hurricanes.

The problem had to do with inconsistencies between old satellite data and the more complex data collected by modern satellites. The researchers resolved the problem by simplifying new satellite data and helping to make it more compatible with the older data.

What the new data set showed them was that global warming has led to stronger hurricanes in the Atlantic. But the trend does not seem to hold up in any of the world's other oceans.

"The data says the Atlantic has been trending upwards in hurricane intensity quite a bit," said James Kossin, a research scientist at the Cooperative Institute for Meteorological Satellite Studies. "But the trends appear to be inflated or spurious everywhere else, meaning we still can't make any global statements."

One reason for the differences in the Atlantic, the scientists theorized, is that the ocean is colder than others and small, man-made changes may be having more of an impact there. But more computer modeling and study will be necessary to truly understand what climate change is doing in the Atlantic, Kossin said.