Conjecture Proof Leads to Pólya Prize
Posted by Rob Knies
JUL 17, 2014 01:17 AM
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It was almost a year ago, in this space, that you might have learned the astounding news that a team of two researchers from Yale University and one from Microsoft Research had announced a proof of a riddle that had eluded mathematicians for more than half a century.

The Kadison-Singer conjecture, first proposed by Richard Kadison and Isadore Singer in 1959, pertains to the mathematical foundations of quantum mechanics. At the time, experts suggested that the implications could be significant. That, says Nikhil Srivastava of Microsoft Research India, is starting to come true.

Now, during the 2014 annual meeting of the Society for Industrial and Applied Mathematics (SIAM), being held in Chicago from July 7 to 11, the breakthrough is earning a more immediate reward. The 2014 George Pólya Prize will be presented to Srivastava and colleagues Adam W. Marcus and Daniel A. Spielman by Irene Fonseca, professor of mathematics at Carnegie Mellon University and current SIAM president.

Pólya (1887-1985) was a Hungarian mathematician who served as a professor for four decades, first at ETH Zurich, then at Stanford University. He is credited with fundamental advances in combinatorics, numerical analysis, number theory, and probability theory.

The Pólya Prize is presented by SIAM every two years, alternating between two categories: notable application of combinatorial theory and notable contribution to Pólya's areas of interest, including approximation theory, complex analysis, number theory, orthogonal polynomials, probability theory, and mathematical discovery and learning.

This year's award is in the second category. In the email in which Srivastava learned of the honor, he read that the selection committee wanted to recognize him and his colleagues "for the solution to the Kadison-Singer problem."

The citation continued:

"Not only have Marcus, Spielman, and Srivastava proved an important conjecture, which has consequences in various areas of mathematics, but their elegant methods promise to be applicable to a broad range of other problems, as well."

In a post written by Srivastava on the Windows on Theory blog shortly after the conjecture was proved, he emphasized the discrepancy-theoretic nature of the new result and explained its application for partitioning graphs into expanders.

As you might expect, his explication is detailed and precise. And while the work that led to the proof represents the ultimate reward, the Pólya Prize brings Srivastava plenty of satisfaction.

"It is definitely inspiring and motivating," he says, "to be put on a list with so many great mathematicians."

Srivastava takes a minute to reflect on the year since the proof became public.

"The response has been quite good," he reports. "We were invited to give lots of talks, all over the place, which were quite well received. Other people have already started using the theorem to prove new things."

This is the second time that a Microsoft researcher has won the George Pólya Prize, established in 1969 and first presented in 1971. In 2006, the award, also in the category of notable contribution to Pólya's areas of interest, went to Gregory Lawler, then of Cornell University; Wendelin Werner of Université Paris-Sud; and the late Oded Schramm of Microsoft Research.

Winning awards is rewarding, but for Srivastava, the true reward for his research is in extending its relevance.

"My focus is mainly to better understand the techniques that went into this proof," he says. "I suspect it is an instance of much more general phenomena, rather than a one-off."

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