Source: Quanta, Nov 2015

Word spread quickly through the mathematics community that one of the paramount problems in C*-algebras and a host of other fields had been solved by three outsiders — computer scientists who had barely a nodding acquaintance with the disciplines at the heart of the problem.

Mathematicians in these disciplines greeted the news with a combination of delight and hand-wringing. The solution, which Casazza and Tremain called “a major achievement of our time,” defied expectations about how the problem would be solved and seemed bafflingly foreign. Over the past two years, the experts in the Kadison-Singer problem have had to work hard to assimilate the ideas of the proof. Spielman, Marcus and Srivastava “brought a bunch of tools into this problem that none of us had ever heard of,” Casazza said. “A lot of us loved this problem and were dying to see it solved, and we had a lot of trouble understanding how they solved it.”

“The people who have the deep intuition about why these methods work are not the people who have been working on these problems for a long time,” said Terence Tao, of the University of California, Los Angeles, who has been following these developments. Mathematicians have held several workshops to unite these disparate camps, but the proof may take several more years to digest, Tao said. “We don’t have the manual for this magic tool yet.”

no one realized just how ubiquitous the Kadison-Singer problem had become until Casazza found that it was equivalent to the most important problem in his own area of signal processing. The problem concerned whether the processing of a signal can be broken down into smaller, simpler parts. Casazza dived into the Kadison-Singer problem, and in 2005, he, Tremain and two co-authors wrote a paper demonstrating that it was equivalent to the biggest unsolved problems in a dozen areas of math and engineering. A solution to any one of these problems, the authors showed, would solve them all.

Spielman, Marcus and Srivastava suspected that the answer was yes, and their intuition did not just stem from their previous work on network sparsification. They also** ran millions of simulations without finding any counterexamples**. “A lot of our stuff was led by experimentation,” Marcus said. “Twenty years ago, the three of us sitting in the same room would not have solved this problem.”

The proof, which has since been thoroughly vetted, is highly original, Naor said. “What I love about it is just this feeling of freshness,” he said. “That’s why we want to solve open problems — for the rare events when somebody comes up with a solution that’s so different from what was before that it just completely changes our perspective.”

using ideas from the proof of the Kadison-Singer problem, Nima Anari, of the University of California, Berkeley, and Shayan Oveis Gharan, of the University of Washington in Seattle, have shown that this algorithm performs exponentially better than people had realized. The new result is “major, major progress,” Naor said.

The proof of the Kadison-Singer problem implies that all the constructions in its dozen incarnations can, in principle, be carried out — quantum knowledge can be extended to full quantum systems, networks can be decomposed into electrically similar ones, matrices can be broken into simpler chunks. The proof won’t change what quantum physicists do, but it could have applications in signal processing, since it implies that collections of vectors used to digitize signals can be broken down into smaller frames that can be processed faster. The theorem “has potential to affect some important engineering problems,” Casazza said.

When mathematicians import ideas across fields, “that’s when I think these really interesting jumps in knowledge happen.”