‘Outsiders’ Crack A 50-Year-Old Math Problem by Erica Klarreich.
From the post:
In 2008, Daniel Spielman told his Yale University colleague Gil Kalai about a computer science problem he was working on, concerning how to “sparsify” a network so that it has fewer connections between nodes but still preserves the essential features of the original network.
Network sparsification has applications in data compression and efficient computation, but Spielman’s particular problem suggested something different to Kalai. It seemed connected to the famous Kadison-Singer problem, a question about the foundations of quantum physics that had remained unsolved for almost 50 years.
Over the decades, the Kadison-Singer problem had wormed its way into a dozen distant areas of mathematics and engineering, but no one seemed to be able to crack it. The question “defied the best efforts of some of the most talented mathematicians of the last 50 years,” wrote Peter Casazza and Janet Tremain of the University of Missouri in Columbia, in a 2014 survey article.
As a computer scientist, Spielman knew little of quantum mechanics or the Kadison-Singer problem’s allied mathematical field, called C*-algebras. But when Kalai, whose main institution is the Hebrew University of Jerusalem, described one of the problem’s many equivalent formulations, Spielman realized that he himself might be in the perfect position to solve it. “It seemed so natural, so central to the kinds of things I think about,” he said. “I thought, ‘I’ve got to be able to prove that.’” He guessed that the problem might take him a few weeks.
Instead, it took him five years. In 2013, working with his postdoc Adam Marcus, now at Princeton University, and his graduate student Nikhil Srivastava, now at the University of California, Berkeley, Spielman finally succeeded. 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.
Why all the excitement?
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.
Just so you know, the same people who are saying it will be years before practical results emerge from this breakthrough are the same ones who assumed the answer to this problem was negative. 😉
I first saw this in a post by Lars Marius Garshol