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How Computer Scientists Cracked a 50-Year-Old Math Problem (quantamagazine.org)
An anonymous reader writes: 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, Daniel Spielman knew little of quantum mechanics or the Kadison-Singer problem's allied mathematical field, called C*-algebras. But when Gil 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.
As a computer scientist, Daniel Spielman knew little of quantum mechanics or the Kadison-Singer problem's allied mathematical field, called C*-algebras. But when Gil 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.
I don't even understand what's been solved!
They describe how. Five different, round-about ways of deriving positive intersecting matrices are described. They develop a method of defining boundary equations for the matrices, so as to prove an interesting algorithm that hadn't been able to be solved via an algorithm, just conjectures. They define this interesting boundary equation to box-in the conjectures, so to speak, and by defining the algorithmic domain, offer a proof that it works.
Profit!
I'm not sure how just yet.... but Profit!
If someone else can explain it succinctly, give it a shot.
---- Teach Peace. It's Cheaper Than War.
Kalai and Spielman are both very talented and have done a lot of work in many different branches of mathematics. Moreover, in this particular context they proved an equivalent version of the conjecture that was much closer to their own sort of work. The problem in question has many different equivalent formulations such as that described here http://arxiv.org/abs/math/0209078 is essentially a statement about vector spaces that anyone with some basic linear algebra background could understand. This is a very common tactic in mathematics if one has a tough problem: try to find equivalent problems that are in other subfields of math where their might be techniques to handle them.
This is old news, the proof was announced several years ago.
They use some cool theory initially developed by two Swedish mathematician, (one who sadly passed away a few years back),
dealing with polynomials and families of polynomials with only real roots.
The title "Mixed characteristic polynomials" has to do with matrices, and the characteristic polynomial of these.
A central concept is interlacing families of polynomials. Two polynomials with real roots are interlacing if the roots are interlacing, meaning when plotted on the real line, every other root belong to say the first polynomial.
It is actually pretty cool, since the original conjecture sounds really far from polynomials, matrices, and realrootednes.
Dude, we get the first story reminiscent of the Old Slashdot in freaking forever... and you're asking for practical applications?
#DeleteChrome
No one has a sense of humor anymore, especially when ti comes to algorithms proven by boundary equations.
---- Teach Peace. It's Cheaper Than War.
What are you doing here? Where are your parents!?
Inheritance is the sincerest form of nepotism.
Indeed. That is why I am interested in different schools of thought in mathematics. For example, the ancient Greeks were builders rather than mathematicians, and therefore solved different problems or similar problems in another way. I would not know how to prove Pythagoras' theorem without the Greek school of thought. On the other hand, the Arabic school of thought brought us abstract thinking. It took aerodynamics to add boundary layer theory to computational mathematics.
The most interesting thing can occur when those schools of thought are mixed. Hodographic transformation as used in aerodynamics is very similar to a Burrows-Wheeler transform in computer science, but the application is totally different. Who knows what other differential equations solving techniques could yield better data compression, for example?
Nae king! Nae laird! Nae yurrupiean pressedent! We willna be fooled again!
The difference in the way of thinking is simple.
Mathematician: "This is too complex for my brain. I can grasp the outer layer of the problem, but the underlying thing is beyond any human's capacity."
CompSci guy: "Oh, I can write a program that handles the outer layer of this problem; I have no clue what that underlying thing is but I bet it can be brute-forced."
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