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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.
By injecting its SQL
Table-ized A.I.
I don't even understand what's been solved!
"How Computer Scientists Cracked a 50-Year Old Math Problem". Great! How?
Those who can't, go into more traditional scientific disciplines.
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.
It might be the DMT kicking in.
You are welcome on my lawn.
What practical manifestations will this have?
Will it enable faster/better/bigger/smaller/higher/lower/longer/shorter/hotter/colder? What?
When Fascism comes to America, it will call itself Anti-Fascism, and tell you to give up your guns.
... Just file a bugzilla defect. We will fix it in the next release.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
To paraphrase a popular programming axiom:
"There's more than one way to think of it"
Math physics, etc. is our own physically limited observation and description of how math and the universe functions, doesn’t mean it's the correct way or able to get achieve all the answers, Kind of like Einstein, Tesla, and many other discoverers, its not always thinking the ways everyone else has been taught to think but coming at it from a wholly different perspective... and not always by such seemingly brilliant individuals.
"Enjoy what you're doing! If it becomes drudgery, you're doing it wrong!" - Jim Butterfield
[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.
I've underestimated ever-so-slightly like that. Now I don't feel so bad about being dumb!
If the universe is someone's simulation -- does that mean the stars are just stuck pixels?
Can we please try to name drop some more schools into the teaser?
I am SURE somewhere we missed somebody's school affiliation that, while having jack shit to do with anything, merits a mention. Surely a janitor who attended night classes at Yale Lock Academy or somebody's third cousin Louis who once had cheese from a shop near Rutgers deserves the same accolades.
Seriously, why the hell does it matter where all these people went to school and why does this need to be in the teaser? You know what was MISSING from the teaser? A reason why anyone should care.
Sig for hire.
It comes down to the fact that some problems need outsiders, whose thinking have yet to be confined inside that proverbial box, in order to attain the correct solution
Muchas Gracias, Señor Edward Snowden !
Those Mathematicians are just butthurt that some problems can "simply" be solved by enumeration. They are more then welcome to come up with an alternative proof.
Which is more important? The answer or the one of the solutions? Why "elegance of the proof" even matter?
It doesn't say *anything* about the conjecture in the first place! You read on and on and on. You hope that at one point the vapid author comes close to describing what the problem they solved was all about and every time. Every. Single. Time. The author manages to deflect away from it. Bah!
Religion is what happens when nature strikes and groupthink goes wrong.
" computer scientists who had barely a nodding acquaintance with the disciplines at the heart of the problem."
If it took 5 years and actually worked it's because they were becoming very, very well acquainted with disciplines at the heart of the problem.
"computer scientists who had barely a nodding acquaintance with the disciplines at the heart of the problem"
I don't think who wrote this has any idea how much math is in the university curriculum for computer science in different parts of the world. While far from "proper" mathematicians, there are lots of places where CS grads have much more than a nodding acquaintance.
I am putting myself to the fullest possible use, which is all I can think that any conscious entity can ever hope to do.
I think people who write summaries that do not contain the most important piece of information should be killed.
A sentence describing the problem and a sentence describing the breakthrough would be great. Not a life story about some people I care nothing about.
Math is a huge subject. I've heard most mathematicians can't read most mathematics papers. I believe we will increasingly see math problems solved by people who see new isomorphisms; people who realize that two problems in two fields are actually the same problem. I'm just waiting for the day that an important math problem is discovered to have been solved long ago in a different formulation.
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Output of sudo apt-get moo maybe was the inspiration? See:
http://paste.ubuntu.com/135042...
Unless it's fortune | cowsay or something. See:
http://paste.ubuntu.com/135042...
Given the relevance, often oblique, I'm inclined to believe this is manually done. I've not seen it mentioned on Slashdot's hidden thread.
"So long and thanks for all the fish."
I mean, are these guys working on this between Star Wars Battlefront matches, or when they get burned out collecting items in Fallout 4? I mean, 15 years is a long time, so there had to be some sort of routine to this, right? What about when Everquest was popular, did that push this out for an extra five years or what?