DeepMind Cracks 'Knot' Conjecture That Bedeviled Mathematicians For Decades (livescience.com) 21
The artificial intelligence (AI) program DeepMind has gotten closer to proving a math conjecture that's bedeviled mathematicians for decades and revealed another new conjecture that may unravel how mathematicians understand knots. Live Science reports: The two pure math conjectures are the first-ever important advances in pure mathematics (or math not directly linked to any non-math application) generated by artificial intelligence, the researchers reported Dec. 1 in the journal Nature. [...] The first challenge was setting DeepMind onto a useful path. [...] They focused on two fields: knot theory, which is the mathematical study of knots; and representation theory, which is a field that focuses on abstract algebraic structures, such as rings and lattices, and relates those abstract structures to linear algebraic equations, or the familiar equations with Xs, Ys, pluses and minuses that might be found in a high-school math class.
In understanding knots, mathematicians rely on something called invariants, which are algebraic, geometric or numerical quantities that are the same. In this case, they looked at invariants that were the same in equivalent knots; equivalence can be defined in several ways, but knots can be considered equivalent if you can distort one into another without breaking the knot. Geometric invariants are essentially measurements of a knot's overall shape, whereas algebraic invariants describe how the knots twist in and around each other. "Up until now, there was no proven connection between those two things," [said Alex Davies, a machine-learning specialist at DeepMind and one of the authors of the new paper], referring to geometric and algebraic invariants. But mathematicians thought there might be some kind of relationship between the two, so the researchers decided to use DeepMind to find it. With the help of the AI program, they were able to identify a new geometric measurement, which they dubbed the "natural slope" of a knot. This measurement was mathematically related to a known algebraic invariant called the signature, which describes certain surfaces on knots.
In the second case, DeepMind took a conjecture generated by mathematicians in the late 1970s and helped reveal why that conjecture works. For 40 years, mathematicians have conjectured that it's possible to look at a specific kind of very complex, multidimensional graph and figure out a particular kind of equation to represent it. But they haven't quite worked out how to do it. Now, DeepMind has come closer by linking specific features of the graphs to predictions about these equations, which are called Kazhdan-Lusztig (KL) polynomials, named after the mathematicians who first proposed them. "What we were able to do is train some machine-learning models that were able to predict what the polynomial was, very accurately, from the graph," Davies said. The team also analyzed what features of the graph DeepMind was using to make those predictions, which got them closer to a general rule about how the two map to each other. This means DeepMind has made significant progress on solving this conjecture, known as the combinatorial invariance conjecture.
In understanding knots, mathematicians rely on something called invariants, which are algebraic, geometric or numerical quantities that are the same. In this case, they looked at invariants that were the same in equivalent knots; equivalence can be defined in several ways, but knots can be considered equivalent if you can distort one into another without breaking the knot. Geometric invariants are essentially measurements of a knot's overall shape, whereas algebraic invariants describe how the knots twist in and around each other. "Up until now, there was no proven connection between those two things," [said Alex Davies, a machine-learning specialist at DeepMind and one of the authors of the new paper], referring to geometric and algebraic invariants. But mathematicians thought there might be some kind of relationship between the two, so the researchers decided to use DeepMind to find it. With the help of the AI program, they were able to identify a new geometric measurement, which they dubbed the "natural slope" of a knot. This measurement was mathematically related to a known algebraic invariant called the signature, which describes certain surfaces on knots.
In the second case, DeepMind took a conjecture generated by mathematicians in the late 1970s and helped reveal why that conjecture works. For 40 years, mathematicians have conjectured that it's possible to look at a specific kind of very complex, multidimensional graph and figure out a particular kind of equation to represent it. But they haven't quite worked out how to do it. Now, DeepMind has come closer by linking specific features of the graphs to predictions about these equations, which are called Kazhdan-Lusztig (KL) polynomials, named after the mathematicians who first proposed them. "What we were able to do is train some machine-learning models that were able to predict what the polynomial was, very accurately, from the graph," Davies said. The team also analyzed what features of the graph DeepMind was using to make those predictions, which got them closer to a general rule about how the two map to each other. This means DeepMind has made significant progress on solving this conjecture, known as the combinatorial invariance conjecture.
"DeepMind" is not an artificial intelligence (Score:5, Informative)
DeepMind is a company, owned by Google, which employs extraordinarily skilled natural intelligences known as human scientists and mathematicians.
They thought up all the ideas, and had the insights and programmed the computers, which executed artificial models designed by human intelligences which are aware of the vast history of related fields.
"What we were able to do is train some machine-learning models that were able to predict what the polynomial was, very accurately, from the graph," Davies said. The team also analyzed what features of the graph DeepMind was using to make those predictions, which got them closer to a general rule about how the two map to each other. This means DeepMind has made significant progress on solving this conjecture, known as the combinatorial invariance conjecture.
The natural intelligences used machine learning models as pattern matching & invariant extraction tools and analyzed those models' results to further inform mathematical conjectures.
Re: (Score:3, Interesting)
Re: (Score:2, Insightful)
How many grandmasters were defeated by the Kitty Hawk Flyer?
Re: (Score:3)
Holy fuck, you got whooshed by the Kitty Hawk. *roflcopter*
Re: (Score:1)
how many grandmasters can't fly?
Re: (Score:3)
How many grandmasters were defeated by the Kitty Hawk Flyer?
We're talking about flying right? The best human flier in the world was thoroughly defeated by the Kitty Hawk Flyer, so... all of them?
Re: (Score:2)
I get the impression that it's more like they're selling a 747 assembly kit. Meaning you still need to find the engineers to bolt it all together, not to mention the entire flight crew...
Re:Kinf of like the 4 color map problem (Score:2)
The four color map problem was proved using a computer, but the design of the proof was done by humans. That was back in the 1970s. Is this knot proof qualitatively different?
BTW Although I'm not a mathematician, from what little I know I understand about these things, it sounds like an impressive achievement nonetheless.
Re: Kinf of like the 4 color map problem (Score:2)
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This. DeepMind is a useful tool, much as a hammer is. But we don't say that a hammer built a custom cabinet.
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An ant den is intelligent; a fish is intelligent. How is machine learning not artificial intelligence? It isn't sentient intelligence, sure, but it's accumulation of data across facts to uncover nonlinear relationships. What other definition of intelligence are you looking for?
Re: (Score:2)
Intelligence is a dumb word. I prefer cognition. Cognition is modular. Intelligence is an attribute of the whole system. Cognition is an engineering domain. Intelligence is a judgement domain.
In Other Words. (Score:2)
Finally ... (Score:2)
"Here kid. Read this."
What the fuck is up with the false headlines? (Score:5, Insightful)
Re: What the fuck is up with the false headlines? (Score:2)
Why are you attacking the author specifically, do you frequent that site?
livescience.com is an online pop science magazine? The article is totally in line with everything else published on the site. Are jackalopes real? Gold: The rich element. That's what that site publishes.
So why are you going at the author? Why are you calling her out?
Re: What the fuck is up with the false headlines? (Score:2)
Re:What the fuck is up with the false headlines? (Score:5, Insightful)
And often the headlines aren't written by the writer of the article, so it's not necessarily their fault the headline is so misleading.
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It did crack one conjecture and it advanced the other, both of which may contribute to unraveling a larger problem than both conjectures.