{"id":3471,"date":"2022-01-04T22:00:00","date_gmt":"2022-01-04T22:00:00","guid":{"rendered":"https:\/\/modernsciences.org\/staging\/4414\/?p=3471"},"modified":"2021-12-14T07:09:11","modified_gmt":"2021-12-14T07:09:11","slug":"alphabets-deepmind-ai-cracks-a-decades-old-graphs-problem-and-mathematical-knots","status":"publish","type":"post","link":"https:\/\/modernsciences.org\/staging\/4414\/alphabets-deepmind-ai-cracks-a-decades-old-graphs-problem-and-mathematical-knots\/","title":{"rendered":"Alphabet\u2019s \u201cDeepMind\u201d AI Cracks a Decades-Old Graphs Problem and Mathematical Knots"},"content":{"rendered":"\n

If there\u2019s anything that we likely know for sure about mathematics, it\u2019s that it reaches much further from what we know at a surface level. True, we encounter a lot of stuff during our years in school; our teachers will likely be among those who tell us that there\u2019s far more about mathematics than just algebraic expressions and equations, though.<\/p>\n\n\n\n

That statement couldn\u2019t be any more true in the realm of topology<\/em>, or the mathematical study of geometric objects that remain as such despite \u201ccontinuous\u201d deformations that might otherwise ruin its shape but not tear through its surface. Taking deep dives through this field often requires more than just human intellect, as explorations in this field may require several complex calculations at once.<\/p>\n\n\n\n

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DeepMind is also responsible for the Go-playing computer program AlphaGo; this very program was the first to ever beat a human player, professional Go player Fan Hui, at a full-on game of Go back in 2015. (Goban, 2007) <\/figcaption><\/figure><\/div>\n\n\n\n

Enter DeepMind<\/em>, an artificial intelligence (AI) created by Alphabet Inc.\u2019s similarly-named AI subsidiary. Alphabet is perhaps best known as the parent company of the tech giant Google. Now with DeepMind in their hands, they\u2019ve just taken down one of mathematics\u2019 challenges in knot theory, and their results were published in the journal Nature<\/em><\/a>.<\/p>\n\n\n\n

“What hasn’t happened before is using [machine learning] to make significant new discoveries in pure mathematics,” said co-author and DeepMind machine learning specialist Alex Davies\u2014and yes, that\u2019s \u201cdiscoveries,\u201d as they employed the help of DeepMind to peer into two problems: one with knots and topology, and another that\u2019s decades old concerning graphs in multiple dimensions.<\/p>\n\n\n\n

The first aspect of pure mathematics that DeepMind focused on for this study is knot theory<\/em>, one of the topics under the umbrella of topology. In particular, DeepMind focused on mathematical concepts known as invariants<\/em>, which news source LiveScience<\/a> defines as \u201calgebraic, geometric or numerical quantities that are the same.\u201d<\/p>\n\n\n\n

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Artificial intelligence, of course, goes beyond the usual fare of scientific exploits; they too have now encroached the realm of user electronics and devices, and can be found from specially-designed robots to assistants you can speak to from your nearest smartphone. (Beard, 2018) <\/figcaption><\/figure><\/div>\n\n\n\n

In particular, DeepMind peered into the relationships surrounding both geometric<\/em> and algebraic<\/em> invariants surrounding equivalent<\/em> knots, or knots that are shaped in such a way that one can be distorted to make it look like the other without breaking the knot. In turn, a knot\u2019s geometric invariants define its shape, while algebraic invariants define how it \u201ctwists.\u201d<\/p>\n\n\n\n

Churning calculations within itself, DeepMind managed to identify a new way of measuring knots\u2014a geometric measurement the research team dubbed  a knot\u2019s natural slope<\/em>. What\u2019s interesting is the fact that the knot\u2019s natural slope was \u201cmathematically related\u201d to another invariant called the signature<\/em>, an invariant which \u201cdescribes certain surfaces on knots.\u201d<\/p>\n\n\n\n

This, to the team, meant that their newly-found natural slope of a knot connected geometric invariants to algebraic invariants\u2014a connection that had not been found before. This means that they found a new conjecture<\/em> in mathematics, which are ideas that may be true but nevertheless require proving in all circumstances.<\/p>\n\n\n\n

The second problem that DeepMind tackled is one that was found back in the 1970s. You see, the mathematicians then found a particular conjecture concerning graphs in multiple dimensions called the combinatorial invariance conjecture<\/em> (CIC)\u2014a problem that, one might immediately think, is far beyond what we humans can process in our measly three-dimensional brains.<\/p>\n\n\n\n

This is where DeepMind came in; mathematicians, with the help of machine learning, were also able to crack the equations needed to describe these multidimensional graphs, which are called Kazhdan-Lusztig polynomials<\/em>.<\/p>\n\n\n\n

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Some of the most basic forms of polynomials were taught to us as students; of course, with more complexity in these equations comes more processing power needed to parse them into an understandable form\u2014power which machine learning can provide to us in large amounts. (Malecki, 2015) <\/figcaption><\/figure><\/div>\n\n\n\n

“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,” said Davies. In short, DeepMind has made considerable progress towards solving the CIC.<\/p>\n\n\n\n

While Davies and team recognize that there are no immediate practical applications to their discoveries, they do hope that their findings may convince other mathematicians to give the help of artificial intelligence a shot when faced with seemingly insurmountable hurdles.<\/p>\n\n\n\n

Davies concluded in a statement to LiveScience: “The first thing we’d like to do is go out there into the mathematical community a little bit more and hopefully encourage people to use this technique and go out there and find new and exciting things.\u201d<\/p>\n\n\n\n

(For more mathematics topics, check out the discovery of the solution to the \u201cn-queens\u201d problem in chess<\/a>; after that, read on how pi has been determined up to 62.8 trillion digits<\/a>.)<\/p>\n\n\n\n

References<\/h2>\n\n\n\n