If there’s anything that we likely know for sure about mathematics, it’s 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’s far more about mathematics than just algebraic expressions and equations, though.
That statement couldn’t be any more true in the realm of topology, or the mathematical study of geometric objects that remain as such despite “continuous” 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.
Enter DeepMind, an artificial intelligence (AI) created by Alphabet Inc.’s 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’ve just taken down one of mathematics’ challenges in knot theory, and their results were published in the journal Nature.
“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—and yes, that’s “discoveries,” as they employed the help of DeepMind to peer into two problems: one with knots and topology, and another that’s decades old concerning graphs in multiple dimensions.
The first aspect of pure mathematics that DeepMind focused on for this study is knot theory, one of the topics under the umbrella of topology. In particular, DeepMind focused on mathematical concepts known as invariants, which news source LiveScience defines as “algebraic, geometric or numerical quantities that are the same.”
In particular, DeepMind peered into the relationships surrounding both geometric and algebraic invariants surrounding equivalent 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’s geometric invariants define its shape, while algebraic invariants define how it “twists.”
Churning calculations within itself, DeepMind managed to identify a new way of measuring knots—a geometric measurement the research team dubbed a knot’s natural slope. What’s interesting is the fact that the knot’s natural slope was “mathematically related” to another invariant called the signature, an invariant which “describes certain surfaces on knots.”
This, to the team, meant that their newly-found natural slope of a knot connected geometric invariants to algebraic invariants—a connection that had not been found before. This means that they found a new conjecture in mathematics, which are ideas that may be true but nevertheless require proving in all circumstances.
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 (CIC)—a problem that, one might immediately think, is far beyond what we humans can process in our measly three-dimensional brains.
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.
“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.
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.
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.”
(For more mathematics topics, check out the discovery of the solution to the “n-queens” problem in chess; after that, read on how pi has been determined up to 62.8 trillion digits.)
References
- Davies, A., Veličković, P., Buesing, L., Blackwell, S., Zheng, D., Tomašev, N., Tanburn, R., Battaglia, P., Blundell, C., Juhász, A., Lackenby, M., Williamson, G., Hassabis, D., & Kohli, P. (2021). Advancing mathematics by guiding human intuition with AI. Nature, 600(7887), 70–74. https://doi.org/10.1038/s41586-021-04086-x
- Pappas, S. (2021, December 6). DeepMind cracks ‘knot’ conjecture that bedeviled mathematicians for decades. LiveScience. https://www.livescience.com/deepmind-artificial-intelligence-pure-math
