AI’s helping us solve giant math puzzles

Research in mathematics is a deeply imaginative and intuitive process. This might come as a surprise for those who are still recovering from high-school algebra.

What does the world look like at the quantum scale? What shape would our universe take if we were as large as a galaxy? What would it be like to live in six or even 60 dimensions? These are the problems that mathematicians and physicists are grappling with every day.

To find the answers, mathematicians like me try to find patterns that relate complicated mathematical objects by making conjectures (ideas about how those patterns might work), which are promoted to theorems if we can prove they are true. This process relies on our intuition as much as our knowledge.

Over the past few years I’ve been working with experts at artificial intelligence (AI) company DeepMind to find out whether their programs can help with the creative or intuitive aspects of mathematical research. In a new paper published in Nature, we show they can: recent techniques in AI have been essential to the discovery of a new conjecture and a new theorem in two fields called “knot theory” and “representation theory”.

Machine intuition

Where does the intuition of a mathematician come from? One can ask the same question in any field of human endeavour. How does a chess grandmaster know their opponent is in trouble? How does a surfer know where to wait for a wave?

The short answer is we don’t know. Something miraculous seems to happen in the human brain. Moreover, this “miraculous something” takes thousands of hours to develop and is not easily taught.

The past decade has seen computers display the first hints of something like human intuition. The most striking example of this occurred in 2016, in a Go match between DeepMind’s AlphaGo program and Lee Sedol, one of the world’s best players.

AlphaGo won 4–1, and experts observed that some of AlphaGo’s moves displayed human-level intuition. One particular move (“move 37”) is now famous as a new discovery in the game.

How do computers learn?

Behind these breakthroughs lies a technique called deep learning. On a computer one builds a neural network – essentially a crude mathematical model of a brain, with many interconnected neurons.

At first, the network’s output is useless. But over time (from hours to even weeks or months), the network is trained, essentially by adjusting the firing rates of the neurons.

Such ideas were tried in the 1970s with unconvincing results. Around 2010, however, a revolution occurred when researchers drastically increased the number of neurons in the model (from hundreds in the 1970s to billions today).