“Advancing mathematics by guiding human intuition with AI,” by Alex Davies, Petar Velickovic and colleagues at DeepMind together with mathematicians Marc Lackenby and Andras Juhasz at Oxford and Geordie Williamson at Sydney, was published in Nature in December 2021. It demonstrated a way to use machine learning not to prove theorems automatically, but to help human mathematicians notice relationships they would otherwise miss.
The framework trains a model to predict one mathematical quantity from others, then uses attribution techniques to reveal which inputs the model relied on, pointing the mathematician toward a candidate relationship to investigate and prove by hand. The team applied this to two areas of pure mathematics. In knot theory they uncovered a previously unknown link between a knot’s algebraic invariant (its signature) and its geometry, captured by a new quantity they named the natural slope. In representation theory the approach produced evidence and an algorithm bearing on the combinatorial invariance conjecture, an open problem for roughly forty years, which Williamson then advanced and verified across millions of examples.
The work is significant as one of the first cases where machine learning contributed to genuine results in pure mathematics, and because of how it did so: the AI guided human intuition rather than replacing it, suggesting a collaborative model for using these tools in deep theoretical research.