The news that DeepMind had helped mathematicians in research (one in representation theory, and one in knot theory) certainly got many thinking, what other projects could AI help us with? See MO question What are possible applications of deep learning to research mathematics?

Geordie Williamson has been giving talks on his experience with the DeepMind team (his lectures in Machine Learning for the Working Mathematician are also great), and his answer would be whenever we can produce lots of data or examples which we wish to look for patterns. He gave graphs and high-dimensional manifolds as examples.

For high-dimensional manifolds, it would be great if AI could provide the kind of geometric intuition that we acquire “easily” for 2 and 3 dimensions (and struggle even for 4). But what would be a good task for AI to learn? Predicting the (co)homology of manifolds or CW-complexes?

In fact, in the latest advances in natural language processing, they found that the vectors that certain words are assigned to satisfy some curious relations, like “queen - woman + man = king”. Maybe AI can learn that cycles can be added too.

One task that I’m having in mind now: given the complement of a complex hypersurface in $\mathbb C^N$, decide if two cycles are homologous. That’s something we could just “see” in $\mathbb C^1$, but somehow it doesn’t feel like a “fundamental” question for geometric intuition, but what geometric intuition helps to answer (if there’s a difference).

So, my question is, what tasks would be “fundamental” to forming our geometric intuition? And for those who acquired some intuition in higher dimensions (regular polytopes? sphere-packing? singular locus?), what would be a good dataset and task for acquiring that intuition, whether or not it’s feasible for current machine learning techniques?