Combinatorial Invariance: a Case Study of Pure Math / Machine Learning Interaction

Combinatorial Invariance: a Case Study of Pure Math / Machine  Learning Interaction

The combinatorial invariance conjecture is a fascinating conjecture in Representation Theory. Basically it says that certain important polynomials ("Kazhdan-Lusztig polynomials") are determined in a very non-trivial way by a directed graph. About 2 years ago, I began working on this problem with DeepMind. Our goal, also discussed in the talks of Marc and Andras, was to try to discover whether modern Machine Learning techniques are helpful in approaching problems in Pure Mathematics. I will outline how the Machine Learning models work, how successful they were and (by far the trickiest part) how we went about extracting new mathematics from these models. The result is a formula which sheds light on the combinatorial invariance conjecture. This is joint work with the DeepMind team: Charles Blundell, Lars Buesing, Alex Davies and Petar Veličković.