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, an AI lab based in London. (DeepMind is famous for their AlphaGo program which was the first program to defeat the best human Go players.) Our goal 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.
Geordie Williamson is a Professor at the University of Sydney, and the Director of its Mathematical Research Institute. He has made several fundamental contributions to the field including his proof (with Ben Elias) of the Kazhdan-Lusztig positivity conjecture, his algebraic proof of the Jantzen conjectures, and his discovery of counter-examples to the expected bounds in the Lusztig conjecture in modular representation theory. This last result came as a shock to a whole community of researchers, and has since shifted the focus away from old conjectures.
For his work he has been awarded the Chevalley Prize of the American Mathematical Society, the European Mathematics Society Prize, the Clay Research Award and the New Horizons Prize in Mathematics (with Ben Elias). In 2018 he addressed the International Congress of Mathematicians as a plenary speaker. More recently, he received the medal of the Australian Mathematical Society and the Christopher Heyde Medal of the Australian Academy of Science. He is the youngest living fellow of both the Royal Society and the Australian Academy of Science.