The curious ubiquity of ADE graphs, and the Numbers and Mutation games

Abstract:

Coxeter-Dynkin graphs feature prominently in dozens of topics in modern mathematics, including Lie algebras and Lie groups, reflection groups, regular polytopes, lattice theory, singularities, root systems, von Neumann algebras, quantum groups, knot theory, and many areas of combinatorics. The ADE graphs and their affine invariants are particularly intriguing.

Explaining this remarkable ubiquity is a tantalising problem, first enunciated by Arnold in 1976. In this talk we consider the graphs as central via two remarkable games, the Numbers game and the Mutation game.