I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to $\mathbb{E}_n$-algebras.
I will also give the main ingredients of the proof of Kontsevich's conjecture, which is a joint work in progress with Damien Lejay.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/HAMP