Special Math Colloquium
The notion of roots is absolutely central to Lie theory and, in its classical version, very much tied to groups generated by reflections. My goal in this talk it to explore how this notion may be broadened to incorporate what is currently happening on the frontiers of Lie theory and mathematical physics. One interesting new phenomenon, which I am hoping to discuss, is the possibility that roots and dual roots live in lattices of different rank. This makes the Langland-like duality that exchanges them a particularly dramatic operation.