PIMS MSS Colloquium: Terry Gannon

Vertex operator algebras (VOAs) are a mathematical approach to quantum field theories which won their creator a Fields medal. But they are highly complicated structures -- if you haven't already learned what they are, it is surely best if you never do. Fortunately you don't need to know anything about them to follow my talk, though they'll be the shadows lurking in the background. In a sense I'll sketch, VOAs behave as an upside-down version of groups. This helps, because groups are much easier to understand. Now, one of the greatest results in Algebra in the 20th century was identifying the finite groups which, like lego pieces, snap together to form all other groups. With a handful of exceptions, these lego pieces are all of a common form (called `Lie-type'). The upside-down metaphor then predicts that the VOAs of Lie-type (with a handful of exceptions) are maximal, and this was just proven to be true. But something unexpected and quite interesting happens when you look at those exceptions, and that is the punchline of my talk. No knowledge of VOAs and groups will be assumed. These exceptions lead us to A-D-E, which is a simple pattern permeating math, then to curves associated to Fermat's Last Theorem, and after that ...???