Local Normal Forms of Noncommutative Functions

In algebraic terms, the purpose of the talk is to classify finite dimensional Jacobi algebras arising on the d-loop quiver.  The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters.  I will spend most of my time explaining what the algebras are, why they classify, and how to intrinsically extract ADE information from them.  I will also say a little on why this should be viewed as an extension of classical singularity theory, since many of the ideas are inspired by Arnold and others.  At the end, I’ll briefly explain why I’m really interested in this problem, the connection with different quivers, and the applications of the above classification to curve counting and birational geometry.  This is all joint work with Gavin Brown.