In this talk we will try to make a little tour from the classical theory of singularities in the sense of Felix Klein to modern developments in representation theory.
The main idea behind it is the concept of categorification. This idea will not be explained in detail, but illustrated by an example. The so called Kleinian singularities arise from finite subgroups of SL(2,C) and have beautiful explicit descriptions. We will present a generalization of them (certain special examples of Springer fibres) and indicate their role in symplectic geometry and Fukaya categories.
As an application we will sketch a construction of Khovanov homology, which is a knot invariant categorifying the Jones polynomial.