Noncommutative geometry, founded by Connes, is roughly a generalization of (algebraic or topological) geometry to associative algebras, viewed as functions on a "noncommutative space". Recently, noncommutative geometry has become widely applicable in the area of algebraic geometry and topology, partly thanks to mirror symmetry. One main idea is that (derived) categories of sheaves on complicated spaces can be replaced by modules over a much simpler associative algebra. For example, a result of Van den Bergh and Bondal shows that sheaves on any projective variety are derived-equivalent to modules over a certain dg algebra with finite homology.
In this talk I will explain these notions in the context of quotients of affine space, Cn, by the action of finite subgroups of GLn. This is called the "McKay correspondence", and in the case n=2 these correspond to the Dynkin diagrams. I will also give other examples.