A surface and a threefold with equivalent singularity categories

We start with an introduction to singularity categories and equivalences between them. In particular, we recall known results about singular equivalences between commutative rings, which go back to Knörrer, Yang, Kawamata and a joint work with Karmazyn. Then we explain a new singular equivalence between an affine surface and an affine threefold. This seems to be the first (non-trivial) example of a singular equivalence involving rings of even and odd Krull dimension.