On the Landau-Ginzburg/conformal field theory correspondence

In this brief review we introduce the Landau-Ginzburg/conformal field theory correspondence, a result from the physics literature of the late 80s and early 90s which predicts a relation between categories of matrix factorizations and categories of representations of vertex operator algebras. To date there is no precise mathematical statement for it. Here, we explain some recent examples of this correspondence, some standing conjectures and future directions of research.