The MNOP conjecture

I will review the Maulik-Nekrasov-Okounkov-Pandharipande conjecture, relating two different ways of counting curves in an algebraic variety X of dimension 3 or less. The first counts holomorphic maps of curves into X, or parameterised curves, and gives Gromov-Witten theory. The second counts embedded curves in X cut out by equations, or unparameterised curves, and is called stable pair theory. The conjecture is that these theories contain the same information, but with a complicated, mysterious transformation between the two sets of invariants. I will describe an application, and the recent proof of MNOP for most Calabi-Yau 3-folds X by Pandharipande-Pixton.