Given an r-dimensional family of degree d plane curves, it is a classical (Victorian) question how many there are with r nodes. I will attempt to explain what this means, what form Goettsche and others conjectured for the answer (for curves on arbitrary complex surfaces), and a short proof.

These will be introductory lectures surveying GW, MNOP and GV invariants -- all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

These will be introductory lectures surveying GW, MNOP and GV invariants - all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

These will be introductory lectures surveying GW, MNOP and GV invariants -- all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.