Intersection theory on deligne-mumford compactifications (after Witten and Kontsevich)

Physicists have developed two approaches to quantum gravity in dimension two One involves an a priori ill de ned integral over all conformal structures on a surface which after a suitable renormalization procedure produces a well de ned integral over moduli spaces of curves In another they consider a weighted average over piecewise at metrics on that surface and take a suitable limit of such expressions The belief that these two approaches yield the same answer led Witten to make a number of conjectures about the intersection numbers of certain natural classes that live on the moduli space of stable pointed curves One of these conjectures has been rigourously proved by Kontsevic