David Gay: Maps from 4-manifolds to the 2-sphere

There has been a recent flurry of activity surrounding various kinds of smooth maps from 4-manifolds to the 2-sphere, starting with Lefschetz fibrations and ending up with the rather obvious idea of simply studying generic smooth maps to S^2 (as a generalization of Morse theory). The idea that we should be able to get some interesting information about 4-manifolds this way may seem naive, but I hope it is not and I will try to present some evidence that it is not. At the very least there are some interesting pictures to think about. The part of the work that is original is joint with Rob Kirby, and otherwise this will be an exposition of work by Perutz, Lekili, Baykur and Williams, among others.