Michael Usher

Lefschetz fibrations and pseudoholomorphic curves

I'll give an overview of work by Donaldson-Smith and myself aimed at gaining insight into Gromov-type invariants of symplectic four-manifolds through the use of bundles of symmetric products associated to a Lefscehtz fibration. This gives rise to new proofs of some results about embedded curves that Taubes had obtained using Seiberg-Witten theory, and also to an approach for constructing integer-valued invariants that count immersed curves with prescribed numbers of nodes.