Counting pseudo-holomorphic sections of Lefschetz fibrations

A sequence of simple closed curves on a closed surface determines a symplectic 4-manifold with boundary, the total space of the Lefschetz fibration over the disc with the specified fiber and the curves as vanishing cycles. One can ask how to compute the invariants of this symplectic 4-manifold – notably the relative Seiberg-Witten invariants or their symplectic avatars - in terms of the surface data. One such symplectic avatar is the Donaldson-Smith count of pseudo-holomorphic multi-sections of the Lefschetz fibration representing a prescribed second homology class. In this talk I will describe a project, joint with my UT Austin Ph.D. student Riccardo Pedrotti, that addresses the computation of the count of pseudo-holomorphic sections (not yet multi-sections) in a prescribed homology class.