Vortices, Seiberg-Witten monopoles and Heegaard Floer theory

I will give an overview of the roles, known and expected, of Lagrangian
submanifolds in vortex moduli spaces of Riemannian surfaces in the
construction of extended TQFT-type structures in Seiberg-Witten theory and
its cousin, Heegaard Floer theory. Part of the talk will discuss a theory,
under construction by the speaker with Yanki Lekili, of an extension of the full
Heegaard Floer 3-manifold invariants from closed 3-manifolds to
3-manifolds with boundary, via Fukaya categories of symmetric products of
Riemann surfaces.