Supersymmetric vortices, 3d BPS states, and 3-manifolds

I will review the relation between vortex partition functions for
supersymmetric 2d theories and BPS indices in three dimensions, and
overview some of the more interesting mathematical and physical
properties enjoyed by the latter. In particular, BPS indices provide
a building block for quantum invariants of (auxiliary) knots and
three-manifolds, and suggest a new approach to categorification of
these objects.