Vortices and the L^2 volume of spaces of holomorphic maps

I will review how Baptista uses his recent work on the L^{^2}
geometry of vortex moduli spaces in gauged linear sigma-models to
extract a wide-ranging conjecture on the L^{^2} volume of various
spaces of holomorphic maps. The latter spaces are of independent
interest as soliton moduli spaces in ungauged nonlinear sigma-models,
and their volumes have been computed independently in some
exceptionally symmetric special cases, providing a nontrivial check
on Baptista's conjecture.