Currents are dual to curves

The space of geodesic currents on a closed surface is a simultaneous

generalization of Teichmüller space, the space of measured

laminations, and the space of (not necessarily simple) curves. But

they form an infinite-dimensional space that can be hard to get a

handle on. We characterize geodesic currents in terms of their

intersection number with curves: a functional on curves that satisfies

a few simple axioms, most notably a smoothing condition on

resolving a crossing, is the intersection number with a (unique)

geodesic current, and conversely.

This is joint work with D$\'i$dac Mart$\'i$nez-Granado, extending our

previous criterion for extending curve functionals to geodesic

currents, and has applications to (eg) counting problems.