  We construct a scattering theory for harmonic one-forms on Riemann surfaces,
obtained from boundary value problems through systems of curves and the jump
problem. We obtain an explicit expression for the scattering matrix in terms of
integral operators which we call Schiffer operators, and show that the matrix
is unitary. As a consequence of this scattering theory, we prove index theorems
relating these conformally invariant integral operators to topological
invariants. We also obtain a general association of positive polarizing
Lagrangian spaces to bordered Riemann surfaces, which unifies the classical
polarizations for compact surfaces of algebraic geometry with the
infinite-dimensional period map of the universal Teichmueller space.
