I will describe the derivation of a formula of Gauss-Bonnet
type involving the renormalized area of a minimal submanifold of a
Poincaré-Einstein space. This requires development of several new
ingredients: a special compactification coming from scattering theory
on the minimal submanifold, conformally invariant powers of the
Laplacian and Q-curvature on submanifolds of conformal manifolds, and a
conjectured analog of a celebrated result of Alexakis asserting a
decomposition of integrands of conformally invariant integrals. This is
joint work with Jeffrey Case, Tzu-Mo Kuo, Aaron Tyrrell, and Andrew Waldron.