Renormalized Volume and Area for Poincaré-Einstein Spaces

Poincaré-Einstein spaces are a generalization of the Poincaré model of hyperbolic space. They were introduced to study holographically the conformal geometry of the boundary at infinity. This geometric correspondence underlies the AdS/CFT correspondence in physics. Physical renormalization procedures have led to geometric invariants called the renormalized volume and renormalized area. This talk will review these constructions and describe some recent progress concerning formulas of Gauss-Bonnet type involving the renormalized area of minimal submanifolds of Poincaré-Einstein spaces.