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There has been an intense interplay between the AdS/CFT correspondence and conformal geometry in mathematics. In particular bulk ``Poincar'e--Einstein structures'' where one solves the cosmological Einstein equations with boundary data at conformal infinity, have proven crucial for understanding boundary conformal geometries. More recently, the conformal geometry of hypersurfaces has begun to play a role in the physics of hypersurfaces. Here the relevant ``bulk problem'' is the singular Yamabe problem: given a metric find a conformally related metric with constant scalar curvature and prescribed data at a conformal infinity. We show how obstructions to solving the singular Yamabe problem form the basis for a calculus of conformal hypersurface invariants. Results include a new obstruction to solving Einstein's equations in four space-time dimensions with data at a conformal infinity, as we as a novel variational calculus for hypersurface energy functionals.

Note this talk is at the Physics Department.