In this talk we discuss new techniques for taking ineffective local, e.g. tangent cone, understanding and deriving from this effective estimates on regularity. Our primary applications are to Einstein manifolds, harmonic maps between Riemannian manifolds, and minimal surfaces.For Einstein manifolds the results include, for all $p0:sup_{B_r(x)}|Rm|\leq r^{-2}}$. If we assume additionally that the curvature lies in some $L^q$ we are able to prove that $r^{-1}_{|Rm|}$ lies in weak $L^2q$. For minimizing harmonic maps $f$ we prove $W^{1,p}\cap W^{2,p/2}$ estimates for $p2$ and the first $L^p$ estimates on the hessian for any $p$. The estimates are sharp. For minimizing hypersurfaces we prove $L^p$ estimates for $p