We revisit the “state-dependence” of the map that we proposed recently between bulk operators in the interior of a large AdS black hole and operators in the boundary CFT. By refining recent versions of the information paradox, we show that this feature is necessary for the CFT to successfully describe local physics behind the horizon — not only for single-sided black holes but even in the eternal black hole. We show that state-dependence is invisible to an infalling observer who cannot differentiate these operators from those of ordinary quantum effective field theory. Therefore the infalling observer does not observe any violations of quantum mechanics. We successfully resolve a large class of potential ambiguities in our construction. We analyze states where the CFT is entangled with another system and show that the ER=EPR conjecture emerges from our construction in a natural and precise form. We comment on the possible semi-classical origins of state-dependence.

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Stochastic gradient (SG) method has been popularly applied to solve optimization problems with objective that is stochastic or an average of many functions. Most existing works on SG assume that the underlying problem is unconstrained or has an easy-to-project constraint set. In this paper, we consider problems that have a stochastic objective and also many functional constraints. For such problems, it could be extremely expensive to project a point to the feasible set, or even compute subgradient and/or function value of all constraint functions. To find solutions of these problems, we propose a novel SG method based on the augmented Lagrangian function. Within every iteration, it inquires a stochastic subgradient of the objective, a subgradient and function value of one randomly sampled constraint function, and function value of another sampled constraint function. Hence, the per-iteration complexity is low. We establish its convergence rate for convex and also strongly convex problems. It can achieve the optimal O (1/\sqrt {k}) convergence rate for convex case and nearly optimal O (1/\sqrt {k}) rate for strongly convex case. Numerical experiments on quadratically constrained quadratic programming are conducted to demonstrate its efficiency.

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of\cite {sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension 1 or 1 the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.