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.

We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de- veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well- posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (Cn , 0) → (C, 0). The Yau algebra L(V ) is defined to be the Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), ∂x1 ∂xn i.e., L(V ) = Der(A(V ), A(V )) and plays an important role in singularity theory. It is known that L(V ) is a finite dimensional Lie algebra and its dimension λ(V ) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras Lk(V) which are defined to be the Lie algebras of derivations of the moduli algebras Ak(V) = On/(f,mkJ(f)),k ≥ 0, i.e., Lk(V) = Der(Ak(V),Ak(V)) and where m is the maximal ideal of On. In particular, it is Yau algebra when k = 0. The dimension of Lk(V ) is denoted by λk(V ). These numbers i.e., k-th Yau numbers λk(V ), are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras Lk(V ) and also compute the Lie algebras L1(V ) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the λk(V ) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k = 1 for large class of singularities.

For Komatu-Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of \cite{S} to find the SDEs satisfied by the induced motion $\xi(t)$ on $\partial\HH$ and the slit motion $\s(t)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogenous functions. Next with solutions of such SDEs, we study the corresponding stochastic Komatu-Loewner evolution, denoted as ${\rm SKLE}_{\alpha,b}$. We introduce a function $b_{\rm BMD}$ measuring the discrepancy of a standard slit domain from $\HH$ relative to BMD. We show that ${\rm SKLE}_{\sqrt{6},-b_{\rm BMD}}$ enjoys a locality property.

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is <i>O</i>(<i>x</i>^(<i>d</i>+1)) with <i>d</i> the dimension of the physical space. Since there are the same number of mesh points in the <i>x</i><i>t</i> space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only <i>O</i>(1) time. We also extend the algorithm to compute the FTLE