For a state in a quantum field theory on some spacetime, we can associate a density matrix to any subset of a given spacelike slice by tracing out the remaining degrees of freedom. In the context of the AdS/CFT correspondence, if the original state has a dual bulk spacetime with a good classical description, it is natural to ask how much information about the bulk spacetime is carried by the density matrix for such a subset of field theory degrees of freedom. In this note, we provide several constraints on the largest region that can be fully reconstructed, and discuss specific proposals for the geometric construction of this dual region.

https://iopscience.iop.org/article/10.1088/0264-9381/29/15/155009/pdf

The gravity dual of a density matrix

Introduction: the problem of determining a metric by its hodograph and a linearization of the problem the kinetic equation in a Riemannian manifold. Part 1 The ray transform of symmetric tensor fields on Euclidean space: the ray transform and its relationship to the Fourier transform description of the kernel of the ray transform in the smooth case equivalence of the first two statements of theorem 2.2.1 in the case n=2 proof of theorem 2.2.2. the ray transform of a field-distribution decomposition of a tensor field into potential and solenoidal parts a theorem on the tangent component a theorem on conjugate tensor fields on the sphere primality of the ideal ([x]2, ) description of the image of the ray transform integral moments of the function I f inversion formulas for the ray transform proof of theorem 2.12.1 inversion of the ray transform on the space of field-distributions the Plancherel formula for the ray transform applications of the ray transform to an inverse problem of photoelasticity further results. Part 2 Some questions of tensor analysis. Part 3 The ray transform on a Riemannian manifold. Part 4 The transverse ray transform. Part 5 The truncated transverse ray transform. Part 6 The mixed ray transform. Part 7 The exponential ray transform (Part contents)

/pdf/integral-geometry-of-tensor-fields-2b2isl6v6s.pdf

Integral Geometry of Tensor Fields

We survey mathematical developments in the inverse method of electrical impedance tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature, this is also known as Calderon's problem from Calderon's pioneer contribution (Calderon 1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) p 65 (Soc. Brasil. Mat.)). We concentrate this review around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section, we review some counterexamples to Calderon's problems that have attracted a lot of interest because of connections with cloaking and invisibility.

http://www.dim.uchile.cl/~axosses/calderoniprevised.pdf

Electrical impedance tomography and Calderón's problem

We present a mathematical framework which underlies the connection be- tween information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts - points, distances and angles - are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entan- glement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.

/pdf/integral-geometry-and-holography-i6x5kev8v6.pdf

Integral geometry and holography

Inverse transport consists of reconstructing the optical properties of a domain from measurements performed at the domain's boundary. This review concerns several types of measurements: time-dependent, time-independent, angularly resolved and angularly averaged measurements. We review recent results on the reconstruction of the optical parameters from such measurements and the stability of such reconstructions. Inverse transport finds applications e.g. in medical imaging (optical tomography, optical molecular imaging) and in geophysical imaging (remote sensing in the Earth's atmosphere).

Inverse transport theory and applications

We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p12.pdf

Two dimensional compact simple Riemannian manifolds are boundary distance rigid

Integral geometry of tensor fields on a manifold of negative curvature

We describe a relation between the scattering relation, the Hilbert transform in frequency space, and the geodesic ray transform for simple, two-dimensional compact Riemannian manifolds with boundary. We use this relation to give a characterization of the range of the geodesic X-ray transform acting on scalar functions and vector fields in terms of the scattering relation which involves only measurements at the boundary. We also give inversion formulas for the geodesic X-ray transform acting on scalar functions and vector fields.

On characterization of the range and inversion formulas for the geodesic X-ray transform

In this paper we develop the numerical algorithm for solving the
inverse problem for the wave equation by the Boundary Control
method. The results of numerical experiments are presented.

Numerical recovering of a density by the BC-method

In this paper we develop numerical algorithm for solving inverse problem for the wave equation using Boundary Control method. The results of numerical experiments are represented.

/pdf/numerical-recovering-a-density-by-bc-method-4vpsw9nq70.pdf

Leonid Pestov

Papers

Two dimensional compact simple Riemannian manifolds are boundary distance rigid

Integral geometry of tensor fields on a manifold of negative curvature

On characterization of the range and inversion formulas for the geodesic X-ray transform

Numerical recovering of a density by the BC-method

Numerical recovering a density by BC-method