Whitfield Diffie and Martin Hellman startled the computer security world in 1975 with their paper “New Directions in Cryptography”, which introduced public-key cryptography [7]. There is now evidence that three cryptographers at the British Government Communications Headquarters (GCHQ), the British equivalent of the National Security Agency, may have predated this work as well as the discovery of the Rivest-Shamir-Adleman (RSA) public-key encryption algorithm [9]. It appears, however, that the British researchers did not appreciate the significance of their discovery. Since the mid 1970s, cryptography has become big business, a bestseller (Applied Cryptography by Bruce Schneier has sold over two hundred thousand copies), and an extremely active area of research at the intersection of mathematics and computer science. A plethora of new books in the area currently floods the bookstores. This review cannot discuss them all. Instead I will delineate the subareas of this burgeoning field and focus on some of the texts that I believe will be of more interest to Bulletin readers.

The Yamabe problem

We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass, scalar curvature and harmonic maps are described and the positive mass theorem for n-dimensional spin manifolds is proved.

/pdf/the-mass-of-an-asymptotically-flat-manifold-33wzjfwsog.pdf

The mass of an asymptotically flat manifold

This paper gives a mathematically rigorous proof of the positive energy theorem using spinors. This completes and simplifies the original argument presented by Edward Witten. We clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant Dirac operator.

/pdf/on-witten-s-proof-of-the-positive-energy-theorem-2kbkhtazrb.pdf

On Witten's proof of the positive energy theorem

Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularityfree, vacuum space-times which are stationary in a neighborhood of i; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global I ; we prove existence of initial data for many black holes which are exactly Kerr – or exactly Schwarzschild – both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to r−m terms, for any fixed m, and with multipole moments freely prescribable within certain ranges. ∗Partially supported by a Polish Research Committee grant 2 P03B 073 24; email piotr@ gargan.math.univ-tours.fr †Visiting Scientist. Permanent address: Departement de Mathematiques, Faculte des Sciences, Parc de Grandmont, F37200 Tours, France. ‡Partially supported by the ACI program of the French Ministry of Research; email delay@ gargan.math.univ-tours.fr

/pdf/on-mapping-properties-of-the-general-relativistic-2d951ia0jf.pdf

On Mapping Properties of the General Relativistic Constraints Operator in Weighted Function Spaces, with applications

${\bf L}_p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type

https://projecteuclid.org/download/pdf_1/euclid.acta/1485890098

Elliptic systems in Hs,δ spaces on manifolds which are euclidean at infinity

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Elliptic systems in Hs,δ spaces on manifolds which are euclidean at infinity