This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.

It has been conjectured that every (2+1)-TQFT is a Chern-SimonsWitten (CSW) theory labeled by a pair (G, λ), where G is a compact Lie group, and λ ∈ H(BG;Z) a cohomology class. We study two TQFTs constructed from Jones’ subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the E6 subfactor, and the other is the quantum dou...

We find a different approach to define convex functions in the subRiemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofalo-Nieuhn (equivalent to that by Lu-Manfredi-Stroffolini). Nonholonomic geodes...

In this paper, we try to generalize to the case of compact Riemannian orbifolds Q some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds M. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics. In the classical case the solution of those problems involve the consideration of the ...

Optical vortices generically arise when optical beams are combined. Recently, we reported how several laser beams containing optical vortices could be combined to form optical vortex loops, links and knots embedded in a light beam (Leach et al 2004 Nature 432 165). Here, we describe in detail the experiments in which vortex loops form these structures. The experimental construction follows a th...

Let X be a nonsingular algebraic curve of genus g ≥ 3, and let M ξ denote the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d are coprime and d > n(2g − 2). We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. Let W ξ (L) denote the vector bundle over M ξ defined by the direct image p M ξ * (U ξ ⊗ p * X L) where U ξ is a ...

The different methods of reducing decoherence in quantum devices are discussed from the unified point of view based on the energy conservation principle and the concept of forbidden transitions. Minimal decoherence model, " bang-bang " techniques, Zeno effect and decoherence-free subspaces are studied as particular examples.

In this paper we study a storage process or a liquid queue in which the input process is the local time of a positively recurrent stationary diffusion in stationary state and the potential output takes place with a constant deterministic rate. For this storage process we find its stationary distribution and compute the joint distribution of the starting and ending times of the busy and idle per...

We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to “class P” of Hof-Knill-Simon. We also give ...

We consider partial differential operators H = − div(C∇) in divergence form on R with a positive-semidefinite, symmetric, matrix C of real L∞-coefficients and establish that H is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.