A spin foam model for pure gauge theory coupled to quantum gravity

We propose a spin foam model for pure gauge fields coupled to Riemannian quantum gravity in four dimensions. The model is formulated for the triangulation of a four-manifold which is given merely combinatorially. The Riemannian Barrett--Crane model provides the gravity sector of our model and dynamically assigns geometric data to the given combinatorial triangulation. The gauge theory sector is a lattice gauge theory living on the same triangulation and obtains from the gravity sector the geometric information which is required to calculate the Yang--Mills action. The model is designed so that one obtains a continuum approximation of the gauge theory sector at an effective level, similarly to the continuum limit of lattice gauge theory, when the typical length scale of gravity is much smaller than the Yang--Mills scale.Comment: 18 pages, LaTeX, 1 figure, v2: details clarified, references adde

Frequency reassignment in cellular phone networks

In cellular communications networks, cells use beacon frequencies to ensure the smooth operation of the network, for example in handling call handovers from one cell to another. These frequencies are assigned according to a frequency plan, which is updated from time to time, in response to evolving network requirements. The migration from one frequency plan to a new one proceeds in stages, governed by the network's base station controllers. Existing methods result in periods of reduced network availability or performance during the reassgnment process. The problem posed to the Study Group was to develop a dynamic reassignment algorithm for implementing a new frequency plan so that there is little or no disruption of the network's performance during the transition. This problem was naturally formulated in terms of graph colouring and an effective algorithm was developed based on a straightforward approach of search and random colouring

On the chromatic number of a random hypergraph

We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.Comment: 45 pages, 2 figures, revised versio

Hyperbolic manifolds that fiber algebraically up to dimension 8

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$, and this leads to the first examples of hyperbolic $n$-manifolds $\widetilde M_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M_n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P_n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$.Comment: 40 pages, 21 figure