Three dimensional quantum gravity with torus universe, T 2 ×R, topol-ogy is reformulated as the motion of a relativistic point particle moving in an Sl(2, Z) orbifold of flat Minkowski spacetime. The latter is precisely the three dimensional Milne Universe studied recently by Russo as a background for Strings. We comment briefly on the dynamics and quantization of the model. Stanley has made ma...

A local classification of all locally conformal flat Riemannian 4-manifolds whose Ricci tensor satisfies the equation ∇ ( ρ− 1 6 sg ) = 1 18 ds⊙ g as well as a local classification of all locally conformal flat Riemannian 4-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The problem of local descripti...

There have been some recent claims that brane-worlds of co-dimension two in a 6D bulk with compact extra dimensions may lead to self-tuning of the effective 4D cosmological constant. Here we show that if a phase transition occurs on a flat brane, so as to change its tension, then the brane will not remain flat. In other words, there is really no self-tuning in such models, which can in fact be ...

The Polyakov bosonic string is quantised in Euclidean Anti-de Sitter space-time using functional methods. Regularisation of the functional determinants using both heat-kernel and zeta function techniques shows that the AdSD+1 string reduces to the Liouville field theory, as in the flat space-time case. A Feynman graph expansion is used to evaluate (approximately) the coefficient multiplying the...

Full proofs of results announced here are to be found in [lO]. We consider the problem of classifying germs of neighbourhoods of locally flat topological submanifolds. Let M be a fixed topological manifold. An r-neighbourhood of M is a pair (i, N) where N is a topological manifold of dimension n+r and i:M—*N a locally flat embedding satisfying i^ÇdN) —dM. Denote the set of germs of r-neighbourh...

We want to generalize this to cones. Here again is the definition of a cone on a scheme X. Let S = ⊕i≥0S i be a sheaf of graded OX-algebras. Assume OX → S 0 is surjective, S is coherent, and S is generated (as an algebra) by S. Then you can define Proj(S), which has a line bundle O(1). Proj(S) → X is a projective (hence proper) morphism, but it isn’t necessarily flat! (Draw a picture, where the...

The control of spatial propagation properties of narrow light beams such as divergence, focusing or imaging are main objectives in optics and photonics. In this letter, we propose and demonstrate experimentally a flat focusing mirror, based on an especially designed dielectric structure without any optical axis. More generally, it also enables imaging any light pattern in reflection. The flat f...

We construct flat Laguerre planes that admit the universal covering group of SL2R, extended by a factor R, as a group of automorphisms. These planes contain the affine Moulton planes as derived planes, and they are constructed by gluing together two copies of a Moulton plane, folded up along their fixed lines. The action of the Moulton group carries over (modulo Z2) to the Laguerre planes, and ...

We prove existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a numerical criterion, in terms of spectral behavior of differential operators, under which one can obtain a decomposition of a formal flat connection in arbitrary di...