Oxidation = group theory

Dimensional reduction of theories involving (super-)gravity gives rise to sigma models on coset spaces of the form G/H, with G a non-compact group, and H its maximal compact subgroup. The reverse process, called oxidation, is the reconstruction of the possible higher dimensional theories, given the lower dimensional theory. In 3 dimensions, all degrees of freedom can be dualized to scalars. Given the group G for a 3 dimensional sigma model on the coset G/H, we demonstrate an efficient method for recovering the higher dimensional theories, essentially by decomposition into subgroups. The equations of motion, Bianchi identities, Kaluza-Klein modifications and Chern-Simons terms are easily extracted from the root lattice of the group G. We briefly discuss some aspects of oxidation from the E_{8(8)}/SO(16) coset, and demonstrate that our formalism reproduces the Chern-Simons term of 11-d supergravity, knows about the T-duality of IIA and IIB theory, and easily deals with self-dual tensors, like the 5-tensor of IIB supergravity.Comment: LaTeX, 8 pages, uses IOP style files; Talk given at the RTN workshop ``The quantum structure of spacetime and the geometric nature of fundamental interactions'', Leuven, September 200

Almost group theory

The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups

Group theory in cryptography

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.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U