The BKL analysis of spacelike singularities in cosmology gives
evidence for chaotic behavior of the four-dimensional spacetime
metric near the singularity. This can be understood as a billiard
motion in the Weyl chamber of a hyperbolic Kac-Moody algebra, which is
an infinite-dimensional extension of the Lie algebra sl(2,R). Thus the
billiard motion is controlled by the Weyl group of the hyperbolic
algebra, which can be realized as the modular group PGL(2,Z). The
billiard description can be applied also to 11-dimensional
supergravity, the low energy limit of M-theory. In this case sl(2,R)
is replaced by the exceptional Lie algebra E8, with the hyperbolic
extension E10. I will in my talk explain how the modular realization
of the Weyl group can be extended to the E10 case by generalizing the
rational integers Z to an integer version of the eight-dimensional
algebra of octonions. The talk is based on the paper 1010.2212 with Axel Kleinschmidt and Hermann
Nicolai.
