Spacelike Singularities and Hidden Symmetries of Gravity

We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.Comment: 228 pages. Typos corrected. References added. Subject index added. Published versio

Newton flows for elliptic functions III & IV:Pseudo Newton graphs: bifurcation and creation of flows

An elliptic Newton flow is a dynamical system that can be interpreted as a continuous version of Newton’s iteration method for finding the zeros of an elliptic function f. Previous work focuses on structurally stable flows (i.e., the phase portraits are topologically invariant under perturbations of the poles and zeros for f), including a classification/representation result for such flows in terms of Newton graphs (i.e., cellularly embedded toroidal graphs fulfilling certain combinatorial properties). The present paper deals with non-structurally stable elliptic Newton flows determined by pseudo Newton graphs (i.e., cellularly embedded toroidal graphs, either generated by a Newton graph, or the so-called nuclear Newton graph, exhibiting only one vertex and two edges). Our study results into a deeper insight in the creation of structurally stable Newton flows and the bifurcation of non-structurally stable Newton flows. As it requires the classification of all third order Newton graphs, we present this classification