ar X iv : m at h / 03 04 18 6 v 1 [ m at h . Q A ] 1 5 A pr 2 00 3 TRIPLE AFFINE ARTIN GROUPS

ar X iv : m at h / 98 04 13 2 v 1 [ m at h . Q A ] 2 8 A pr 1 99 8 AFFINE WEYL GROUPS , DISCRETE DYNAMICAL SYSTEMS AND PAINLEVÉ EQUATIONS

A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed.

ar X iv : m at h / 05 04 37 3 v 1 [ m at h . Q A ] 1 9 A pr 2 00 5 Lax Operator for the Quantised Orthosymplectic

ar X iv : m at h / 05 04 08 2 v 4 [ m at h . D G ] 1 5 A pr 2 00 6 COMPLETE PROJECTIVE CONNECTIONS

The first examples of complete projective connections are uncovered: on surfaces, normal projective connections whose geodesics are all closed and embedded are complete. On manifolds of any dimension, normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature are complete.

ar X iv : m at h / 05 04 16 3 v 1 [ m at h . D G ] 8 A pr 2 00 5 AFFINE CURVATURE HOMOGENEOUS 3 - DIMENSIONAL LORENTZ MANIFOLDS

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous , but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing...

ar X iv : m at h / 05 04 55 2 v 1 [ m at h . N T ] 2 7 A pr 2 00 5 Multiple logarithms , algebraic cycles and trees

This is a short exposition—mostly by way of the toy models " double logarithm " and " triple logarithm " —which should serve as an introduction to the article [3] in which we establish a connection between multiple polylogarithms, rooted trees and algebraic cycles.