Mathematics Colloquium

The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann's 1936 paper ``Sur les espaces localement homog`enes''. Their classification leads to interesting dynamical systems. For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z) on the upper half¬plane. This action is dynamically trivial, with a quotient space the familiar modular curve. In contrast, the classification of other simple geometries on the torus leads to the standard linear action of SL(2,Z) on R^2, with chaotic dynamics and a pathological quotient space. This talk describes such dynamical systems, where the moduli space is described by the nonlinear symmetries of cubic equations like Markoff's equation x^2 + y^2 + z^2 = x y z. Both trivial and chaotic dynamics arise simultaneously,relating to possibly singular hyperbolic metrics on surfaces of Euler characteristic equals ¬1.