PIMS-UManitoba Distinguished Lecture: Steven P. Lalley

The geodesics on a sphere (the prototypical positively curved surface) are the great circles, all of which are closed (that is, each great circle eventually returns to its initial point, in the same direction as it left). It is a remarkable fact that on a compact, negatively curved surface, only countably many geodesics are closed; these form a discrete sequence whose lengths diverge to infinity. The sequence of closed geodesics on a hyperbolic surface (that is, a surface of constant negative curvature -1) have remarkable number-theoretic and geometric properties, and obey striking statistical laws. I will discuss some of these in detail, with particular emphasis on the pattern of self-intersections of a typical closed geodesic.