Abstract

Felix Klein said that the study of new transcendental functions defined by differential equations was “the central problem of the whole of modern mathematics”. The same impetus drove the search that led to differential equations called the Painlevé equations, more than a century ago. The subtle properties of their solutions are still being discovered and new applications are growing in physics. The solutions of the discrete Painlevé equations show tantalizingly similar properties, but the study of these transcendental functions remains far from complete. In this talk, I will give an overview of three major mathematical threads in this area, which starts with dynamics of planets, the study of linear differential and difference equations, and gives a glimpse of the geometry underlying these alluring equations. At the end, I will briefly mention some new work with Pieter Roffelsen (Sydney) on the solutions of the q-discrete version of the sixth Painlevé equation.