The Painlevé differential equations are important nonlinear
ordinary differential equations that define new special functions, known
as the Painlevé transcendents. They have wide-ranging applications
across many areas of mathematics and provide canonical examples of
integrable systems in one dimension, with many beautiful properties
including affine Weyl group symmetries and relations to certain rational
While the theory of discrete analogues of the Painlevé equations is much more recent, these systems have already been observed to play a similarly important role in applications to their differential counterparts.
This talk will provide an introduction to discrete Painlevé equations through the geometric framework initiated by H. Sakai, which makes use of the geometry of rational surfaces associated with affine Weyl groups.