Analysis of the Discrete Painlevé Equations and Their Degenerations.
The discrete Painlevé equations have been of great interest in recent years, due to their applications in many combinatorial and physical settings. In the 1990's, Ramani, Grammaticos, and Hietarinta showed that autonomous forms of these equations were instances of QRT mappings. A decade later, Sakai studied these equations from a different perspective, using rational surfaces. We will examine concrete examples using both viewpoints, and discuss how certain techniques could translate to degenerations of the discrete Painlevé equations.