Stokes' Phenomenon refers to apparently discontinuous changes in exponentially small asymptotic terms in different regions of the complex plane separated by Stokes curves. Studying this change directly is challenging, as conventional asymptotic series methods cannot capture exponentially small terms. I will discuss exponential asymptotic methods, which are able to examine these effects. These methods reveal that Stokes' Phenomenon is not discontinuous; it is a smooth effect that occurs in the neighbourhood of Stokes curves.

I will show how exponential asymptotic methods can be extended to study discrete problems. Using these ideas, I will show how Stokes' Phenomenon can be studied certain solutions to the first discrete Painlevé equation, and a few other related discrete problems. I will finish by speculating on how I hope to exploit integrability to improve this type of analysis. Speaker(s): Christopher Lustri (Macquarie University)