Mathematics & Statistics Departmental Research Seminar

An Introduction to the Stokes Phnomenon from the Perspective of Borel Resummation

Abstract:

The Stokes phenomenon is a fundamental feature of asymptotic analysis: the same formal series can describe different actual functions in different regions of the complex plane. In this talk, we give an introductory explanation of this phenomenon from the viewpoint of Borel resummation.

We begin by discussing why divergent series appear naturally in the study of differential equations, and why such series can still carry useful information. We then introduce the basic idea of Borel resummation as a method for recovering genuine functions from formal expansions.

From this perspective, the Stokes phenomenon can be understood as a consequence of how the resummation depends on direction in the complex plane. Time permitting, we will illustrate these ideas with classical examples and briefly mention their connection with exact WKB analysis.