The search for complex singularities in exponential asymptotics – Philippe Trinh (University of Bath)

December 13, 2019 @ 4:30 pm – 5:00 pm
Seminar Room 1
Newton Institute

In the last two decades, the development of specialized techniques in mathematics known as exponential asymptotics has led to the successful resolution of long-standing problems in topics as varied as quantum mechanics, crystal growth, dislocations, pattern formation, turbulence, thin film flow, and hydrodynamics. These developments have emerged from the realization that in many such problems, exponentially small effects, linked to the presence of singularities in the analytic continuation of a perturbative solution, can significantly change the solutions of the underlying mathematical models. However, the majority of problems studied have been one-dimensional in nature (essentially ordinary differential or difference equations). The extension of exponential asymptotics to multi-dimensional problems requires some fundamental advances in our ability to compute numerical analytic continuations of single- and multi-variate complex functions. In this talk, I shall present the basic theory of exponential asymptotics and discuss these open challenges in the context of problems in fluid mechanics.