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Philippe H. Trinh : The role of exponentially small effects in the physical sciences



Recently, 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 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 can significantly change the solutions of the underlying mathematical models.
In this talk, we will introduce the audience to the history, ideas, and basic techniques of exponential asymptotics, with particular emphasis on how to recognize when such approaches are necessary. We will discuss the 19th century struggles of the great Cambridge physicist G.G. Stokes to better understand what is now known as the Stokes Phenomenon. We will then show how this understanding would provide the key insight into resolving two famous problems: the problem of modelling dendritic crystal growth, and the Saffman-Taylor viscous fingering problem.
Our discussion will conclude with a glimpse of the present and future applications of exponential asymptotics, notably within the context of hydrodynamics and ship waves, and for the mathematical modelling of rupture and singularity formation in fluid flows.

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