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"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.
The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, in which exponentially small (so-called non-perturbative) terms suddenly appear. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems.
Examples in applied maths will be drawn from the areas of crystal growth, surface waves on fluids, dislocation dynamics, Hele-Shaw flow, thin film rupture, quantum mechanics, and atmospheric dynamics. There are also many applications in pure maths, for example, in exact WKB analysis, topological recursion, algebraic geometry, and dynamical systems.
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