## Exponential asymptotics

Accurate approximations can often be derived by expanding a function in terms of a small (or large) asymptotic parameter involved in the problem. Often such series diverge for any value of the asymptotic parameter. Poincaré's definition of an asymptotic expansion provided a calculus for their rigorous treatment. However, his definition ignores exponentially smaller scales which are fundamental for resolving ambiguities as well as controlling the divergence and range of validity of the series. Exponential asymptotics addresses all these problems. Rooted in the theory of resurgence, it exploits universal relationships between the components of asymptotic expansions to incorporate the missing exponentially small terms with little extra effort. The mathematical successes of this extremely powerful approach include: exact, remainder terms for wide classes of expansions; improved rigorous error bounds; hyper-exponential improvement in numerical approximations; systematic calculation of Stokes constants; geometric information from spectral functions; long time-asymptotics of nonlinear PDEs; higher-order Stokes phenomena taking place at sub-sub-dominant scales; solvability conditions for PDE problems. Notable applications to physical systems include the explanation of why smoothed nonlinear shocks are not caustics, the discovery of forward beaming of noise from sources in the wake of jet engine flow as well as to string theory. Southampton is a key centre for research in exponential asymptotics with active collaborations in the US, UK, Europe, Asia and Australia.