Highly oscillatory problems, ubiquitous in applications, are typically consider to be very difficult theoretically and demanding computationally. However, recent developments provide highly effective means to treat problems with rapid oscillation. The Isaac Newton Institute program brings together professionals concerned in highly oscillatory phenomena, with an emphasis on multiscale modeling, homogenization, symplectic integration, Riemann-Hilbert techniques, highly oscillatory quadrature and exponential integrators, together with theoreticians and with workers in a wide range of applications. Typical grand challenges that we seek to address by this multidisciplinary collaboration are the solution of Schroedinger, Helmholtz and Maxwell equations, as well as long-term integration of equations originating in molecular dynamics.
High oscillation pervades a very wide range of applications: electromagnetics, fluid dynamics, molecular modelling, quantum chemistry, computerized tomography, plasma transport, celestial mechanics, medical imaging, signal processing. . . . It has been addressed by a wide range of mathematical techniques, ranging from asymptotic theory, harmonic analysis, theory of dynamical systems, theory of integrable systems and differential geometry. The computation of highly oscillatory problems has spawned a large number of different numerical approaches and algorithms. The purpose of this program is to foster research into different aspects of high oscillation, including the theoretical, the computational and the applied, from a united standpoint and to promote the synergy implicit in an interdisciplinary activity.
