Dynamical systems with Hamiltonians changing in time, can be studied by averaging over the fast oscillations, or by adiabatic approximation in the case of slowly changing dynamics. In both cases the averaged dynamics is useful up to finite time only, given by the inverse of the large parameter. I will describe a recently developed new approach, in which the averaging can be done on larger and larger time scales, in a way that gives rigorously controlled errors to all orders in the small parameter, and to arbitrary large time intervals. As such, it is a new way to derive and understand Nekhoroshev type theorems. I then present applications: new proof of adiabatic theorems and scattering for time dependent potentials. Finally, I will present numerical scheme for dispersive wave equations with controlled errors to very large time scales (10³-10⁷).

Professor Avy (Avraham) Soffer obtained his PhD in mathematical physics from Tel-Aviv University in 1984. He is now a distingguished professor at Department of Mathematics, Rutgers University. Professor Soffer receives many awards, including ICM (International Congress of Mathematicians) invited speaker 2006, AMS fellow 2012. His research interests are in the fields of Partial Differential equations and mathematical Physics, include Quantum Dynamics and Scattering, Spectral Theory and Resonances. It also includes Dispersive Wave Equations, Linear and Nonlinear, Soliton Dynamics , Scattering on Manifolds.