This applied mathematics research program is highly motivated by strong laser physics and Quantum ElectroDynamics (QED), which are very dynamic and fast evolving research fields in theoretical and applied physics. The latter are also the source of many current and future applications. They have even led to the birth of a new science, the celebrated Attosecond Science dedicated to control of electrons inside atoms and molecule using very short and intense laser pulses! Canada is a leader in those fields, and this research program intends to contribute to their development from the mathematical and computational point of view. Indeed, the highly nonlinear and complex phenomena occurring in strong laser-medium interaction require advanced and modern mathematical models and tools for their analysis, and the development of efficient and accurate computational methods for their approximation and simulation. Although, the program will mainly focus on mathematical analysis and properties, code developments and applications in fundamental and applied physics will be explored within interdisciplinary collaborations.