This research project addresses three main topics: (1) The well-posedness theory of the initial value problem for the Gross-Pitaevskii (GP) hierarchy, based on the approach initiated by the PI jointly with N. Pavlovic (UT Austin). A main motivation underlying our work is our aim at generalizing methods from nonlinear dispersive PDE's for the analysis of quantum field theories (QFT's). (2) Problems in non-relativistic Quantum Electrodynamics (QED), in particular relating to the semiclassical motion of electrons, radiation damping, and the problem of ultraviolet renormalization in QED. (3) Derivation of kinetic equations from kinetic scaling limits of the thermal momentum distribution function for interacting electron gases, modeled in dynamical Hartree-Fock theory. These projects involve several ongoing collaborations, including international ones.
Study of the GP hierarchy, which emerges in the analysis of interacting Bose gases, bridges some of the most exciting recent developments in physics (Bose-Einstein condensation) with some of the most impressive recent developments in mathematics (nonlinear dispersive PDE's). Non-relativistic QED is the physical theory of non-relativistic quantum mechanical matter (electrons, atoms, molecules) interacting with the energy quanta (photons) of light, and describes processes in a wide spectrum of quintessential areas in technology (chemistry, electronics, modeling of solar panels, etc). The derivation of kinetic equations from quantum dynamics leads to a precise mathematical understanding of classical physics (fluid dynamics) from quantum physics. The educational component of this project involves the training of graduate students in highly multidisciplinary annual thematic programs that provide them with a integrative and specialized understanding of links between Analysis, Applied and Computational Mathematics, nonlinear PDE's, and Mathematical Physics. The goal is to provide graduate students with an exceptionally broad understanding of their research fields.
