Since the first realization of Bose-Einstein condensate (BEC) in atomic gases over two decades ago, the study of many-body boson systems has gained significant attention in both the physics and math communities. Shortly after the discovery, Lieb and his collaborators proved the existence of BEC in a dilute trapped gas at the absolute zero temperature. Later, Erd\"os , Schlein and Yau gave a qualitative proof of the persistence of condensate under time evolution and showed that the dynamics of the condensate, in the absence of quantum fluctuations, is well approximated by the Gross-Pitaevskii equation. Despite the success founded in the mean-field theory, recent experiments suggest that mean-field dynamics may not account for the depletion of the condensate, the phenomenon where particles in the condensate escape to higher energy states. Thus, this prompts the question: what lies beyond the mean-field approximation?

In this talk, we present a rigorous formulation of Bogoliubov theory for interacting

bosons and use it to derive the time-dependent Hartree-Fock-Bogoliubov equations (a system of nonlinear Schr\"odinger-type equations). We will then give a brief sketch of the proof of the well-posedness of the system in 3D and show how the system can be used to get an approximation of the dynamics of the many-body boson system in Fock space. If time permits, we will also discuss some open problems in the field.

### Analysis Seminar

Thursday, November 8, 2018 - 3:00pm

#### Jacky Chong

University of Maryland, College Park