Since the experimental realization of Bose—Einstein condensation (BEC) in Bose gases, it has been an ongoing challenge in mathematical physics to derive the phenomenon from the first principles of quantum mechanics. This has first been achieved in 2002 by Lieb and Seiringer for dilute, trapped systems. More recently, BEC has been proven for weakly interacting systems within the so-called mean-field limit.

In my talk, I will both review existing and present recent results on the dynamics of weakly interacting Bose gases in the mean-field limit. It turns out that, to leading order, the dynamics of the condensate are governed by the nonlinear Schrödinger equation. The fluctuations around the BEC are described by Bogoliubov's theory. This talk is based on joint work with Phan Thanh Nam.