"Version: 20251101"--Title page verso.Includes bibliographical references.1. Introduction -- 1.1. History of Bose-Einstein condensates -- 1.2. Bose-Einstein condensation : an experimental and theoretical perspective -- 1.3. Mean field description and Gross-Pitaevskii equation -- 1.4. Fermionic condensates -- 1.5. Beyond mean-field description -- 1.6. Outline2. Analytical methods -- 2.1 Inverse …