Mathematics of Bose-Einstein Condensation

We propose a project in mathematics with a focus on many-body theory in mathematical physics. We are especially interested in themathematical tools involved in the description and analysis of the recent experimental realizations of Bose-Einstein Condensation. Itremains one of the most important challenges of mathematical physics to rigorously understand the formation of condensatesin interacting systems. This project aims to address that challenge.Progress on the problem of condensation has been made on certain length scales, and we aim to push the boundaries of theselengths with a view towards the end-goal of actually having a mathematical proof of condensation in a continuum system ofinteracting quantum particles in the thermodynamic limit. To approach this objective we will study various related systems andproblems with the expectation of getting improved understanding by seeing the methods in a new light. To fully solve these simpler problems will require the development of new mathematical tools and the gain of critical insight. Some of these simplified problems are concerned with the energy of the Bose gas in the dilute limit, also in dimensions different from 3, as well as LHY-physics—specially prepared systems where the normally lower order correction terms become dominant.