Periodic Reporting for period 1 - DEBOGAS (Dilute Bose Gases at Positive Temperature) Reporting period: 2019-10-01 to 2021-09-30 Summary of the context and overall objectives of the project Context: In our description of nature with quantum mechanics we encounter two sorts of particles, bosons and fermions. From the Pauli exclusion principle we know that already two non-interacting fermions (e.g. electrons, protons, certain atoms) can never occupy the same quantum state (they can for example never be at the same point in space) — we say they have a statistical repulsion. Bosons (e.g. photons, certain atoms) in contrast have a statistical attraction and already the ideal (non-interacting) Bose gas shows an interesting phenomenon called the Bose—Einstein condensation (BEC) phase transition: Below a certain critical temperature a macroscopic fraction of all particles in the gas starts to behave in exactly the same way. The BEC phase transition is a purely quantum mechanical effect and has no classical counterpart.Bose gases, and more generally quantum gases, play a prominent role in modern physics because they allow for the simulation of a large variety of complex quantum many-particle systems with room-size experimental set-ups. With these experiments it is possible to obtain precise information about the physics of systems consisting of a huge number of degrees of freedom (e.g. particles), whose properties are not computable even with the largest supercomputers on earth. The persisting goal of these studies is to obtain a better understanding of how the microscopic constituents of such systems determine their macroscopic properties as e.g. their electrical conductivity, and to shed light on the physical origin of such important effects as e.g. high-temperature superconductivity. Mathematical physicists contribute in this endeavour by proving the existence of prominent physical effects as generally as possible starting from their fundamental quantum mechanical description, and by rigorously deriving effective equations used by physicists to describe them. Apart from their mathematical content, such proofs usually also allow us to learn more about the physics of the problem. The overall objective of the project “Dilute Bose Gases at Positive Temperature” was to develop new mathematical tools to study thermodynamic properties of dilute Bose gases at positive temperature (in contrast to zero temperature) as well as the dynamics of approximate thermodynamic equilibrium states after external electric and/or magnetic fields have been changed (e.g. a trapping potential has been switched off). The project, which has been intended for a period of two years, has been ended ahead of schedule after nine months in favour of a lecturer position (4 years) at the Institute of Mathematics of the University of Zurich financed by an Ambizione Grant of the Swiss National Science Foundation (SNSF). Nevertheless, one main result and several partial results could be obtained. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far Performed work and main results: The main result of the project concerns the rigorous computation of the mean-field (MF) shift of the critical temperature of a trapped Bose gas starting from the many-particle quantum mechanical description in a parameter regime called the semi-classical mean-field (SCMF) limit. Let me explain in some more detail what this means. The SCMF limit is not a dilute limit that is directly relevant for the description of current experiments but it serves as an almost realistic toy model to develop techniques for the more challenging dilute regime. Apart from that, it is interesting in its own right. Bose gases in experiments are usually captured in a trapping potential. The interplay between the trapping potential and the repulsive interaction between the particles in the gas leads to a decreased critical temperature for BEC w.r.t. to the trapped ideal (non-interacting) Bose gas. The difference between these two critical temperatures is called the MF shift of the critical temperature that I managed to compute in collaboration with my co-author and that is mentioned above. To the best of my knowledge this is the first proof of a BEC phase transition for a realistic continuum model, where the critical temperature for BEC can be shown to depend on the potential describing the interaction between the particles to leading order. Concerning the dynamics of approximate thermodynamic equilibrium states and other objectives related to the thermodynamic properties of dilute quantum gases I could obtain partial results. Their completion remains work in progress.Overview exploitation and dissemination of results: The results on the critical temperature shift are collected in a research article that will soon be submitted to a mathematical journal. They concern basic research at the border of mathematical analysis and the physics of many-particle quantum systems and as that have no direct applications outside the scientific context. The results will be communicated on conferences in mathematical physics as soon as the Covid-19 pandemic allows for the implementation of conferences again. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Progress beyond state of the art: As already mentioned, our result seems to be the first proof of a BEC phase transition for a realistic continuum model, where the critical temperature is shown to depend explicitly on the interaction potential to leading order. The main technical novelty of our work is the introduction of a novel semi-classical free energy functional. We show that the full interacting quantum mechanical free energy can be approximated by this effective functional, and that the full system shows BEC if and only if the semi-classical free energy functional shows BEC. The simpler mathematical structure of the semi-classical free energy functional (compared to the full quantum mechanical description of the system) is then used to prove the statement for the critical temperature shift.