Periodic Reporting for period 3 - RSPDE (Regularity and Stability in Partial Differential Equations) Reporting period: 2020-02-01 to 2021-07-31 Summary of the context and overall objectives of the project This project aims to use several deep techniques in mathematics to progress in a series of fundamental problems.These are:1) Optimal transport: what is the cheapest possible way of transporting resources from one place to another? By studying the regularity of optimal maps, we aim to better understand their structure.2) Functional inequalities: these are mathematical tools that allow one to study/understand equilibrium configurations for crystals and for several important dynamical evolution problems.3) Stability in PDEs: this broad question can be used to understand the behaviour of physical systems at critical configurations, for instance in equations which describe combustion phenomena where a possible blow-up may happen.These questions/problems are mathematically related, and their understanding will allow one to have a better knowledge (and therefore control) on the physical systems that they aim to describe. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far During the last years, the group has obtained already a series of major advancement in all the problems described above.Particularly relevant is a new result on the stability of critical semilinear PDEs, and their smoothness in low dimensions. In the next years, these results will play a crucial role in pursuing the proposed project thanks to the innovative character of the techniques that we just introduced. 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) Already the results obtained until now (as one can see from the list of publications of the group) have shed a lot of light.We are confident that, by the end of the project, all the problems discussed in the original proposal will be solved.