The various work packages have evolved independently.
First, we worked intensively on the Morrey conjecture. This famous open question on quasiconvexity is central to understanding which energy functionals are stable under minimization procedures, an issue basic to many aspects of physics. Here we achieved the first steps towards quasiconvexity of the Burkholder functional, and proved the property for the functional restricted to the set where it takes negative values. This particular condition is consistent with the classical assumptions of models arising in Non-linear Elasticity, where the matter-collapsing deformations are severely penalised.
In the context of fluid mechanics, we have succesfully modelled both Kelvin-Helmholtz and Rayleigh-Taylor instabilities (publications in Communication in Pure and Applied Math. and in Inventiones Mathematicae) predicting the size and shape of the so-called mixing zone of fluids. It turned out that such solutions are a manifestation of spontaneous stochasticity and describe what is called the strong butterfly effect.
In a complementary direction, we solved a conjecture of Taylor on conservation of magnetic helicity, even in the physically most relevant multiply connected case. We also constructed non-trivial bounded solutions preserving arbitrary magnetic helicity but dissipating energy and cross-helicity, and we found the exact integrability threshold to dissipate helicity. This resulted in 2024 a publication in the journal Communications in Pure and Applied Mathematics. In a related work we gave a short proof of the celebrated non-uniqueness result for weak solutions to the forced Euler equation (Crelle's Journal, 2025).
Inverse scattering asks for a practical way to describe an electromagnetic quantum potential from its diffracted waves. During the project we discovered that our former approach could be improved by taking several averages. In addition, we showed that in the presence of magnetic potential, one can describe the electric potential from the diffracted waves. We were finally able to produce a reconstruction algorithm to determine the conductivity from boundary measurements, at the regularity class of conductivities conjectured to be extremal.
Understanding scaling limits of random tilings, models of statistical physics for atomic and crystal structures, was one of the major achievements of the project. The main issue was to understand the boundary between the liquid and frozen regimes. For this we had to develop a completely new approach to the singular free boundary problem related to the tilings, but in the end we achieved a full understanding of scaling limits in high generality, for all so-called dimer coverings. Here specific quasiconformal mappings, one of the basic themes of the project, were indispensable. The result was published in the journal Communications in Pure and Applied Mathematics. Similarly, random quasiconformal mappings were analysed in great detail via random singular integral operators.
