Periodic Reporting for period 4 - QUAMAP (Quasiconformal Methods in Analysis and Applications)
Periodo di rendicontazione: 2024-03-01 al 2025-08-31
Scaling invariance occurs in different forms in many natural phenomena. A well known example of a rough scale invariance is the fractal nature of a shoreline. From a mathematical point of view, explaining and making use of the scale invariance leads to the method of conformal (shape preserving) and quasiconformal mappings (which preserve shapes only roughly). Fourier analysis, where one decomposes a signal into its constituent frequencies, provides a related approach to analyse complicated natural phenomena. In brief, the goal of the project was to develop mathematical methods towards understanding different scale-invariant phenomena and their related geometric properties.
QUAMAP was devoted to applying aspects of modern mathematical analysis, from quasiconformal mappings to Fourier analysis, to several problems arising in mathematical physics. Using these tools, along with nonlinear analogues, the project aimed to shed light on diverse challenging problems including the geometry of energy minimisers, scaling limits in random geometry, an imaging technique based on electrical surface measurements, as well as the behaviour of fluids in turbulent regimes.The use of delicate quasiconformal methods, in conjunction with convex integration and/or nonlinear Fourier analysis, was the common theme of the project.
A number of important outstanding problems were susceptible to attack via these methods. First and foremost, Morrey's fundamental question on energy minimizers in two dimensional vectorial calculus of variations was studied. Understanding the geometry of conformally invariant random structures was similarly one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calderon's inverse conductivity problem was also among the goals of the project, as well as related applications to imaging. Further goals were found in fluid mechanics and scattering, as well as in the fundamental properties of the quasiconformal mappings, interesting in their own right.
QUAMAP was devoted to applying aspects of modern mathematical analysis, from quasiconformal mappings to Fourier analysis, to several problems arising in mathematical physics. Using these tools, along with nonlinear analogues, the project aimed to shed light on diverse challenging problems including the geometry of energy minimisers, scaling limits in random geometry, an imaging technique based on electrical surface measurements, as well as the behaviour of fluids in turbulent regimes.The use of delicate quasiconformal methods, in conjunction with convex integration and/or nonlinear Fourier analysis, was the common theme of the project.
A number of important outstanding problems were susceptible to attack via these methods. First and foremost, Morrey's fundamental question on energy minimizers in two dimensional vectorial calculus of variations was studied. Understanding the geometry of conformally invariant random structures was similarly one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calderon's inverse conductivity problem was also among the goals of the project, as well as related applications to imaging. Further goals were found in fluid mechanics and scattering, as well as in the fundamental properties of the quasiconformal mappings, interesting in their own right.
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.
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.
The project was structured around a number of high risk, high-gain objectives. Being a project in pure mathematics, the nature of the success is necessarily somewhat technical. We made serious progress in relation with the famous Morrey conjecture, which concerns the nature of possible energy functionals in non-linear elasticity. We have also been able to find mathematical descriptions for the evolution of fluids in certain very unstable situations, while simultaneously prescribing their macroscopic behaviour. Such phenomena are intertwined with the so-called strong butterfly effect. Put briefly, this means that it is not possible to prescribe the pointwise evolution of the fluids, due to inherent spontaneous stochastiticy.
In the context of magnetohydrodynamics, governing the evolution of plasma, such situations are constrained by the so-called the self-organization conjecture. We have developed a method - the first one in the literature on this topic - to construct weak solutions compatible with this phenomenon, and solved a conjecture of Taylor from 1976 about the unexpected conservation of such quantities in a turbulent regime.
Another theme of the project was the interaction between analysis and probability. Here, for scaling limits of random tilings we were able to understand in high generality the geometry of the phase boundary between liquid and frozen regions. This aspect of statistical physics relates to various phenomena, for example crystal growth.
Central to the proposal was also the study of sophisticated techniques aimed at new effective algorithms for questions in scattering and tomography. The best known such process is perhaps MRI, but there are many others, and their mathematical study is very challenging. We discovered that our previous work in recovering quantum potentials improves greatly by taking suitable averages of the process. In addition, with an eye on applications to inverse problems, our group improved the state of art for classical questions such as the Kakeya and Fourier restriction problems. Remarkably, techniques from the apparently unrelated field of semialgebraic geometry were key tools here. Finally, we were able to show that Lipschitz conductivities can indeed be reconstructed from voltage-to-current boundary measurements, reaching the level of regularity conjectured by Uhlmann.
In the context of magnetohydrodynamics, governing the evolution of plasma, such situations are constrained by the so-called the self-organization conjecture. We have developed a method - the first one in the literature on this topic - to construct weak solutions compatible with this phenomenon, and solved a conjecture of Taylor from 1976 about the unexpected conservation of such quantities in a turbulent regime.
Another theme of the project was the interaction between analysis and probability. Here, for scaling limits of random tilings we were able to understand in high generality the geometry of the phase boundary between liquid and frozen regions. This aspect of statistical physics relates to various phenomena, for example crystal growth.
Central to the proposal was also the study of sophisticated techniques aimed at new effective algorithms for questions in scattering and tomography. The best known such process is perhaps MRI, but there are many others, and their mathematical study is very challenging. We discovered that our previous work in recovering quantum potentials improves greatly by taking suitable averages of the process. In addition, with an eye on applications to inverse problems, our group improved the state of art for classical questions such as the Kakeya and Fourier restriction problems. Remarkably, techniques from the apparently unrelated field of semialgebraic geometry were key tools here. Finally, we were able to show that Lipschitz conductivities can indeed be reconstructed from voltage-to-current boundary measurements, reaching the level of regularity conjectured by Uhlmann.