Periodic Reporting for period 3 - QUAMAP (Quasiconformal Methods in Analysis and Applications)
Reporting period: 2022-09-01 to 2024-02-29
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 (that 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 is to develop mathematical methods towards understanding different scale invariant phenomena and the related geometric properties.
QUAMAP is 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 hopes 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, is the common theme of the project.
A number of important outstanding problems are susceptible to attack via these methods. First and foremost, Morrey's fundamental question on energy minimizers in two dimensional vectorial calculus of variations is studied, as well as the related conjecture of Iwaniec regarding the sharp $L^p$ bounds for the Beurling transform. Understanding the geometry of conformally invariant random structures is similarly one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calder\'on's inverse conductivity problem is also among the goals of the project, as well as related applications to imaging. Further goals are to be found in fluid mechanics and scattering, as well as in the fundamental properties of quasiconformal mappings, interesting in their own right, such as the outstanding deformation problem for chord-arc curves.
QUAMAP is 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 hopes 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, is the common theme of the project.
A number of important outstanding problems are susceptible to attack via these methods. First and foremost, Morrey's fundamental question on energy minimizers in two dimensional vectorial calculus of variations is studied, as well as the related conjecture of Iwaniec regarding the sharp $L^p$ bounds for the Beurling transform. Understanding the geometry of conformally invariant random structures is similarly one of the central goals of the proposal. Uhlmann's conjecture regarding the optimal regularity for uniqueness in Calder\'on's inverse conductivity problem is also among the goals of the project, as well as related applications to imaging. Further goals are to be found in fluid mechanics and scattering, as well as in the fundamental properties of quasiconformal mappings, interesting in their own right, such as the outstanding deformation problem for chord-arc curves.
The various work packages have evolved independently. First, we have intensively worked 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 has negative values. This particular condition is consistent with the classical assumptions of models arising in Non-linear Elasticity, where the deformations collapsing matter are severely penalised. As a byproduct we obtained the surprising result that Morrey conjecture holds for a class of stored energy functionals with many symmetries.
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 turns out that such solutions are a manifestation of spontaneous stochasticity and describe what is called the strong butterfly effect. This is an extremal form of chaotic behaviour, leading to non uniqueness of solutions for the relevant partial differential equations.
On a complementary direction, we solved a conjecture of Taylor on conservation of magnetic helicity, even in the physically most relevant multiply connected case. This is the most famous example of the so-called self-organization conjecture, that in the presence of various conserved quantities in turbulent regimes physical systems might evolve to an organized state.
Inverse scattering asks for a practical way to describe an electromagnetic quantum potential from its diffracted waves. Along the project we discovered that our former approach can be improved by taking several averages. In addition, we have shown that in the presence of magnetic potential, one can describe the electric potential knowing the diffracted waves. As a promising step towards recovering simultaneously both electric conductivity and magnetic permeability we combined the problem with the so called unique continuation for certain embeddings of surfaces in higher dimensions.
Understanding scaling limits of random tilings, models for crystal structures, has been one of the major achievements in the first part of the project. Here 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.
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 turns out that such solutions are a manifestation of spontaneous stochasticity and describe what is called the strong butterfly effect. This is an extremal form of chaotic behaviour, leading to non uniqueness of solutions for the relevant partial differential equations.
On a complementary direction, we solved a conjecture of Taylor on conservation of magnetic helicity, even in the physically most relevant multiply connected case. This is the most famous example of the so-called self-organization conjecture, that in the presence of various conserved quantities in turbulent regimes physical systems might evolve to an organized state.
Inverse scattering asks for a practical way to describe an electromagnetic quantum potential from its diffracted waves. Along the project we discovered that our former approach can be improved by taking several averages. In addition, we have shown that in the presence of magnetic potential, one can describe the electric potential knowing the diffracted waves. As a promising step towards recovering simultaneously both electric conductivity and magnetic permeability we combined the problem with the so called unique continuation for certain embeddings of surfaces in higher dimensions.
Understanding scaling limits of random tilings, models for crystal structures, has been one of the major achievements in the first part of the project. Here 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 project is 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 have 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, and simultaneously prescribed their macroscopic behavior. Such phenomena are intertwined with the so called the strong butterfly effect. Put briefly, this means that it is not possible to prescribe the pointwise evolution of the fluids, due to the inherent spontaneous stochastiticy. However, we managed to develop a new technique to prescribe its macroscopical evolution in this setting.
In the context of magnetohydrodynamics, governing the evolution of plasma, such situations are constraint by the so called the self-organization conjecture. This says that certain systems organize themselves towards a prefered state, due to the presence of unexpected conserved quantitities in addition to energy or momentum. 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 is the interaction between analysis and probability. We have been able to prove very general homogenization results, describing in the context of partial differential equations the macroscopic behaviour of random structures and understanding their behavior when the scale of discretness tends to zero. Similarly, in scaling limits of random tilings, models for crystal structures, we have been able to understand in high generality the geometry of the phase boundary between liquid and frozen boundaries.
Central to the proposal is also the study of sophisticated techniques which aim to new effective algorithms in 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. Here we have discovered that our previous work in the recover quantum potentials greatly improves by taking suitable averages of the process. The relevant algorithms rely on deep results in fundamental questions in harmonic analysis and partial differential equations. In addition, with eye on the applications to Inverse Problems, our group has improved the state of the art in the classical questions as the Kakeya problem. It is remarkable that techniques from appararently totally unrelated fields such as Gromov Algebraic Lemma or the Tarski-Seidenberg theorem are the key tools here.
In the context of magnetohydrodynamics, governing the evolution of plasma, such situations are constraint by the so called the self-organization conjecture. This says that certain systems organize themselves towards a prefered state, due to the presence of unexpected conserved quantitities in addition to energy or momentum. 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 is the interaction between analysis and probability. We have been able to prove very general homogenization results, describing in the context of partial differential equations the macroscopic behaviour of random structures and understanding their behavior when the scale of discretness tends to zero. Similarly, in scaling limits of random tilings, models for crystal structures, we have been able to understand in high generality the geometry of the phase boundary between liquid and frozen boundaries.
Central to the proposal is also the study of sophisticated techniques which aim to new effective algorithms in 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. Here we have discovered that our previous work in the recover quantum potentials greatly improves by taking suitable averages of the process. The relevant algorithms rely on deep results in fundamental questions in harmonic analysis and partial differential equations. In addition, with eye on the applications to Inverse Problems, our group has improved the state of the art in the classical questions as the Kakeya problem. It is remarkable that techniques from appararently totally unrelated fields such as Gromov Algebraic Lemma or the Tarski-Seidenberg theorem are the key tools here.