Nonlinear geometric inverse problems
Lead Research Organisation:
University of Cambridge
Department Name: Pure Maths and Mathematical Statistics
Abstract
This proposal considers the mathematical study of inverse problems for non-linear wave equations arising in theoretical physics and differential geometry. The main problem we wish to address is the following: can the geometric structures governing the wave propagation be globally determined from local information, or more physically, can an observer do local measurements to determine the geometric structures in the maximal region where the waves can propagate and return back? There has been recent progress on this question when the geometric structure is space-time itself and the relevant partial differential equations are the Einstein equations. Here we propose the study of the Yang-Mills-Higgs model when the Lorentzian background is fixed and the goal is the reconstruction of the Yang-Mills field and the Higgs field. The main difference between the inverse problems for the
Einstein and Yang-Mills-Higgs equations is that the geometric structures to be reconstructed appear in the leading order terms in the former case and in the lower order terms in the latter case. This difference poses novel challenges, since a perturbation in the
leading order is stronger and therefore easier to see from the data. Our proposal relates the reconstruction of the lower order terms
to the previously unstudied problem to invert a broken non-abelian X-ray transform.
Einstein and Yang-Mills-Higgs equations is that the geometric structures to be reconstructed appear in the leading order terms in the former case and in the lower order terms in the latter case. This difference poses novel challenges, since a perturbation in the
leading order is stronger and therefore easier to see from the data. Our proposal relates the reconstruction of the lower order terms
to the previously unstudied problem to invert a broken non-abelian X-ray transform.
Planned Impact
The direct beneficiaries of this research will be mathematicians working in geometric inverse problems, including those interested in nonlinear wave equations, non-abelian geodesic ray transforms, the boundary rigidity problem, the Calderon problem and transport equations. Potential secondary beneficiaries include applied mathematicians in the UK and abroad interested in inverse problems and imaging.
The proposed research has potential applications outside mathematics,
including Ultrasound Elastography and Optical Tomography. A wider impact will be achieved
via communication and engagement with applied mathematicians.
The proposed project will help to maintain and expand the UK's excellence in the area of Inverse Problems.
It also builds connections with other areas, most notably Differential Geometry and Partial
Differential Equations.
In particular, it supports the Centre for Inverse Problems at UCL, the Doctoral Training Centre in Analysis at Cambridge
and the London School in Geometry and Number Theory. It also strengthens the cluster of excellent researchers in linear partial differential equations
and spectral theory in the London area, as mentioned by the 2010 International
Review of Mathematical Sciences.
The project team includes a Research Associate and one of the two Co-PIs is an early career researcher. An integral part of the impact is the direct knowledge transfer to them.
The proposed research has potential applications outside mathematics,
including Ultrasound Elastography and Optical Tomography. A wider impact will be achieved
via communication and engagement with applied mathematicians.
The proposed project will help to maintain and expand the UK's excellence in the area of Inverse Problems.
It also builds connections with other areas, most notably Differential Geometry and Partial
Differential Equations.
In particular, it supports the Centre for Inverse Problems at UCL, the Doctoral Training Centre in Analysis at Cambridge
and the London School in Geometry and Number Theory. It also strengthens the cluster of excellent researchers in linear partial differential equations
and spectral theory in the London area, as mentioned by the 2010 International
Review of Mathematical Sciences.
The project team includes a Research Associate and one of the two Co-PIs is an early career researcher. An integral part of the impact is the direct knowledge transfer to them.
Organisations
People |
ORCID iD |
| Gabriel Paternain (Principal Investigator) |
Publications
Bohr J
(2023)
The Transport Oka-Grauert principle for simple surfaces
in Journal de l'École polytechnique - Mathématiques
Cekic M
(2022)
The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
in Inventiones mathematicae
Chen X
(2022)
The Sobolev inequalities on real hyperbolic spaces and eigenvalue bounds for Schrödinger operators with complex potentials
in Analysis & PDE
Chen X
(2021)
Inverse Problem for the Yang-Mills Equations
Chen X
(2020)
Inverse problem for the Yang-Mills equations
Chen X
(2021)
Inverse Problem for the Yang-Mills Equations
in Communications in Mathematical Physics
Chen X
(2021)
Detection of Hermitian connections in wave equations with cubic non-linearity
in Journal of the European Mathematical Society
Chen X
(2022)
The Semiclassical Resolvent on Conic Manifolds and Application to Schrödinger Equations
in Communications in Mathematical Physics
Chen X
(2021)
Inverse Problem for the Yang-Mills Equations
Ilmavirta J
(2022)
Broken ray tensor tomography with one reflecting obstacle
in Communications in Analysis and Geometry
Ilmavirta J
(2019)
Functions of constant geodesic x-ray transform
in Inverse Problems
Ilmavirta J
(2018)
Functions of constant geodesic X-ray transform
Ilmavirta Joonas
(2022)
Broken ray tensor tomography with one reflecting obstacle
in COMMUNICATIONS IN ANALYSIS AND GEOMETRY
Mettler T
(2020)
Convex projective surfaces with compatible Weyl connection are hyperbolic
in Analysis & PDE
METTLER T
(2021)
Vortices over Riemann surfaces and dominated splittings
in Ergodic Theory and Dynamical Systems
Monard F
(2021)
Consistent Inversion of Noisy Non-Abelian X-Ray Transforms
Monard F
(2021)
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors
in The Annals of Statistics
Monard F
(2020)
Consistent Inversion of Noisy Non-Abelian X-Ray Transforms
in Communications on Pure and Applied Mathematics
Paternain G
(2020)
A sharp stability estimate for tensor tomography in non-positive curvature
in Mathematische Zeitschrift
Paternain G
(2020)
A sharp stability estimate for tensor tomography in non-positive curvature
Paternain G
(2020)
The non-Abelian X-ray transform on surfaces
| Description | We managed to succeed in proving that a Yang-Mills field and a Higgs field in 3+1 dimensions can be recovered from active local measurements. The result is general enough that covers the case of classical fields in the standard model of particle physics. Along the way we also managed to show that one can recover a connection from the source-to-solution map of a wave equation with cubic non-linearity. The approach has two stages: in the first we recover broken parallel transport information along light rays and in the second instance we recover the fields from these transport information. |
| Exploitation Route | The tools and techniques developed in this project can be used for several other relevant geometric non-linear problems, so we expect that see developments in this direction soon, once the techniques and tools are properly digested and processed by the community. |
| Sectors | Education Other |