Nonlinear geometric inverse problems

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.

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.