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Details of Grant 

EPSRC Reference: EP/R001898/1
Title: Nonlinear geometric inverse problems
Principal Investigator: Paternain, Professor G
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Pure Maths and Mathematical Statistics
Organisation: University of Cambridge
Scheme: Standard Research
Starts: 01 December 2017 Ends: 30 November 2020 Value (£): 325,635
EPSRC Research Topic Classifications:
Mathematical Analysis Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
EP/R002207/1
Panel History:
Panel DatePanel NameOutcome
07 Jun 2017 EPSRC Mathematical Sciences Prioritisation Panel June 2017 Announced
Summary on Grant Application Form
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.

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Organisation Website: http://www.cam.ac.uk