Local and global boundary rigidity

Abstract: The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated asa recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local innature. In particular, we can recover locally the compressional and the shearwave speeds for the elastic Earth model, given local information. The talk is based on joint work with G.Uhlmann (UW) and A.Vasy (Stanford). We will alsopresent results for a recovery of a Lorentzian metric from red shifts motivatedby the problem of observing cosmic strings.

analysis of PDEsdynamical systemsfunctional analysisoptimization and controlspectral theory

Audience: advanced learners

Webinar on PDE and related areas

Series comments: The webinar is organised jointly from IIT-Kanpur, TIFR-CAM,Bangalore, IISER-Pune and IISER-Kolkata.

Zoom Link for the talk will be available on iitk.ac.in/math/weekly-webinar-on-pde-and-related-areas

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