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Gunther Uhlmann studies inverse problems, imaging and partial differential equations. His research asks how measurements made outside an inaccessible region constrain its internal coefficients or geometry. His institutional research account identifies applications to medical imaging and the structure of Earth.^[ref]

With Leonid Pestov, he proved a two-dimensional boundary rigidity theorem: for a compact simple Riemannian manifold, boundary distances recover the metric up to a boundary-fixing diffeomorphism. The qualification prevents confusing a recoverable physical geometry with a unique coordinate description.^[ref]

This gives a precise version of the imaging question “can travel times reveal the interior?” It also explains why assumptions about ray geometry matter: if measurements never probe a hidden region, changing that region need not change the data.

Leonid Pestov is a mathematician associated with inverse problems for waves and geometry. With Gunther Uhlmann, he proved boundary rigidity for compact simple two-dimensional Riemannian manifolds: boundary-to-boundary distances determine the interior metric up to a change of coordinates fixing the boundary.^[ref]

The ambiguity is physical as well as mathematical. Relabeling interior points does not change the travel time between labeled boundary points, so no reconstruction using only those times can distinguish the relabeling. The theorem shows that, under its geometric assumptions, this is the entire ambiguity. Pestov's institutional lecture description also concerns reconstructing coefficients of a wave equation from observations, connecting the geometric theorem with travel-time and wave-imaging measurements.^[ref]

By Leonid Pestov, Gunther Uhlmann.

Primary paper: annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p12.pdf

Annals of Mathematics 161, 1093–1110. Introduction and Theorem 1.1: boundary distances determine a simple compact surface metric up to a boundary-fixing diffeomorphism.