Long-awaited mathematics proof could help scan Earth's innards | Nature.com

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"Proposed solution to geometry puzzle allows an object’s structure to be determined from limited information" reports Davide Castelvecchi, staff reporter at Nature in London and a contributing editor.

Earth's inner structure could be revealed by the speed at which waves travel from one edge to another.
Planterary Visions Ltd./Science Photo Library

Mathematicians say that they have solved a major, decades-old problem in geometry: how to reconstruct the inner structure of a mystery object ‘X’ from knowing only how fast waves travel between any two points on its boundary.

The work has implications in real-world situations, such as for geophysicists who use seismic waves to analyse the structure of Earth’s interior.

“Without destroying ‘X’, can we figure out what’s inside?” asked mathematician András Vasy of Stanford University in California, when he presented the work in a talk at University College London (UCL) last week. “One way to do it is to send waves through it,” he said, and measure their properties.

Now, Vasy and two of his collaborators say that they have proved that this information alone is sufficient to reveal an object’s internal structure.

Looking inwards 
The problem is called the boundary-rigidity conjecture. It belongs to the field of Riemannian geometry, the modern theory of curved spaces with any number of dimensions. Albert Einstein built his general theory of relativity — in which mass warps the geometry of space-time — on this branch of mathematics.

Mathematicians already knew that the way in which curvature varies from place to place inside a ‘Riemannian manifold’ — the mathematical jargon for curved space — determines the shortest paths between any two points.

The conjecture flips things around: it says that knowing the lengths of the shortest paths between points on a boundary essentially determines the curvature throughout. (The geometry is therefore said to be ‘rigid’.) Thus, by measuring how fast waves travel inside a space, one could work out the shortest paths, and theoretically, the overall structure.

The conjecture dates back to at least 1981, when the late mathematician René Michel¹ formulated certain technical assumptions about the spaces for which it should be true. (It is not true for Riemannian manifolds in general.)

Vasy’s co-author Gunther Uhlmann, a mathematician at the University of Washington in Seattle, has been working on this problem since the late 1990s, and he and a collaborator had already solved it for two-dimensional Riemannian manifolds — that is, curved surfaces². Now, Vasy and his collaborators have solved it for spaces that have three or more dimensions.
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Source: Nature.com