Long-awaited mathematics proof can help scan Earth’s innards

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.
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“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 proved1 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.  Thus, by measuring how fast waves travel inside a space, one could work out the shortest paths, and theoretically, the overall structure.
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The conjecture dates back to at least 1981, when the late mathematician René Michel2 formulated certain technical assumptions about the spaces for which it should be true.

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 surfaces3. Now, Vasy, Uhlmann and Plamen Stefanov, who is at Purdue University in West Lafayette, Indiana, have solved it for spaces that have three or more dimensions, as well.

Layer by layer
In Einstein’s space-time, curvature produces gravitational lensing, a phenomenon familiar to astronomers, in which the path of light bends around massive objects such as stars. Similar mathematics apply to conventional lensing, or refraction: light rays or sound waves shift direction when the medium through which they are travelling changes.

In the case of seismic waves — generated by events such as earthquakes — the differing properties of Earth at varying depths mean that the shortest path for such waves is usually not a straight line, but a curved one. Since the early 1900s, geophysicists have used this fact to map the planet’s internal structure, and this is how they discovered the mantle and the inner and outer cores.

Those discoveries were rooted in mathematical treatments that had some simplifying assumptions. Until now, it was not clear that one could fully determine Earth’s structure using only wave travel times.
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But that is what Vasy and his team’s proof shows — and the geophysical problem was a key motivation for solving the conjecture.

Their assumption, which differed from Michel’s, was that the curved space, or manifold, is structured with concentric layers. This allowed them to construct a solution in stages. “You go layer by layer, like peeling an onion,” says Uhlmann. For practical applications, this means that researchers will not only know that there is a unique solution to the problem; they will also have a procedure to calculate that solution explicitly.

The three mathematicians circulated their 50-page paper among a small pool of experts and then posted it in the arXiv repository. Depending on the feedback they get, the authors hope to submit it to a journal in the coming weeks.

From theory to reality
Vasy says that the work could be helpful to people who develop medical-imaging techniques such as ultrasound, as well as to seismologists.

But applying the theory to real geophysical data will not happen immediately, says Maarten de Hoop, a computational seismologist at Rice University in Houston, Texas. One difficulty is that the theory assumes that there is information at every point. But in reality, data are collected only at relatively sparse locations. Uhlmann says that he is working on that problem with colleagues who specialize in numerical analysis.
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The improved mathematical approach probably won’t drastically change our picture of Earth’s structure yet, says de Hoop. But it could lead to a better understanding of known features, such as the mantle plumes underneath Iceland or Hawaii, and perhaps, to the discovery of new ones, he says.

As with every meaty mathematical result, “it will take a while to come to grips with it” and to vet the proof thoroughly, says Gabriel Paternain, a mathematician at the University of Cambridge, UK.

Experts are taking the claim seriously, in part because it builds on a technical step from a linear form of the problem that the community had already accepted as a breakthrough4, adds UCL mathematician Yaroslav Kurylev.

So far, says Paternain, the impression is “excellent”.

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