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An open mathematics question for 70 years has finally been solved

An open mathematics question for 70 years has finally been solved

This diagram shows Bessel functions, where the dots correspond to the frequencies of sounds produced by a round drum.

Credit: Michael Levitin

Is it possible to guess the shape of a drum just by hearing the sounds it makes?

This is one of the questions that occupies the interest of Iosif Poltrovic, a professor in the Department of Mathematics and Statistics at the University of Montreal. To understand this type of physical phenomenon related to wave propagation, he used spectral geometry, a branch of mathematics.

Recently, the professor, in collaboration with his foreign colleagues Nikolai Filonov, Michael Levitin and David Shear, demonstrated a special case related to a famous intuition in spectroscopic geometry. It was introduced in 1954 by George Polya – a prominent mathematician – and concerns frequency estimates of a round cylinder, or, in mathematical language, the eigenvalues ​​of the disk.

Polya himself confirmed his conjecture in 1961 about regions tiling the plane, such as triangles or rectangles. Until last year, speculation was known only in these areas, while the disk, despite its apparent simplicity, remained elusive.

“Imagine an infinite floor covered with several tiles of the same shape that fit together to fill the space. The floor can be tiled with squares or triangles, but not with discs. “In fact, it is a very bad way of paving,” explains Iosif Poltrović.

Universality of mathematics

in condition Published in Mathematics Journal Mathematical inventionsThe researchers thus showed that Bolya's conjecture is correct for a disk, a particularly challenging case.

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Although this result has mainly theoretical value, the proof method has applications in computational mathematics and numerical computation. The authors are currently studying this path.

“Mathematics, although a fundamental science, shares some similarities with sports and art. Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. As is often the case, beautiful mathematical discoveries become useful.” -All you have to do is find the right app.