Researchers have made significant progress in spectral geometry by proving a special case of the polya conjecture related to the eigenvalues of a disk. Their work, which combines theoretical elegance and potential practical applications, highlights the universal value and artistic beauty of mathematical research. Photo credit: Issues.fr.com
A professor and his colleagues have proven Pólya's conjecture about the eigenvalues of a disk, a tricky problem in mathematics.
Can you tell what shape a drum is from the sounds it makes?
That's the kind of question Iosif Polterovich, a professor in the Department of Mathematics and Statistics at the University of Montreal, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena in wave propagation.
Breakthrough in mathematical conjecture
Last summer, Polterovich and his international collaborators – Nikolay Filonov, Michael Levitin and David Sher – proved a special case of a famous conjecture in spectral geometry formulated in 1954 by the eminent Hungarian-American mathematician George Pólya.
The conjecture involves estimating the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.
This diagram shows Bessel functions where the points correspond to the frequencies of the tones of a round drum. Photo credit: Michael Levitin
Pólya himself confirmed his conjecture in 1961 for areas that overlap a plane, such as triangles and rectangles. Until last year, conjectures were only known for these cases. Despite its apparent simplicity, the CD remained elusive.
“Imagine an infinite floor covered with tiles of the same shape that fit together to fill the space,” Polterovich said. “It can be tiled with squares or triangles, but not disks. A disk is actually not a good shape for laying tiles.
The universality and impact of mathematics
In a paper published in July 2023 in the mathematics journal Mathematical Inventions, the researchers show that the Pólya conjecture holds for the disk, a case considered particularly difficult.
Although their result is primarily of theoretical value, their proof method has applications in computational mathematics and digital computing. The authors are currently investigating this route.
Joseph Polterovich
“Although mathematics is a fundamental science, in some ways it is similar to sports and the arts,” Polterovich said.
“Trying to prove a long-held conjecture is a sport. Finding an elegant solution is an art. And in many cases, great mathematical discoveries turn out to be useful: you just have to find the right application.