Mathematicians prove Pólya’s conjecture for the eigenvalues of a disk, a 70-year-old math problem
Is it possible to deduce the shape of a drum from the sounds it makes? This is the kind of question that Iosif Polterovich, a professor in the Department of Mathematics and Statistics at Université de Montréal, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena involving wave propagation.
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 bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.
Pólya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. Until last year, the conjecture was known only for these cases. The disk, despite its apparent simplicity, 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 with disks. A disk is actually not a good shape for tiling.”
The universality of mathematics
In an article published in the mathematical journal Inventiones Mathematicae, the researchers show that Pólya’s conjecture is true for the disk, a case considered particularly challenging.
Though their result is essentially of theoretical value, their proof method has applications in computational mathematics and numerical computation. The authors are now investigating this avenue.
“While mathematics is a fundamental science, it is similar to sports and the arts in some ways,” Polterovich said.
“Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. And in many cases, beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.”
More information:
Nikolay Filonov et al, Pólya’s conjecture for Euclidean balls, Inventiones mathematicae (2023). DOI: 10.1007/s00222-023-01198-1
Citation:
Mathematicians prove Pólya’s conjecture for the eigenvalues of a disk, a 70-year-old math problem (2024, March 1)
retrieved 1 March 2024
from https://phys.org/news/2024-03-mathematicians-plya-conjecture-eigenvalues-disk.html
This document is subject to copyright. Apart from any fair dealing for the purpose of private study or research, no
part may be reproduced without the written permission. The content is provided for information purposes only.
Is it possible to deduce the shape of a drum from the sounds it makes? This is the kind of question that Iosif Polterovich, a professor in the Department of Mathematics and Statistics at Université de Montréal, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena involving wave propagation.
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 bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.
Pólya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. Until last year, the conjecture was known only for these cases. The disk, despite its apparent simplicity, 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 with disks. A disk is actually not a good shape for tiling.”
The universality of mathematics
In an article published in the mathematical journal Inventiones Mathematicae, the researchers show that Pólya’s conjecture is true for the disk, a case considered particularly challenging.
Though their result is essentially of theoretical value, their proof method has applications in computational mathematics and numerical computation. The authors are now investigating this avenue.
“While mathematics is a fundamental science, it is similar to sports and the arts in some ways,” Polterovich said.
“Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. And in many cases, beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.”
More information:
Nikolay Filonov et al, Pólya’s conjecture for Euclidean balls, Inventiones mathematicae (2023). DOI: 10.1007/s00222-023-01198-1
Citation:
Mathematicians prove Pólya’s conjecture for the eigenvalues of a disk, a 70-year-old math problem (2024, March 1)
retrieved 1 March 2024
from https://phys.org/news/2024-03-mathematicians-plya-conjecture-eigenvalues-disk.html
This document is subject to copyright. Apart from any fair dealing for the purpose of private study or research, no
part may be reproduced without the written permission. The content is provided for information purposes only.
Denial of responsibility! Techno Blender is an automatic aggregator of the all world’s media. In each content, the hyperlink to the primary source is specified. All trademarks belong to their rightful owners, all materials to their authors. If you are the owner of the content and do not want us to publish your materials, please contact us by email – [email protected]. The content will be deleted within 24 hours.