https://phys.org/news/2024-03-mathematicians-plya-conjecture-eigenvalues-disk.html Phys.org Topics * Week's top * Latest news * Unread news * Subscribe [ ] Science X Account [ ] [ ] [*] Remember me Sign In Click here to sign in with or Forget Password? Not a member? Sign up Learn more * Nanotechnology * Physics * Earth * Astronomy & Space * Chemistry * Biology * Other Sciences * Medical Xpress Medicine * Tech Xplore Technology [INS::INS] * * share this! * 253 * Twit * Share * Email 1. Home 2. Other Sciences 3. Mathematics * * * --------------------------------------------------------------------- March 1, 2024 Editors' notes This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: fact-checked trusted source proofread Mathematicians prove Polya's conjecture for the eigenvalues of a disk, a 70-year-old math problem by Beatrice St-Cyr-Leroux, University of Montreal Mathematicians prove Polya's conjecture for the eigenvalues of a disk, a 70-year-old math problem Credit: Inventiones mathematicae (2023). DOI: 10.1007/s00222-023-01198-1 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 Universite de Montreal, 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 Polya. The conjecture bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk. Polya 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 Polya'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, Polya's conjecture for Euclidean balls, Inventiones mathematicae (2023). DOI: 10.1007/ s00222-023-01198-1 Provided by University of Montreal Citation: Mathematicians prove Polya's conjecture for the eigenvalues of a disk, a 70-year-old math problem (2024, March 1) retrieved 2 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. --------------------------------------------------------------------- Explore further The Riemann conjecture unveiled by physics --------------------------------------------------------------------- 253 shares * Facebook * Twitter * Email Feedback to editors * Featured * Last Comments * Popular Saturday Citations: Will they or won't they? A black hole binary refuses to merge. 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