Have you ever wondered if it is possible to deduce the shape of a drum from the sounds it makes? Iosif Polterovich is a professor at Université de Montréal. He works in the Department of Mathematics and Statistics. He enjoys asking certain types of questions. Recently, he achieved a major breakthrough with his international team. They used spectral geometry, which is a part of mathematics. Their work helps us understand how waves spread in physical phenomena.
Last summer, Polterovich and his team achieved a major milestone. His colleagues, Nikolay Filonov, Michael Levitin, and David Sher, were part of this effort. They proved a special case of a well-known conjecture. This conjecture in spectral geometry dates back to 1954. It was first introduced by George Pólya, a renowned Hungarian-American mathematician. This conjecture deals with estimating the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.
What is Pólya’s Conjecture?
Pólya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. However, until last year, the conjecture remained unproven for the case of a disk. Imagine an infinite floor covered with tiles of the same shape that fit together to fill the space, Polterovich explained. One can tile it with squares or triangles, but not with disks. A disk is actually not a good shape for tiling.
Polterovich and his team published an article in Inventiones Mathematicae. They proved that Pólya’s conjecture is true for disks. Researchers found this case very challenging. At first, this result seems theoretical. However, their proof method could be useful in computational mathematics and numerical computation. The researchers are now exploring this avenue further.
Polterovich remarked that while mathematics is often considered a fundamental science, it shares similarities with sports and the arts. Trying to prove a long-standing conjecture is like a sport, and finding an elegant solution is like an art. And in many cases, beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.
Breakthrough
This breakthrough by Polterovich and his team is a testament to the universality of mathematics and its ability to provide insights into various fields of study. Their work also inspires young mathematicians, especially teenagers interested in STEM education.
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To stay updated on the latest developments in the world of STEM education and be inspired by groundbreaking research such as Polterovich’s work on Pólya’s conjecture, visit ENTECH magazine’s website. You can also follow them on their social media profiles: Facebook, Twitter, Pinterest, and Medium.
Citation: Mathematicians prove Pólya’s conjecture for the eigenvalues of a disk, a 70-year-old math problem (2024, March 1) retrieved 4 March 2024 from https://phys.org/news/2024-03-mathematicians-plya-conjecture-eigenvalues-disk.html
