Mathematicians prove Pólya's conjecture for the eigenvalues of a disk, a 70-year-old math problem

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Abstraction: 70-year-old spectral geometry conjecture about disk eigenvalues finally proven

Key points:

Pólya's conjecture (1954) concerns estimating frequencies of a round drum — eigenvalues of a disk
Pólya himself proved the conjecture in 1961 only for plane-tiling domains (triangles, rectangles); disks cannot tile a plane
Proof achieved by Polterovich, Filonov, Levitin, and Sher; published in Inventiones Mathematicae (2023)
Key difficulty: the disk's non-tiling nature made standard approaches inapplicable for 70 years
Result is primarily theoretical but proof method has applications in computational mathematics and numerical computation
Connected to the classic question "can you hear the shape of a drum?" in spectral geometry

Connections: George Polya · Iosif Polterovich · Spectral Geometry · Eigenvalues