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Pólya's conjecture for Dirichlet eigenvalues
Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.
| 1 | { |
| 2 | "domain": "mathematics", |
| 3 | "topics": [ |
| 4 | "analysis", |
| 5 | "mathematical-physics" |
| 6 | ], |
| 7 | "type": "construction", |
| 8 | "also": [ |
| 9 | "proof" |
| 10 | ], |
| 11 | "source": { |
| 12 | "key": "frontiermath-open", |
| 13 | "name": "FrontierMath: Open Problems", |
| 14 | "url": "https://epoch.ai/frontiermath/open-problems/polya-laplacian-spectrum", |
| 15 | "id": "polya-laplacian-spectrum", |
| 16 | "curator": "Epoch AI", |
| 17 | "poser": "George Pólya", |
| 18 | "status": "unsolved", |
| 19 | "as_of": "2026-10-20", |
| 20 | "catalog": "0521c19bbd38" |
| 21 | }, |
| 22 | "review": true, |
| 23 | "approval": "first-time" |
| 24 | } |