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.

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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	}