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	+}