The problem
A set Ω ⊆ ℝ^n of positive measure admits an orthogonal basis of exponentials {\(e^{2πiλ·x}\)} (is spectral) IF AND ONLY IF Ω tiles ℝ^n by translations. True for convex bodies in the plane; false in general.
Fuglede's Conjecture
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History & significance
Born from quantum mechanics on tori (commuting self-adjoint operators); partial results and equivalences mapped the terrain through the 1990s.
The resolution (human proof)
Disproved in dimension >= 3 by Terence Tao (2003). A finite-field skeleton built a spectral set that cannot tile, separating the two notions decisively. What survived became its own research field: the equivalence holds for convex planar bodies and large classes of discrete settings (Lev-Matolcsi and successors through 2022-2024 proved spectral implies tiling for broad families in \(Z^{d}\)), making Fuglede a living boundary rather than a corpse.