Open-problem statements, with their sources attached.

Benchmark open subproblem: existence of a SIC-POVM in dimension $56$.

Show that every bounded linear operator T : H → H on a separable Hilbert space H of dimension at least 2 has a non-trivial closed T-invariant subspace: a closed linear subspace W of H, which is different from H and from {0}, such that T ( W ) ⊂ W. One needs the assumption that the dimension of H is at least 2 because otherwise any subspace would be either H or {0}.

Benchmark open subproblem: existence of a SIC-POVM in dimension $58$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $59$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $60$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $64$.

Fuglede's conjecture* in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

Benchmark open subproblem: existence of a SIC-POVM in dimension $68$.

Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators with shape parameter $\alpha > 0$, $\alpha \neq 1$.

The source asks for sufficiently smooth functions. This concrete version uses ContDiffOn ℝ 2 f I as a readable baseline regularity assumption; since the domain is the compact interval $[0,1]$, this also explains why no separate boundedness assumption is included here. The variants below record the unknown smoothness threshold more explicitly.

Fuglede's conjecture* in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.

Benchmark open subproblem: existence of a SIC-POVM in dimension $69$.

Variant of the Bézier-Bernstein Voronovskaja problem which treats "sufficiently smooth" as an eventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every $C^m$ function on $[0,1]$ should have the asserted asymptotic formula.

Benchmark open subproblem: existence of a SIC-POVM in dimension $70$.

Existence-only version of the eventual-smoothness variant. This separates the first part of the source problem, proving that the scaled sequence has some limit, from the stronger task of finding an explicit expression for that limit.

Benchmark open subproblem: existence of a SIC-POVM in dimension $71$.

Variant of the Bézier-Bernstein Voronovskaja problem with the required smoothness order itself left as an answer. Replacing (answer(sorry) : ℕ × ((ℝ → ℝ) → ℝ → ℝ)) by a concrete value lets one state the conjecture for a chosen regularity threshold.

Benchmark open subproblem: existence of a SIC-POVM in dimension $72$.

Benchmark open subproblem: existence of a SIC-POVM in dimension $75$.

Do SIC-POVMs exist in every finite dimension?