Problems

Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are L^p-integrable for 1 < p <…

Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its L^2 space? No. The Cantor measure with base b admits no Fourier frame for any odd integer b > 1,…

For a positive projection P on a Dedekind complete Banach lattice whose largest central operator below P is α id, Wickstead conjectured α must be 0 or 1/n for some natural n, and proved the finite-dimensional case. The paper proves the…

Let p be a complex polynomial of degree n ≥ 2 whose zeros all lie in the closed unit disk. Then for every zero a of p, there exists a critical point ζ of p such that |ζ-a| ≤ 1. This is the standard Sendov statement and exactly matches the…

Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with…

For s ∈ (1/4,1) and any degree, the only W^s,1/s-minimizers among maps S^1 → S^1 are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4…

Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as p → 1^+ or δ → 0^+. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.

Nazarov conjectured that for s ∈ (1, 3/2) the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under u ↦ |u| when u changes sign. Proved and substantially generalized, with the same conclusion for the…

Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several…

If A generates a bounded C_0-semigroup on a Hilbert space and has dense range, does A^-1 also generate a bounded C_0-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator…

Crouzeix conjectured in 2004 that for every square complex matrix A and every polynomial p, lVert p(A)rVert ≤ 2 max_z ∈ W(A) |p(z)|, where W(A) is the numerical range of A - that is, the numerical range is a 2-spectral set. Crouzeix proved…

For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for…

A countable discrete group with a proper length function carries a natural spectral triple on its reduced group C*-algebra. A well-studied question in non-commutative metric geometry asks whether the associated Connes pseudo-metric always…

Pełczyński's duality between strictly singular and strictly cosingular operators fails without weak compactness. Beanland asked, in work with Androulakis and later on MathOverflow, for the separable-range case: the paper answers it…

Let L^nf be the Lagrange interpolation polynomials of a continuous f on the Chebyshev nodes. Prove that, for any closed A⊆ [-1,1], there exists a continuous function f such that A is the set of limit points of L^nf(x).

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Is there an entire non-zero function f:C→ C such that, for any infinite sequence n_1<n_2<…, the set z: f^(n_k)(z)=0 for some k≥ 1 is everywhere dense? The literal question is trivial for polynomials, so the claims address the…

Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar…

For unit-modulus complex numbers z_i, let p_n(z)=∏_i≤ n(z-z_i) and M_n=max_|z|=1|p_n(z)|. Erdős's prize question: is there c>0 with ∑_k≤ n M_k > n^1+c?

Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives κ ≤ 1 + √1 - ℜ⟨ E_1…