Problems

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…

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…

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 a measurable set Ω⊂ R^3, let E(Ω)=P(Ω)+frac12iint_Ω×Ωdx dy/|x-y|, where P is De Giorgi perimeter, and set V_*=5frac2-2^2/32^2/3-1≈3.51. The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove…

Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑_n σ_n,ε n |hat f(n)|^2 → deg f hold for Holder maps below the threshold? No. For every 0 < α < 1/3 there is an f ∈…

For the three-dimensional paraboloid P_3 over a prime field in which -1 is not a square, the Fourier extension operator maps L^2 to L^r for r > 176/51 = 3.45098…, improving the exponent by combining a bilinear approach with point-line…

Denjoy's 1932 theorem says a C^1+bv circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity ω weaker than Lipschitz, there is a C^1+ω…

The Smith-Ward theorem realizes the first k essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of k,…

Let n_1<n_2<… be a lacunary sequence of integers and f∈ L^2([0,1]) with nth Fourier partial sum f_n. Is there an absolute constant C>0 such that if | f-f_n|_2 ≪ (logloglog n)^-C then 1/N∑_k≤ Nf(α n_k)→∫_0^1 f for almost every α? A preprint…

A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of L_p(L_1) as a prominent remaining open…

The realisation problem asks which unital Banach algebras arise as the Calkin algebra B(X)/K(X) of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be…

Vinzant conjectured, in a form later restated by Bandeira, that the 4M-4 threshold for injective complex phase retrieval is sharp. Part (1) holds: for A ∈ C^N × M with N = 4M-5 and i.i.d. standard complex Gaussian entries, the phase…

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below (1+ε)n to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.

Let μ be a probability measure on the unit circle with Verblunsky coefficients α. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of α into components localized at…

The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero…