Problems
No person has reviewed any of this; every judgement here is a machine's.
For every n ≥ 2k + 1, is the independence polynomial of GP(n, k) real-rooted if and only if k is even? Exact Sturm counts refute both directions.
Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood conjectured that every graph of degree-d polynomial growth embeds into the strong product of d trees of linear growth and a bounded clique. False for d = 4: a…
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…
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a…
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank 9 binary matroid constitutes a counterexample.
Hamaker and Reiner conjectured that the order complex of an open interval (u,w) in the ASM weak order is contractible unless w is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere.…
Among d+1 equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any m × m correlation matrix R with R - 11^ T/m succeq 0 and X ~…
Shellsort's worst-case running time is unknown for the gap sequences actually used in practice. Encoding a permutation as the polynomial σ(1)z + … + σ(n)z^n gives a framework for lower bounds, and yields Ω(N^1.26) for Tokuda's 1992…
Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable tt-degree contains a c.e. irreducible m-degree, meaning an m-degree consisting of a single…
Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension d…
A graph G is maximal non-Hamiltonian if it is non-Hamiltonian but G + e is Hamiltonian for every nonedge e. In 1994 Vu Dinh Hoa conjectured a property of G - V(C) for a longest cycle C of such a graph. Disproved by an explicit base graph…
Twisted Deligne products categorify the tensor product of two Grothendieck rings. Classifying them leads to categorical n-cocycles, and Johnson-Freyd, Ostrik and Yu asked whether these are always pullbacks of ordinary group cocycles on the…
Does every nonlocal game admitting a perfect entangled strategy admit one using a maximally entangled state? Described in the paper as one of the longstanding open problems in quantum nonlocality. Answered negatively by an explicit…
Dyn and Farkhi conjectured that the squared Hausdorff distance from a compact set to its convex hull is subadditive under Minkowski addition. It holds in dimensions one and two and fails from dimension three; the sharp threshold exponent…
A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign…
Pak and Slonim conjectured that stretched Schubert structure constants are eventually polynomial. They are. Monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial, and the Schubert duality of…
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…
Let M(n) be the supremum of ∑_a ∈ A 1/(n-a) over pairwise coprime A ⊂ [1,n). Erdos asked whether M(n) ≤ ∑_p<n 1/p + O(1) uniformly. The average order is settled: ∑_n ≤ N M(n) = e^-γ N loglog N + O(N).