Problems
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Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent…
Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?
Mossel and Peres showed that a single-variable function admits a finite-automata Bernoulli factory exactly when it is rational, and their Theorem 2.9 claimed the same extension to multivariable functions. The multivariable claim is false,…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Douglas and Yang attach to each nonzero vector x of a quasinilpotent operator T a local resolvent-growth exponent k_x, giving the power set Λ(T) = k_x : x ≠ 0. Ji and Zhang asked whether 1 always belongs to Λ(T). It does, for every…
Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for…
Along the heat flow, do the successive time derivatives of the entropy of X + √t Z alternate in sign, as conjectured by Cheng and Geng? An explicit measure on R has a fifth derivative with the forbidden sign.
For the ergodic problem tfrac12|Dφ^ε|^2 + F(x) - εΔφ^ε = c(ε) on the torus, normalized by φ^ε(0) = 0, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit lim_ε → 0φ^ε always exists. It need not: there is a one-dimensional…
The near-quadratic Elekes-Ronyai expander conjecture over R predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of…
The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate…
Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the…
A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable…
Let m_1≤…≤ m_k and n be sufficiently large. If T is a tree on n vertices and G is the complete multipartite graph with vertex class sizes m_1,…,m_k, prove that R(T,G)≤ (χ(G)-1)(R(T,K_m_1,m_2)-1)+m_1.
Do the tree-level amplitudes A_n(1^-, 2^+, …, n^+) vanish identically, or can they be nonzero in half-collinear kinematics - and if nonzero, what is their all-n closed form?
Hall and Ho conjectured how the zeros of the heat-flow-evolved characteristic polynomial of a random matrix behave in the large-n limit. General cases are proved; in particular, for a complex Ginibre matrix the empirical measure of those…
Borsuk's conjecture asked whether every bounded set in R^n can be partitioned into n+1 subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in R^63 whose smaller-diameter subsets have at most 5 points, so…
Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.
For an extension G = A rtimes B of elementary abelian p-groups with a ∈ A satisfying C_B(a) = 1, must H = ⟨ a, B⟩ satisfy rank(Z(H) ∩ H') ≤ rank(B)? An explicit extension violates the bound.
Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.
Bounds the weighted sum ∑ 1/(a log a) taken over primitive sets of integers (sets where no element divides another).