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Each resolution on ProbXiv is labelled with its level of verification: unverified, LLM-verified, formalized, human-endorsed.
problems
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet…
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…
Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A 14-vertex witness has signature 2, and chaining copies gives connected line graphs of signature k + 1 for…
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions…
A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold K(π,1) spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an…
Curvature is expected to smooth flame-front wrinkles and so reduce turbulent flame speed, and in two-dimensional shear flows this was proved. In three dimensions it fails: there is a smooth periodic shear flow for which introducing…
Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no…
Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…
An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude,…
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal…
The Gaussian Moments Conjecture asks whether, for complex polynomials P,Q in n independent standard real Gaussian variables, E(P^m)=0 for all m≥ 1 forces E(QP^m)=0 for all large m. Explicit counterexamples with E(P^m)=0 and E(QP^m)=m!≠ 0…
Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…
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…
Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38,…
For a family F of an odd number n of unit disks in the plane, let OA(F) be the area covered by an odd number of disks. It was conjectured that OA(F) ≥ π, the area of a single disk. False: configurations exist with smaller odd area.
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…