Problems

Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.

For a simple 3-polytope with at least three faces of size at least 7, must p_6 ≥ 39/20 + p_3/2 - p_5/4 - ∑_k ≥ 7 p_k? Five minimal ten-face counterexamples refute the printed inequality.

Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in…

Lassak conjectured that a reduced planar convex body of thickness Δ has area at most (π/4)Δ^2, the value for the disc. False: an explicit reduced body of thickness 1 has area 0.786215… > π/4 = 0.785398…, given by a closed-form support…

Sabok asked whether the compact convex set S'(X) attached to a separable metric space of diameter at most one is always a simplex, and whether S'(U_1) is the Poulsen simplex. Both answers are negative, with obstructions already visible for…

Kusner conjectured in 1983 that the maximum number of points in R^n that are pairwise at ℓ_p-distance one is exactly n+1 for every 2 < p < ∞, as in the Euclidean case. False: an explicit configuration of n+2 equilateral points exists for…

Conjectured upper bound on how many pairs among n points in the plane can be exactly one unit apart.

For an arrangement of n hyperplanes in P^3_C with ℓ intersection lines and p intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects p - ℓ + n + 2 ≥ 0. An explicit arrangement built from…

For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, are stable tame isomorphism, quasi-isomorphism, or derived Morita equivalence algorithmically decidable? All three are…

Wegner conjectured in 1965 that every finite family R of axis-parallel rectangles satisfies τ(R) ≤ 2ν(R) - 1, where τ is the minimum number of piercing points and ν the largest pairwise-disjoint subfamily. False, by an explicit…

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…

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

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.

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

Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since…

Let U_n⊂ R^n be the difference polytope of a regular n-simplex such that U_n circumscribes a sphere of diameter 1. Then every set of diameter 1 in R^n is covered by a rotated copy of U_n.

Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective…

Kinoshita conjectured that every embedded projective plane in S^4 is reducible. False: an irreducible embedded projective plane exists in S^4. The construction also answers both parts of Problem 4.37 of the Kirby problem list.

Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension 36 as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-18 modular forms for Γ_0(24), shows the…

Ziegler proved every simplicial d-dimensional 0/1-polytope has at most 2d vertices, and asked whether attaining 2d vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for d ≤ 6; open since ~2000.