Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
60 problems
Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is Var⟨ MX, X⟩ ≤ C E|∇⟨ MX, X⟩|^2 for every symmetric M?
Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.
Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and…
Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the…
Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below 1/4. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen…
Shokurov's global index conjecture, in the setting of foliations. Proved for foliations in dimension at most three, which also answers a question of Liu, Meng and Xie in dimension three.
Does a general pencil of plane cubics over C have exactly 12 common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.
If origin-symmetric convex bodies K, L ⊂ R^n satisfy vol_m(K ∩ E) ≤ vol_m(L ∩ E) for every m-dimensional subspace E with 1 < m < n, does vol_n(K) ≤ vol_n(L) follow? Answered affirmatively for subspace dimensions m = 2 and m = 3.
What is the maximum volume of a convex body in R^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.
Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space overlineM_0,n of stable n-pointed rational curves has only real roots. True, with simple roots and strict interlacing between…
Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles…
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.
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
How dense can a sphere packing in R^n be as n → ∞? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.
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.
A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of…