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?

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…

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.

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…

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.

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…

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

For every n≥2 the paper exhibits an n-dimensional K-polystable toric Q-Fano variety whose alpha invariant is exactly 2/2n+1, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between 1/n+1…

How well separated can a family of point-line pairs in the unit square be? For every ε > 0 there are arbitrarily large families (x_1,ℓ_1),…,(x_n,ℓ_n) in [0,1]^2 with x_i ∈ ℓ_i and dist(x_i,ℓ_j) ≥ n^-2/3-ε for all i ≠ j. Combined with…

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…

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

For odd k with gcd(n,k) = gcd(n+1,k) = 1, is N_k(n) ≡ ⌊ (k+1)/4 ⌋ pmod 2, where N_k(n) counts pairs 1 ≤ b_i ≤ (k-1)/2 with b_1 + b_2 ≥ (k+1)/2 and b_2 ≡ n b_1 pmod k? Conjectured by Chen and Gendron; its proof removes a conditional step in…

Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for…

The total Chern class of Sym^d(C^n) as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.

What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides 1 and apex angle 108^∘ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path…

Ehrhart conjectured that a full-dimensional compact convex body in R^n whose barycenter is its unique interior lattice point has volume at most (n+1)^n/n!. With the inequality itself settled, the remaining question was which bodies attain…

da Silva Machado and Seade conjectured that weighted homogeneous isolated hypersurface singularities are exactly those admitting a logarithmic vector field transverse to the link. True: for a reduced isolated hypersurface germ in C^n+1…

Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as ∇_K(z) = f(z)f(-z) for an integer polynomial f. Hartley proved it for negative amphicheiral knots and Ermotti, Hongler and Weber published the first…