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problems
Whether the real Kalton-Peck space Z_2 is isomorphic to its hyperplanes. It is not: no hyperplane of Z_2 is isomorphic to Z_2, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.
A question of Averkov, Hofscheier and Nill on whether the Ehrhart h^*-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and…
What is the largest A⊆1,…,N such that all subset sums ∑_n∈ S1/n (over S⊆ A) are distinct?
A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of…
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.
Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an n-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed s ≥ 3 and L ≥ 2s+2 and…
Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line,…
Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant α_⋆ ≈ 0.8330786. Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional…
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian…
For a perfect field k and a representation-infinite finite-dimensional k-algebra A, the Auslander–Reiten quiver of A has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for…
Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable, while making only finitely many errors almost surely. Answered affirmatively…
For the switch-walk-switch lamplighter walk on Z_2 wr T_d, prove the sharp asymptotic p_2n(e,e) = ρ_d^2n exp[-(π^2 (log(d-1))^2 + o(1)) n/log^2 n] with ρ_d = frac2√d-1d.
For a continuous bounded-variation path with signature g, logarithmic signature l and increment v, the modified Lyons–Sidorova conjecture predicts the structure of g when R(l)=∞. The paper proves it: g=1 when v=0, and otherwise a prefix α…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
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…
Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in n^2 log n steps, the conjectured optimal rate,…
Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime p ≡ 3 pmod 4 it equals ⌊ (p-2)/3 ⌋^2 x, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating…
For an irreducible crystallographic root system of rank r with Coxeter number h, the paper proves that Au's normalized Witten zeta function has a simple pole at 2/h and evaluates its residue in closed form in terms of the Cartan…
For deterministically minimizing a convex 1-Lipschitz function on the d-dimensional ball using only exact function values, the query complexity sat between Ω(d) and O(d^2 log^2 d) since 1996. The paper proves a near-quadratic lower bound…
Can the k-distinct language - words over [n] of length at most k with no repeated symbol - be recognized by an acyclic NFA of size c^k n^O(1) for some c < 4? A construction of size 2^1.96992k n^O(1) < 3.918^k n^O(1) answers yes.