Problems
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The paper constructs an exact cluster F⊆Z^2 of cardinality 8 with full affine span and an F-tiling whose orbit closure contains no 1-periodic F-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows.…
For an odd prime p, do Sun's normalized trigonometric permanents satisfy s_p < 0 ⇔ p ≡ 5 pmod12 and s'_p < 0 ⇔ p ≡ 7 pmod 8? Exact computation at p = 29 refutes both sign laws.
For a projective variety X with at worst Gorenstein canonical singularities whose stringy E-function E_st(X; u, v) is a polynomial, all stringy Hodge numbers h^p,q_st(X) are non-negative. (Batyrev 1998, Conjecture 3.10.)
We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low…
Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are L^p-integrable for 1 < p <…
R(c) = 40c+41 for every c ≥ 2 such that c+1 is divisible by 3, 4, 5, or 7 (covering ≈ 66% of all c); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases p ≥ 89, all smaller primes settled by SAT. Twenty-eight…
Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting q-colourings on planar graphs is NP-hard for every constant q ≥ 4, and…
Sampling a nearly uniform Eulerian tour of a directed Eulerian multigraph was stuck at mn-type running times coming from arborescence sampling. A randomized algorithm achieves widetildeO(m^3/2) worst case, breaking that barrier on sparse…
Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.
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…
Let X be a set of cardinality ℵ_ω and f a function from the finite subsets of X to X such that f(A)not∈ A for all A. Must there exist an infinite independent Y⊆ X, i.e. with f(B)not∈ Y for all finite B⊂ Y? Claimed resolution: the positive…
Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen…
Espuny Diaz, Lichev and Wesolek conjectured that a Dirac-type minimum degree condition forces Hamiltonicity in spanning subgraphs of cycle powers. Asymptotically true: for every ε > 0 and all large k, any spanning subgraph of the kth power…
Carefully designed stepsize schedules alone accelerate plain gradient descent beyond its textbook O(1/T) rate, without momentum. Whether they can reach the optimal O(T^-2) was open. A lower bound of Omega(T^-1.9319) for last-iterate…
Let H(n) be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than n. With k_1 = 1 and k_n = ⌊ n/2 ⌋ + k_⌊ n/2 ⌋ + k_⌈ n/2 ⌉, prove H(n) ≥ c k_n for some constant c > 1, already for n…
Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal…
Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, Λ^m(P^m) = 0 for all large m forces Λ^m(P^m Q) = 0 for all large m. Refuted by an explicit five-variable counterexample.
What is the largest possible measure of a subset of a radius-R disk in R^2 containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives M(R) ≪ R^1/2; with Sárközy's lower construction, M(R) = R^1/2 +…
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to X, the cone theorem holds for projective log…
Djament asked whether a Grothendieck category satisfying suitable finiteness and exactness conditions must be equivalent to a module category. In the locally noetherian case the answer is no: there is a Grothendieck category with a…