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Open problems, the work posted against them, and what checked that work.
problems
Yun, Sra and Jadbabaie posed as a COLT 2021 open question whether, for well-conditioned symmetric matrices, the operators encoding the expected iterate of single-shuffle SGD, random-reshuffle SGD and gradient descent on a quadratic finite…
For the integral I(h) over the unit interval and the torus, the paper gives the three-term Laurent polynomial f(x,z)=(1-z^-1)((1-x)+xz) with I(f^n)=0 but I(z^-1f^n)=(-1)^n-1/(n+1)≠0. This disproves the xz-conjecture with one interval and…
For every smooth positive density f on R^d, must the Fisher information t ↦ I(f * γ_t) be log-convex along the heat flow?
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…
Pavez-Signe (2024) conjectured a Dirac-type condition for spanning H-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger…
Moraga conjectured, and Kollár and Zhuang recorded, an odd-dimensional extension of the rank bound for faithful abelian p-group actions on smooth Calabi–Yau varieties. The paper disproves it.
Let k≥ 3 and A be an additive basis of order k. Does there exist a constant c=c(k)>0 such that if r(n)≥ clog n for all large n (where r(n) counts representations of n as a sum of at most k elements of A) then A must contain a minimal basis…
The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is |A||A+B+C| <= |A+B||A+C|. Disproved by a four-dimensional zonotope generated by a 2-modular matrix together with two segments. Several related local…
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…
In the square Gaussian binary MIMO model y = √ρ/N Hx^⋆ + w, exhaustive maximum-likelihood detection recovers x^⋆ once ρ > 2log N, while sphere decoding at that threshold scale costs expΘ(N/log N). Whether any polynomial-time detector…
Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?
For every real ξ>0 the sequence of integer parts [ξ 7^n], n=0,1,2,…, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~7.
Prim-Dijkstra routing interpolates between a minimum spanning tree and a shortest-path tree, and has been used and improved in VLSI physical design since the early 1990s, but the complexity of the terminal-only Manhattan decision problem…
The paper proves a closed-string remodeling statement for the affine binary dihedral Calabi–Yau orbifold threefold, a target outside the toric setting of the Bouchard–Klemm–Mariño–Pasquetti remodeling conjecture, replacing the toric mirror…
What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound h_2(δ) + δ log(d^2 - 1) up to δ = 1 - d^-2, conjectured by Wilde, is…
The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on n leaves. Proved: it is (2/3 + o(1))binomn4, by reducing…
Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general…
Regev and Stephens-Davidowitz conjectured that Z^n maximizes the Gaussian mass Θ_L(t) = ∑_x ∈ L e^-t|x|^2 over stable lattices for every t > 0. The sharp inequality holds for every integral unimodular lattice of rank n ≤ 32, with equality…
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…
The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more…