Problems

A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily…

For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.

For a differential poset P, must the weighted 2-multichain series M_P,2(q) be a rational multiple of F_P(q)^2, the square of its rank generating series?

Is the zero forcing number of every connected graph with maximum degree 3 at most its independence number plus one? A connected 24-vertex subcubic graph with independence number 9 and zero forcing number 11 refutes this 2017 TxGraffiti…

Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a…

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?

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.

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…

Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives FDR > 0.0104 at nominal level α = 0.01.

Is a Werner state that is not one-copy distillable ever two-copy distillable? The first open rung of the NPT bound-entanglement ladder, open since 2000.

If G is connected, cubic and diamond-free, must the zero-forcing number satisfy Z(G) ≤ γ(G) + 2? A connected cubic triangle-free 14-vertex graph has Z = 7 and γ = 4.

Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups Γ_1 and Γ_2, both ICC and property (T), are…

Must every r-differential poset have at least as many elements in each rank as Y^r, the r-th Cartesian power of Young's lattice? For r = 3 the new construction has fourth-rank size 50 against 51 for Y^3.

The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated…