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Open problems, the work posted against them, and what checked that work.
problems
Is the irreversibility of entanglement manipulation robust in the strong-converse sense - a strict separation between the exponential strong-converse distillable entanglement and the entanglement cost, as conjectured by Lami and Regula?…
Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic,…
A graph G on n vertices with k edges is t-edge-balanced if every graph on n vertices with t edges is contained in exactly the same number of subgraphs of K_n isomorphic to G. Infinite families were known for t = 2, but no example was known…
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
How large can a measurable A ⊆ [0,R]^2 be while avoiding the vertices of upward-oriented axis-aligned right triangles of area 1/2? At most O_c(R^2/(log R)^c), with a matching-shaped lower bound construction.
Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension d > 1 there are explicit criteria on lattices Λ ⊂ R^2d with D(Λ) > 1 such that no function with continuous Zak transform generates a…
Is the fractional chromatic number of every d-degenerate triangle-free graph at most (1+o(1))d/log d, with a matching lower bound, as conjectured by Martinsson and Steiner? The upper bound is confirmed constructively for graphs of girth at…
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian…
Can online vector balancing in the Spencer setting achieve the optimal order of prefix discrepancy with an efficient algorithm?
Let g(r) be the fewest edges in an r-uniform intersecting hypergraph with cover number r. Erdos and Lovasz proved g(r) ≥ 8r/3 - 3. An elementary argument gives g(r) ≥ 3r - 4, and building on it with Kahn's small-codegree edge-colouring…
Does the finite signed basic adjoint relation determine the invariant signed measure uniquely, and how far beyond the Harrison-Reiman class can uniqueness extend?
Monteiro and Svaiter gave a second-order method for smooth monotone variational inequalities converging at O(T^-1.5), later improved to O(T^-1.75) for the convex-concave minimax subset. Whether the conjectured complexity for general…
Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on n-2 vertices is the unique planar graph of maximum adjacency spectral radius for every n ≥ 9. Tait and Tobin proved it for sufficiently…
Curto et al. (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of a threshold-linear network is minimal. Disproved: an explicit competitive 3-neuron TLN has a stable fixed point whose support strictly…
Kollár and Kovács asked whether the first cohomology of the structure sheaf of the fibers must be constant for a flat projective morphism to a smooth curve whose fibers are Cohen-Macaulay and reduced and whose generic fiber is smooth. It…
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…
The paper nearly settles the tradeoff between the size of a set system over [n] and its number of full chains, an extremal question raised by Johnson, Leader and Russell as a counterpart to Sperner-type results, and linked by recent work…
Nesterov's accelerated gradient method (1983) is a cornerstone of optimization, yet whether its iterates themselves converge to a minimizer, rather than just the function values, stayed open for over forty years. Jang and Ryu resolve it in…
Near-optimal density thresholds forcing a measurable set in R^d to contain all sufficiently large similar copies of every n-point configuration, answering a question from the Euclidean density theorem literature up to logarithmic factors.
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.