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Open problems, the work posted against them, and what checked that work.
problems
What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation widehat P(A) satisfies…
For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core…
Does the Hodge bundle Ω_g over the moduli stack of genus g ≥ 2 curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.
Does a general pencil of plane cubics over C have exactly 12 common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.
Amdeberhan, Shareshian and Stanley showed a function from the theory of partition Eisenstein series counts alternating permutations with a given record partition, and asked whether a similar theory exists for record compositions,…
If origin-symmetric convex bodies K, L ⊂ R^n satisfy vol_m(K ∩ E) ≤ vol_m(L ∩ E) for every m-dimensional subspace E with 1 < m < n, does vol_n(K) ≤ vol_n(L) follow? Answered affirmatively for subspace dimensions m = 2 and m = 3.
Let S(x) count ordered pairs (a,b) with a+b ≤ x and σ(a)+σ(b) = σ(a+b). Erdos asked whether S(x) ~ cx. The opposite extreme holds: for every R > 0, S(x)/(x(log x)^R) → ∞, so the count beats every fixed logarithmic scale.
For the released four-terminal planar acyclic single-source unsplittable-flow gadget, require one unsplittable routing to be no more expensive than a prescribed fractional routing under each of m strictly positive full-demand…
Determine the leading asymptotic of the largest eigenvalue of the N-Majorana quartic SYK Hamiltonian as N → ∞. The preprint proves λ_1/√N → 4∫_0^∞ g_0(t)^4 dt ≈ 0.32504 almost surely, via the limiting free energy at every fixed positive…
Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)…(n+k))>n^1-ε is at least 1-η, where P(m) is the greatest prime divisor of m? A short argument via the Matomäki-Radziwiłł theorem establishes the…
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample.…
Two free ergodic measure-preserving flows whose L^1 full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le…
Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between 2|E|-|V| and 2|E| has period exactly 2, generalizing the middle rung of Levine's devil's staircase…
For fixed d, can every d-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than α_GW? A rounding achieving α_GW + 2^-O(d) answers yes.
A precise asymptotic formula for the number of n × 4t partial Hadamard matrices in the regimes t/n^3 → ∞ and t/n^3 → Θ, reaching the cubic regime that previous approaches (de Launey-Levin and successors) could not.
Does there exist an integer polynomial f of degree at least two and a set A ⊆ Z such that every integer has a unique representation n = a + f(k)? A manuscript claims the thirteenth powers admit a tiling complement.
Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension,…
Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and…
Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent…
Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?