Problems

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is Var⟨ MX, X⟩ ≤ C E|∇⟨ MX, X⟩|^2 for every symmetric M?

Swinnerton-Dyer (1981) proved R-equivalence trivial on smooth cubic surfaces over p-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic…

Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the…

A cyclic meander induces a cyclic permutation on its 2n marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.

For an inclusion-free hypergraph on n vertices, a weight assignment w:[n]→[d] is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least n∑_j=0^d-1 j^n-1,…

Norine conjectured that every red-blue edge-colouring of the n-dimensional hypercube Q_n in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level…

Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL…

Gill introduced the probabilistic automatic complexity A_P(w) of a string: the least number of states of a probabilistic finite automaton for which w is the unique most probably accepted string of its length. He asked whether A_P is…

Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below 1/4. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen…

Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.

Let A(x) count n ≤ x such that every prime p | n has a divisor d > 1 of n with d ≡ 1 pmod p. Erdos asked whether A(x)/x = exp(-(c+o(1))√log xloglog x). It does, with c = 1/(2√log 2).

Regts and Sevenster conjectured that a complex-valued graph parameter f with f(∅)=1 has exponentially bounded edge-connection rank precisely when it is a mixed partition function. The paper proves it, with the numbers of even and odd…

Let D_q(n) be the largest possible least degree of a polynomial omitted by a non-covering family of n distinct-modulus congruence classes in F_q[x]. What is its asymptotic size? The answer is D_q(n) = n/q-1 + O_q(1).

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…

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,…

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…

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.

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,…