Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
11 problems
Shokurov's global index conjecture, in the setting of foliations. Proved for foliations in dimension at most three, which also answers a question of Liu, Meng and Xie in dimension three.
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.
How dense can a sphere packing in R^n be as n → ∞? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.
The kissing number in 19 dimensions is at least 11948, improving the Cohn-Li bound by 256, via a binary code of length 19 and minimum distance 5 fed through the Cohn-Li odd-sign construction.
For every δ > 0 and infinitely many n there is a set of n lines in the plane with no intersecting quadruple such that every subset of size at least n^4/5+δ contains three concurrent lines. This improves the bound for a dual form of a…
How few vertices can a triangulation of RP^5 have? The paper presents a 6-dimensional centrally symmetric simplicial polytope whose antipodal boundary quotient gives a 24-vertex triangulation, far below previous constructions in the…
For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three…
For the least cutoff c(n) after which every k occurs as the number of homothetic cubes in a decomposition of the unit n-cube, is c(n) ≫ n^n? The Lean proof shows c(n) = o(n^n) along odd dimensions.
How large can the difference between the largest and second-largest distance multiplicities be among n planar points?
The lattice triangle problem asks which rational triangles unfold to Veech surfaces; in the hard obtuse window it is conjectured that none do. Via an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, the paper rules out…
We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-n polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary…