Problems
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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…
Let U_n⊂ R^n be the difference polytope of a regular n-simplex such that U_n circumscribes a sphere of diameter 1. Then every set of diameter 1 in R^n is covered by a rotated copy of U_n.
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.
If n planar points have no four concyclic, must some point determine (1 - o(1))n distinct distances? Failing that, can one always force more than (1/3 + c)n?
Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective…
How large can the difference between the largest and second-largest distance multiplicities be among n planar points?
Kinoshita conjectured that every embedded projective plane in S^4 is reducible. False: an irreducible embedded projective plane exists in S^4. The construction also answers both parts of Problem 4.37 of the Kirby problem list.
Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension 36 as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-18 modular forms for Γ_0(24), shows the…
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…
Ziegler proved every simplicial d-dimensional 0/1-polytope has at most 2d vertices, and asked whether attaining 2d vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for d ≤ 6; open since ~2000.
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
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…
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.
What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides 1 and apex angle 108^∘ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path…
Ehrhart conjectured that a full-dimensional compact convex body in R^n whose barycenter is its unique interior lattice point has volume at most (n+1)^n/n!. With the inequality itself settled, the remaining question was which bodies attain…
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…
da Silva Machado and Seade conjectured that weighted homogeneous isolated hypersurface singularities are exactly those admitting a logarithmic vector field transverse to the link. True: for a reduced isolated hypersurface germ in C^n+1…
Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as ∇_K(z) = f(z)f(-z) for an integer polynomial f. Hartley proved it for negative amphicheiral knots and Ermotti, Hongler and Weber published the first…
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…