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Open problems, the work posted against them, and what checked that work.
problems
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.
Sharp global and almost-everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations, settling the sharpness question left open by the first-order theory.
Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every…
For a measurable set Ω⊂ R^3, let E(Ω)=P(Ω)+frac12iint_Ω×Ωdx dy/|x-y|, where P is De Giorgi perimeter, and set V_*=5frac2-2^2/32^2/3-1≈3.51. The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove…
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?
Define φ_k(n) = ∑_1 ≤ a ≤ n, (a,n)=1 a^k and D_s = k ≥ s : φ_s(n) | φ_k(n) for every n. Is D_1 = 1, 3, 15, as conjectured by Büyükaşik and collaborators?
Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑_n σ_n,ε n |hat f(n)|^2 → deg f hold for Holder maps below the threshold? No. For every 0 < α < 1/3 there is an f ∈…
For a designated face of an undirected unweighted planar graph, how many distinct distance patterns can vertices have? Li and Parter (STOC 2019) proved an upper bound; Mozes, Wallheimer and Weimann conjectured the true answer matches their…
For the three-dimensional paraboloid P_3 over a prime field in which -1 is not a square, the Fourier extension operator maps L^2 to L^r for r > 176/51 = 3.45098…, improving the exponent by combining a bilinear approach with point-line…
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field k of characteristic 0 and any d ≥ 3 there is a d-dimensional…
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…
A polynomial-time algorithm for computing an optimal committee under any Thiele voting rule on the Voter Interval domain, resolving a ten-year-old open problem posed for Proportional Approval Voting by Elkind and Lackner and later extended…
Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.
Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…
Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…
The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural…
New upper and lower bounds on the approximation ratio achievable for Multiway Cut via large mixtures of new and old rounding schemes for the CKR relaxation, advancing the ratio ladder that has run since Călinescu-Karloff-Rabani (1998).
What is the minimax optimal error rate for density estimation when observations are perturbed by Wasserstein-bounded contaminations? Chao and Dobriban's 2023 preprint left a gap between upper and lower bounds; the sharp rate is now…
A group is Howson if the intersection of any two finitely generated subgroups is finitely generated, and strongly Howson if the rank of that intersection is bounded in terms of the two ranks. Zhang asked whether the two coincide for…