Problems
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The group version of the Matrix Spencer conjecture holds: for every finite group G there are signs ε ∈ ± 1^G with |∑_g ∈ G ε_g ρ(g)| ≤ C√|G|, where ρ is the left regular representation and C is universal.
For an unkilled Levy process ξ drifting to +∞ with all positive exponential moments, let I_ξ = ∫_0^∞ e^-ξ_t dt and X_ξ = 1/I_ξ. Bertoin and Yor proved X_ξ is moment-determinate when ξ has no positive jumps and conjectured that this…
For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For…
Can any single-pass semi-streaming algorithm beat the naive greedy 1/2-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is…
What is the optimal competitive ratio for online vertex cover when edges arrive one at a time? The paper proves a tight factor-2 lower bound via a reduction in the blueprint framework of Assadi, Jiang and Xiang, closing the gap left by…
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.
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…
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?
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…
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.
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…
Can a noetherian ring have a local cohomology module whose support is not closed - equivalently, one with infinitely many minimal primes? Huneke and Lyubeznik asked; the paper constructs such rings, so the answer is yes.
An asymptotic formula for p(k), the limiting probability that a random permutation has an invariant set of size k: it is asymptotically k^-δ(1+o(1)) times a smooth positive function, sharpening a line of estimates running through…
Denjoy's 1932 theorem says a C^1+bv circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity ω weaker than Lipschitz, there is a C^1+ω…
In the list update problem, is the simple transposition rule optimal under IID requests? The question traces to Rivest's 1976 study of self-organizing lists. The paper proves transposition is within a small constant factor of the optimal…
Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and…
Treglown conjectured, in a complementary form, that for every positive integer k every digraph D with mind^+(v), d^-(v) ≤ k-1 for all v has an equitable acyclic k-colouring. This implies the acyclic colouring versions of the…
Whether perfectly complete quantum key agreement can be built from quantumly secure one-way functions in a black-box way. It cannot: for any protocol where Alice and Bob exchange only classical messages, make at most q_A and q_B quantum…
A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of L_p(L_1) as a prominent remaining open…