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problems
For the ergodic problem tfrac12|Dφ^ε|^2 + F(x) - εΔφ^ε = c(ε) on the torus, normalized by φ^ε(0) = 0, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit lim_ε → 0φ^ε always exists. It need not: there is a one-dimensional…
The near-quadratic Elekes-Ronyai expander conjecture over R predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of…
The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate…
Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the…
A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable…
Let m_1≤…≤ m_k and n be sufficiently large. If T is a tree on n vertices and G is the complete multipartite graph with vertex class sizes m_1,…,m_k, prove that R(T,G)≤ (χ(G)-1)(R(T,K_m_1,m_2)-1)+m_1.
Do the tree-level amplitudes A_n(1^-, 2^+, …, n^+) vanish identically, or can they be nonzero in half-collinear kinematics - and if nonzero, what is their all-n closed form?
Hall and Ho conjectured how the zeros of the heat-flow-evolved characteristic polynomial of a random matrix behave in the large-n limit. General cases are proved; in particular, for a complex Ginibre matrix the empirical measure of those…
Borsuk's conjecture asked whether every bounded set in R^n can be partitioned into n+1 subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in R^63 whose smaller-diameter subsets have at most 5 points, so…
Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.
Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.
Can the critical-exponent relation a + b = 1 at the jamming transition, observed numerically to high precision in the full replica-symmetry-breaking solution of hard spheres, be derived analytically from the scaling equations?
Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space overlineM_0,n of stable n-pointed rational curves has only real roots. True, with simple roots and strict interlacing between…
Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles…
Ross introduced S-perfect numbers, integers expressible as 1 + ∑ λ_j d_j over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd…
For a simple 3-polytope with at least three faces of size at least 7, must p_6 ≥ 39/20 + p_3/2 - p_5/4 - ∑_k ≥ 7 p_k? Five minimal ten-face counterexamples refute the printed inequality.
Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field u∈ W^1,∞( T^3; R^3), chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat…
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
Maz'ya and Shaposhnikova introduced a non-classical maximal operator M^diamond, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be…
What is the minimum asymptotic density δ_k of monochromatic k-term arithmetic progressions in every two-colouring of 1, …, n? The exact certificate gives δ_3 = 117/2192, matching the known 548-bead colouring.