combinatoricsAug 26, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\…
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geometry-topologyJul 27, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Bellman's problem for general regions remains open
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geometry-topologyAug 14, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Kusner conjectured in 1983 that the maximum number of points in $\mathbb{R}^n$ that are pairwise at $\ell_p$-distance one is exactly $n+1$ for every $2 < p < \infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polyno…
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combinatoricsMay 1, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
How dense can a sum-free subset of the lattice cube $\{1,\dots,n\}^d$ be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice $\{x : 1 \le L(x) < 2\}$ for a linear map $L$, previously known only for $d \le 4$. Proved for all $d$. The paper also shows the same phenomenon fails if the cube is repl…
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analysisAug 5, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
For every $n>2$, the paper constructs a bounded map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$, smooth on $B^n\setminus\{0\}$ but discontinuous at the origin, together with an antisymmetric potential
$$
\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)
$$
such that
$$
-\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr)
=
\Omega\cdot|\nabla U|^{n-2}\nabla U
\qquad\text{in }D'(B^n).
$$
Moreover, the potential satisfies the…
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algebraJul 24, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
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mathematical-physicsAug 9, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The sam…
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algebraAug 4, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
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 $d \geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails a…
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mathematical-physicsAug 3, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling e…
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combinatoricsMay 13, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
For $A \subset \mathbb{F}_p$ of density $1/2$, call $A$ almost affine invariant under $\varphi(x) = ax+b$ if $|A \triangle \varphi(A)| = o(p)$. Problem 90 asks for the threshold $K$ below which $A$ can be almost affine invariant simultaneously under all such $\varphi$ with $|a|, |b| \le K$ and $a \ne 0$. The threshold is $K = o(\log p)$.
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quantum-information-computingAug 9, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.
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probability-statisticsJun 14, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
a structured special case, proved the same week as the independent group version
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probability-statisticsJun 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
the group case; the full Matrix Spencer conjecture remains open
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geometry-topologyMay 21, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
proved in dimension at most three
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combinatoricsJul 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
asymptotic form only; Kotzig's conjecture itself remains far from solved
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theoretical-computer-scienceJul 22, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
the construction is probabilistic; an explicit uniformly samplable example remains open
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mathematical-physicsJul 29, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Do $n$ point charges whose electrostatic potential has only non-degenerate critical points always have at most $(n-1)^2$ of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least $24 > 16$ non-degenerate critical points, so the conjecture is false.
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combinatoricsJul 2, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
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analysisAug 20, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-mome…
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combinatoricsAug 3, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Mihail and Vazirani conjectured that the graph of every $0/1$-polytope has edge expansion at least one. Disproved by a family of $0/1$-polytopes whose edge expansion decreases exponentially in the dimension.
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probability-statisticsAug 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
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mathematical-physicsAug 6, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full r…
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algorithms-optimizationMay 21, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A)$, giving a deterministic $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon > 0$ and matching the…
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algorithms-optimizationAug 19, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.
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combinatoricsAug 19, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\ell$ collinear points or $k$ points that pairwise see each other. True for $\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.
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analysisAug 25, 2026Significance 27/100Registry: unreviewed
Prior state unknown→proved
If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.
On scope, two th…
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analysisMay 17, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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 separated from the sharp maximal operator $M^\sharp$. It can: there is a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded but…
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combinatoricsMay 25, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Frankl, Peng, Rodl and Talbot asked in 2007 whether the set of Turan densities of families of $r$-graphs contains intervals. It does: for every $r \ge 3$ the set contains non-degenerate intervals, including one of the form $[1-\delta_r, 1]$.
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quantum-information-computingJul 28, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?
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algebraAug 14, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\mathbb{Q}$-factorial fourfold pairs $(X, \Delta)$ with $K_X + \Delta \equiv M \ge 0$, over bases of positive and mixed characteristic $p > 5$.
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analysisAug 3, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Whether the real Kalton-Peck space $Z_2$ is isomorphic to its hyperplanes. It is not: no hyperplane of $Z_2$ is isomorphic to $Z_2$, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.
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geometry-topologyJun 23, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.
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combinatoricsJul 17, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover - a set of hyperedges covering each vertex an even number of times - with no superfluous polylogarithmic factors? Known up to polylog factors since 2022; now proved exactly for every $k \ge 3$.
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combinatoricsJun 15, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
The near-quadratic Elekes-Ronyai expander conjecture over $\mathbb{R}$ 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 real algebraic integers, has image with a fixed power saving from quadratic size.
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quantum-information-computingJul 23, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
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combinatoricsJun 16, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
in polynomial edge-density regimes; the general question remains open
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