number-theoryJul 13, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated as a conjecture by Chiu, on whether height among unit-volume flat tori is minimized by a lattice maximizing its shortest nonzero vector.
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combinatoricsAug 7, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
For a forbidden configuration $F$, the Anstee-Sali conjecture predicts that $\mathrm{forb}(m, F)$ is $\Theta\left(m^{X(F)-1}\right)$, where $X(F)$ comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has $X(F) = 4$, so the conjecture predicts $\Theta(m^3)$, while a random-alteration argument gives $\Omega(m^{\frac{…
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theoretical-computer-scienceJun 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Furthest Pair and its relatives admit $f(d)\,n^{2-\Theta(1/d)}$ algorithms, making them the standard examples of barely subquadratic computation, and whether that is optimal in superconstant dimension was open. Under SETH it is: Furthest Pair requires quadratic time once the dimension is superconstant.
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analysisJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Let $\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\alpha$ into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the correspondi…
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mathematical-physicsJul 30, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
a record upper bound; the exact constant remains unknown
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logic-foundationsAug 25, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\\prec\\mathcal{U}$, and let $\\mu\\in\\mathfrak{M}_{x}(\\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent:\n\n$(i)$ $\\mu$ is a frequency interpretation measure (fim) over $M$;\n\n$(ii)$ $\\mu$ is definable over $M$ and its canonical random extension $r_{\\mu}$ is genericall…
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geometry-topologyMay 20, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
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 need not be: such a morphism exists with non-constant first cohomology.
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analysisJul 19, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
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 case; the paper proves it is primary for $1<p<\infty$.
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analysisJul 19, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper proves it can.
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analysisAug 3, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Denjoy's 1932 theorem says a $C^{1+\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\omega$ weaker than Lipschitz, there is a $C^{1+\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\omega(t) = t\log(1/t)$ settles an open problem going back to He…
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geometry-topologyJul 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
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algorithms-optimizationJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of Goemans' unsplittable-flow conjecture.)
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geometry-topologyJun 26, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below $1/4$. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen two-vertex augmentation.
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combinatoricsAug 12, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Disproves the Bowler-Brown-Fenner bound of 2*floor((n-1)/3) on common cards between nonisomorphic graphs: an explicit connected 78-vertex pair shares at least 51 cards against the predicted 50, and for every even r >= 4 there are families with overlap fraction asymptotically at least 1 - 1/r, so the attainable fraction approaches the full deck. The Kelly-Ulam reconstruction conjecture itself is untouched.
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geometry-topologyApr 28, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
resolves the noncommutative half of the Kirby-list problem
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combinatoricsJun 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
an improved lower bound; the true order of g(r) remains open
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mathematical-physicsJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For the Sachdev-Ye-Kitaev Hamiltonian on $n$ Majorana modes with $k$-body interactions, the paper proves $\mathbb{E}\|H\|_{op} = (1-o(1))\sqrt{2n}/k$ for super-constant $k \le o(\sqrt{n})$, confirming predictions of Garcia-Garcia, Jia and Verbaarschot and answering a question of Feng, Tian and Wei.
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combinatoricsJul 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Norine conjectured that every red-blue edge-colouring of the $n$-dimensional hypercube $Q_n$ in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level Borsuk–Ulam obstruction.
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analysisMay 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as $p \to 1^+$ or $\delta \to 0^+$. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.
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combinatoricsJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
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theoretical-computer-scienceJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
information-theoretic; a polynomial-time implementation remains open
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analysisJul 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more than a decade. It cannot: for Schrodinger operators in dimensions $d \ge 2$ no spectral point becomes invisible, which together with the known one-dimensional theorem settles no-i…
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probability-statisticsMay 12, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet, settled in the ternary case. For independent $X_1, \ldots, X_n$ taking values in $\{0,1,2\}$, the entropy of $S_n = X_1 + \cdots + X_n$ is maximized when $X_1, \ldots, X_{n-1}$ are uniform on $\{0,2\}$ and $X_n$ has an explicitly described three-point distribution. This extends the Shepp…
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analysisJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Two free ergodic measure-preserving flows whose $\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le Maître and the author.
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geometry-topologyJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang
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analysisJun 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension $d > 1$ there are explicit criteria on lattices $\Lambda \subset \mathbb{R}^{2d}$ with $D(\Lambda) > 1$ such that no function with continuous Zak transform generates a Gabor frame along $\Lambda$, which answers the existence problem negatively for Schwartz-class windows.
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geometry-topologyMay 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.
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geometry-topologyJul 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
the mod 4 form; the general conjecture is false by Ermotti, Hongler and Weber
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algebraJul 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.
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probability-statisticsJul 15, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
the conjectured cutoff location of 2n log n is wrong
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analysisJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.
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algebraJul 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Baumslag asked whether a one-relator group $G=F/\langle\langle r\rangle\rangle$ with $r$ a commutator is Hopfian, residually finite or automatic. The paper constructs a family $G_m=\langle a,t \mid [t,a[a,t]^{-m}]\rangle$ answering all three negatively.
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analysisJun 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
a record exponent; the conjectured range is not yet reached
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theoretical-computer-scienceMay 4, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable $tt$-degree contains a c.e. irreducible $m$-degree, meaning an $m$-degree consisting of a single $1$-degree. Answered negatively: there is a c.e. $tt$-degree containing no c.e. irreducible $m$-degree. This also shows Jockusch's 1969 theorem, which produces an irreducible $m$-…
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analysisJul 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of Schmuckenschläger from 2011, via a new log-convexity property of the first eigenvalue under positive definite linear deformations.
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geometry-topologyMay 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed rational curves has only real roots. True, with simple roots and strict interlacing between consecutive $n$.
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