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Baumslag asked whether a one-relator group with a commutator is Hopfian, residually finite or automatic. The paper constructs a family answering all three negatively.
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.
If a subgroup of a product of groups of type virtually surjects onto every -tuple of factors, must it be of type itself? Yes, for discrete groups, and likewise for . The homological -- Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.
an improved lower bound; the true order of g(r) remains open
Norine conjectured that every red-blue edge-colouring of the -dimensional hypercube 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.
For the Sachdev-Ye-Kitaev Hamiltonian on Majorana modes with -body interactions, the paper proves for super-constant , confirming predictions of Garcia-Garcia, Jia and Verbaarschot and answering a question of Feng, Tian and Wei.
Is the irreversibility of entanglement manipulation robust in the strong-converse sense - a strict separation between the exponential strong-converse distillable entanglement and the entanglement cost, as conjectured by Lami and Regula? Yes: there are states for which any attempt to restore reversibility incurs an error growing exponentially in the number of copies, and the irreversibility persists even at polynomia…
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 taking values in , the entropy of is maximized when are uniform on and has an explicitly described three-point distribution. This extends the Shepp-O…
Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity a…
For every zonotope and vectors , there are signs with for a universal constant . This resolves a 2002 conjecture on vector balancing in zonotopes.
Steurer conjectured in 2010 that any family of unit vectors with polynomially small average correlation contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional expanders.
Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in steps, the conjectured optimal rate, closing the gap left by a long line of successive improvements.
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable -degree contains a c.e. irreducible -degree, meaning an -degree consisting of a single -degree. Answered negatively: there is a c.e. -degree containing no c.e. irreducible -degree. This also shows Jockusch's 1969 theorem, which produces an irreducible -de…
An -Oddtown is a family of subsets of an -element set whose set sizes are not divisible by while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is for prime , Babai and Frankl extended this to prime powers and asked whether still holds for other moduli, a question open even for . Bukh, Chao and Zheng answer it negatively with an explicit supe…
one construction disproves both the product conjecture and its weak form
Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting -colourings on planar graphs is NP-hard for every constant , and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.
The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra on two generators cannot occur. Using Bellissima’s representation , the authors construct an upward-closed subset with . They show that if some elementary topos satis…
An -divisor set contains a multiple of every integer from 1 to . Umans and Wang proposed, as the arithmetic-progression form of their Strong -Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.
An average-case claim about the Gaussian model, not a contradiction of the worst-case NP-hardness of integer least squares. If it holds, no computational-statistical gap separates polynomial-time detection from exhaustive maximum likelihood at first order in this model.
Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.
A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular lattices, which also yields a new algebraic bijective proof of Dilworth's theorem; conjectures of Hopkins on parking function statistics studied by Stanley and Yin; and two conjectures…
Conjecture 13 of King, Gosset, Kothari and Babbush asserts that for the set of Pauli observables with expectation value at least in magnitude, the fractional chromatic number of the induced anticommutation graph is ; it would give a triply efficient Pauli shadow tomography algorithm. False: there are states and observables for which no finite constant bounds…
Disproved on the Hopf threefold : Xia and Zhang construct a smooth Hermitian form and smooth functions with whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass…
Conjectures 2a and 2b of Kauers and Zeilberger, on the asymptotics of a family of restricted lattice walks. Both are obtained from a local limit theorem for excursions of Markov-modulated random walks in cones.
Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.
Yun, Sra and Jadbabaie posed as a COLT 2021 open question whether, for well-conditioned symmetric matrices, the operators encoding the expected iterate of single-shuffle SGD, random-reshuffle SGD and gradient descent on a quadratic finite sum satisfy . They do.
The total Chern class of as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.
Both bounds move, and the gap stays enormous: the lower bound rises from to and the upper falls from to , so is still undetermined between an exponent of and one of . The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness,…
For an unkilled Levy process drifting to with all positive exponential moments, let and . Bertoin and Yor proved is moment-determinate when has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on leaves. Proved: it is , by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.
The interchange graph has the -matrices with row sums and column sums as vertices, adjacent when they differ by a single interchange. Brualdi asked whether is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.
Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how little coherent memory suffices for the advantage to appear.
Must every -differential poset have at least as many elements in each rank as , the -th Cartesian power of Young's lattice? For the new construction has fourth-rank size against for .
Exponent improvements toward Pach's conjecture, which remains open.
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions in the Mizohata-Takeuchi circle of problems.