Clay weekly context brief for the Maths category (ISO week 2026-W38).

Clay tracks publications from the Maths feed list. Below are recent items from this category, each with its source and a short description of what the publication covers when one is available in the source feed.

Recent publications:
1. Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of $S^2\times S^2$
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2609.11727
   We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces.
2. Categories of Multigraded Local Cohomology Modules: Serre Filtrations and Nakayama Duality
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2609.11219
   Let $\Bbbk$ be a field, let $S=\Bbbk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading, and let $\mathfrak m=(x_1,\ldots,x_n)$.
3. The Reye geometry inside the 64 lines of the Schur quartic
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2609.10751
   We identify the classical geometry hidden in the Naskr\k{e}cki--Pokora $(24_4,32_3)$ configuration on the Schur quartic.
4. Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2609.10819
   We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1 2p/(p+1)$, whereas the energy method gives a lower bound of order $\varrho^{1-p}$ only.
5. Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2609.10763
   We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$.
6. Real stability of layer-refined antichain polynomials for three-chain products with a two-element factor
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2609.10609
   For all positive integers $n,k$, we prove that the layer-refined antichain polynomial of the product poset $[2]\times[n]\times[k]$ is real stable.
7. A model structure for cartesian 2-fibrations
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2609.11759
   Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data.
8. A counterexample to an open problem of Dorff
   Source: math.CV (Complex Variables)
   Link: https://arxiv.org/abs/2609.11556
   The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex.
9. The $q$-deformed cross-ratio: modular invariants and Coxeter friezes
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2609.10999
   We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$.
10. The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2609.10847
   We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane.
11. Spectral and numerical radius preservers on pairs of unitarily similar operators
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2609.10635
   Let $\mathcal{H}$ be a complex Hilbert space.
12. A proof of Cairns' conjecture about geodesic triangulations of the 2-sphere
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2609.10586
   This paper contains a detailed proof of a 1941 conjecture of S.S. Cairns concerning the space of geodesic triangulations of the 2-sphere. The methods are neither combinatorical nor Riemannian but belong to differential topology.
13. A continuous $3$-distributive frame that is not $\omega$-distributive
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2609.11671
   We give a negative answer to the question, posed by Ern\'e, whether every $3$-distributive lattice is $\omega$-distributive.
14. Conjugator length in finitely generated groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2609.11880
   We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups.
15. Some Early Results by Tutte Regarding the Cycle Double Cover Conjecture in 1948
   Source: math.HO (History and Overview)
   Link: https://arxiv.org/abs/2609.10618
   OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture.
16. Relative Brauer groups and Kummer Sequence without characteristic constraint in $fppf$-topology
   Source: math.KT (K-Theory and Homology)
   Link: https://arxiv.org/abs/2609.04760
   Let $f: X\rightarrow S$ be a faithful affine map.
17. Inequalities and equalities of means in Constructive Mathematics
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2609.10593
   We study the inequalities and equalities of means of numbers in constructive mathematics.
18. The Nieuwland number of the Octahedron
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2609.10788
   We prove that the Nieuwland number of the Octahedron is $3\sqrt{2}/4$. Along the way, we provide an elementary proof that Rupert's problem for the Octahedron admits no solution in which the outer projection is a hexagon.
19. A Time-Frequency Framework for GKP Codes
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2609.10802
   We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus.
20. Structure-preserving operator splitting for 2.5D ideal MHD with an entropy-stable DGSEM and exactly divergence-free compatible finite elements
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2609.10849
   We develop a high-order, fully explicit operator-splitting method for 2.5D ideal magnetohydrodynamics on Cartesian meshes.
21. Von Neumann algebras as reduced twisted groupoid $C^*$-algebras
   Source: math.OA (Operator Algebras)
   Link: https://arxiv.org/abs/2609.10835
   We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support.
22. Gumbel convergence for maximal packing distances from uniform random samples
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2609.10607
   Let $(X _n )_n$ be a sequence of i.i.d.
23. On $(n,d)$-Coherence and Its Applications to Algebraic $\mathsf{K}$-Theory
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2603.25679
   We introduce a unified approach to finiteness conditions in homological algebra and algebraic $\mathsf{K}$-theory by studying the class of $(n,d)$-coherent rings, defined for $n\in\mathbb{N}^*$ and $d\in\mathbb{N}^*\cup \{\infty\}$.
24. Regular hyperbolic tilings have no $\ell^2$ eigenfunctions
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2609.11857
   We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions.
25. A counterexample to the wrapping number conjecture
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2609.11084
   We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.
26. Associated primes, witnesses, and omega invariants of monomial ideals
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2609.11407
   We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal.
27. Torelli theorem for the product of moduli spaces of vector bundles over a curve
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2609.10764
   We prove a Torelli-type theorem for a product of the moduli spaces of semistable vector bundles over smooth projective curves of genus $g\,\ge\, 4$.
28. Heat equation with singular drift which has bounded solution discontinuous at an interior point
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2609.10828
   The goal of this paper is to present a one dimensional heat equation with a singular drift for which solutions of the Cauchy problem with smooth initial data are bounded and are discontinuous at an interior point (common to all solutions).
29. On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2609.11158
   For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.

Sources in this brief: math-ph (Mathematical Physics); math.AC (Commutative Algebra); math.AG (Algebraic Geometry); math.AP (Analysis of PDEs); math.CA (Classical Analysis and ODEs); math.CO (Combinatorics); math.CT (Category Theory); math.CV (Complex Variables); math.DG (Differential Geometry); math.DS (Dynamical Systems); math.FA (Functional Analysis); math.GM (General Mathematics); math.GN (General Topology); math.GR (Group Theory); math.GT (Geometric Topology); math.HO (History and Overview); math.KT (K-Theory and Homology); math.LO (Logic); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.OA (Operator Algebras); math.PR (Probability); math.RA (Rings and Algebras); math.SP (Spectral Theory).
Selected 29 of 291 available items for this weekly brief.