Clay weekly context brief for the Maths category (ISO week 2026-W37). 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. Enumerating cores of charged multipartitions Source: math.RT (Representation Theory) Link: https://arxiv.org/abs/2609.03738 Granville and Ono proved that there is an $e$-core partition of $n$ for every $n\in\mathbb{N}$ if and only if $e\geq 4$, which translates to a statement about the existence of defect $0$ blocks for symmetric groups in positive characteristic and defect $0$ unipotent blocks of finite general linear groups in positive, non-defining characteristic. 2. A new limit variety of additively idempotent semirings Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2604.18588 We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. 3. Caffarelli Estimates under Lipschitz Perturbations Source: math.PR (Probability) Link: https://arxiv.org/abs/2609.04052 We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. 4. From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra Source: math.NT (Number Theory) Link: https://arxiv.org/abs/2609.03801 We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets. 5. Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost Source: math-ph (Mathematical Physics) Link: https://arxiv.org/abs/2609.03769 We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. 6. A choice-free proof of the Erd\H{o}s--Dushnik--Miller theorem Source: math.LO (Logic) Link: https://arxiv.org/abs/2609.02703 The Erd\H{o}s--Dushnik--Miller theorem states that for every aleph $\kappa$, \[ \kappa\to(\kappa,\omega); \] that is, every coloring $c:[\kappa]^2\to2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. 7. Symbolic Powers and Asymptotic Invariants of GL-Invariant Ideals Source: math.AC (Commutative Algebra) Link: https://arxiv.org/abs/2609.03131 This work concerns ideals invariant under the action of the group of linear base changes on a generic matrix; we call these GL-invariant ideals. 8. Standing waves for Schrodinger equations with Kato class potentials and $L^\infty$-bounded nonlinearities Source: math.AP (Analysis of PDEs) Link: https://arxiv.org/abs/2609.03098 We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. 9. Generalized Telescope Conjecture Source: math.AT (Algebraic Topology) Link: https://arxiv.org/abs/2609.03375 We introduce the atomic smashing frame, extending the Balmer spectrum from tensor-triangular geometry to an arbitrary presentably symmetric monoidal $\infty$-category $\mathcal{V}$. 10. Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations Source: math.CA (Classical Analysis and ODEs) Link: https://arxiv.org/abs/2609.03264 We study an oracle subspace-perturbation model in which the 3D localization procedure is given an \(s\)-dimensional subspace \(\widetilde{\mathcal U}\) and a deterministic error bound $\eps_{\rm sub}$ measuring the sine-theta distance between $\widetilde{\mathcal U}$ and an $s$-dimensional subspace $\cU$ of far-field patterns with wavenumber $\kappa$. 11. Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$ Source: math.CO (Combinatorics) Link: https://arxiv.org/abs/2609.02953 For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2^{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. 12. Override and Update in Restriction Categories Source: math.CT (Category Theory) Link: https://arxiv.org/abs/2609.03472 We study the override and update operators on partial functions from the perspective of restriction categories. 13. A Smooth Counterexample to the Trautman Conjecture Source: math.CV (Complex Variables) Link: https://arxiv.org/abs/2609.03198 The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\mathbb{C}^2$. 14. Maslov Indicies In Symplectic Geometry Revisited Source: math.DG (Differential Geometry) Link: https://arxiv.org/abs/2609.03061 Thirty years ago in ``On the Maslov Index'' the present authors with their late collaborator Prof. 15. SurgeGen: A Hybrid Generative Diffusion Framework for Storm Surge Scenario Synthesis Source: math.DS (Dynamical Systems) Link: https://arxiv.org/abs/2609.03382 Predicting storm surge induced by landfalling tropical cyclones is crucial for flood mitigation and coastal risk management. 16. Variants of order semicontinuity in Banach lattices Source: math.FA (Functional Analysis) Link: https://arxiv.org/abs/2609.03070 In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. 17. On second maximal subgroups of finite groups Source: math.GR (Group Theory) Link: https://arxiv.org/abs/2609.03911 A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup. 18. Primitive stability and the Q-conditions for the rank two free group in hyperbolic d-space Source: math.GT (Geometric Topology) Link: https://arxiv.org/abs/2507.09771 The two largest known domains of discontinuity for the action of Out(F_2) on the PSL(2,C)-character variety of F_2 - defined by Minsky's primitive stability, and Bowditch's Q-conditions - were proven to be equal independently by Lee-Xu and Series. 19. The stealthiest particle configurations in 8 and 24 dimensions Source: math.MG (Metric Geometry) Link: https://arxiv.org/abs/2609.03121 We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, and that they are the unique stealthiest processes. 20. Decay Rates and Domain Dependence of a Coupled Wave-Heat System with Spatially Dependant Heat Coefficients Source: math.SP (Spectral Theory) Link: https://arxiv.org/abs/2609.03980 We study of the long-term behavior of a coupled wave-heat system. 21. Jantzen filtrations for graded affine Hecke algebras Source: math.RT (Representation Theory) Link: https://arxiv.org/abs/2609.03651 We prove the Jantzen conjecture for standard modules of graded affine Hecke algebras with equal parameters, real central character in the dominant deformation direction. 22. Factor-parity Hall sets and controllability Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2609.04056 We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. 23. Universal Pl\"ucker positivity and the Octopus inequality Source: math.PR (Probability) Link: https://arxiv.org/abs/2609.03746 We prove a positivity theorem for the universal Pl\"ucker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. 24. Some generalizations of Oort's conjecture Source: math.NT (Number Theory) Link: https://arxiv.org/abs/2609.03188 For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. 25. Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles Source: math-ph (Mathematical Physics) Link: https://arxiv.org/abs/2609.03618 In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. 26. Nonvanishing derived limits from $\clubsuit$-type principles Source: math.LO (Logic) Link: https://arxiv.org/abs/2609.03038 Combinatorial set theory provides several tools to study derived limits of certain inverse systems of abelian groups. 27. Relative approximation degrees and the henselian rationality problem over perfect fields Source: math.AC (Commutative Algebra) Link: https://arxiv.org/abs/2609.03451 Let $(F|K,w)$ be an immediate valued function field of transcendence degree one over a rank-one perfect valued field $(K,v)$ of characteristic $p>0$. 28. The emergence of the Cauchy horizon from the crease for $3D$ compressible Euler flow Source: math.AP (Analysis of PDEs) Link: https://arxiv.org/abs/2609.03101 We study open sets of nearly plane symmetric data for the $3D$ compressible Euler equations with dynamic entropy and non-trivial vorticity. Sources in this brief: math-ph (Mathematical Physics); math.AC (Commutative Algebra); math.AP (Analysis of PDEs); math.AT (Algebraic Topology); 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.GR (Group Theory); math.GT (Geometric Topology); math.LO (Logic); math.MG (Metric Geometry); math.NT (Number Theory); math.PR (Probability); math.RA (Rings and Algebras); math.RT (Representation Theory); math.SP (Spectral Theory). Selected 28 of 269 available items for this weekly brief.