Clay weekly context brief for the Maths category (ISO week 2026-W41).

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. Mirror symmetry for gCICY threefolds (I): a $[3,-1]$ block and a non-Gorenstein toric phase
   Source: math.SG (Symplectic Geometry)
   Link: https://arxiv.org/abs/2610.03719
   We construct and analyze a mirror family for the generalized complete intersection Calabi-Yau threefold $X_+=[\mathbb{P}^1\,|\,3\ {-1};\ \mathbb{P}^4\,|\,2\ 3]$, with $(h^{1,1},h^{2,1})=(2,46)$.
2. Skew-symmetrizable cluster algebras from surfaces and symmetric quivers
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2512.12247
   We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\varsigma$.
3. Local laws and spectral properties of deformed sparse random matrices
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2507.02298
   We consider deformed sparse random matrices of the form $H= W+ \lambda V$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $\lambda = O(1) $ is a coupling constant.
4. Optimal allocation in combat models with multiple battlefields
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2610.03336
   We study the allocation of a force of a fixed size across independent battlefields against a known distribution of opposing forces.
5. Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems
   Source: math.OA (Operator Algebras)
   Link: https://arxiv.org/abs/2609.13084
   We study dimension-free Bohnenblust--Hille inequalities for local operators on systems of $K$-level qudits.
6. Fast Nanophotonic Inverse Design using Precomputed Numerical Green Functions
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2610.02821
   Inverse design has been transformative in nanophotonics, providing an automated means of realizing high-performance, non-intuitive devices.
7. Parity-Time Symmetry of Plasma Waves
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2610.03433
   The susceptibility tensor of a homogeneous collisionless magnetized plasma possesses parity-time (PT) symmetry.
8. Knot distortion via curve combinatorics
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2510.01095
   Distortion is a geometric knot invariant introduced by Gromov in 1983.
9. An equiconsistency proof for $\mathrm{CZF} + V = L$
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2602.13917
   In many axiomatic set theories, G\"odel's constructible universe $L$ is known as an inner model, that is, a definable class satisfying the same axioms (and containing the same ordinals).
10. Non-isomorphic graphs have distinct vertex-Ramsey classes
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2610.03704
   For a graph $H$, its $k$-colour vertex Ramsey class is the set of all graphs $G$ such that any colouring of the vertices of $G$ in $k$ colours results in a monochromatic (induced) copy of $H$. We prove that for any $k$, Ramsey classes of any non-isomorphic graphs are distinct.
11. Common non-zero graphs of groups with few edges
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2609.39290
   Let \( G \) be a finite group, and consider its common non-zero graph, which we denote by \( \Gamma_{nv}(G) \).
12. A note on power spaces and function spaces of strongly well-filtered spaces
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2610.02725
   Strong well-filteredness is a refinement of well-filteredness arising naturally in non-Hausdorff topology and domain theory.
13. Input Dexterity and Output Negotiation in Feedback-Linearizable Nonlinear Systems
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2603.08480
   We introduce a task-relative taxonomy of actuator inputs for nonlinear systems within the input-output feedback-linearization framework.
14. Stratification of the space of density matrices via invariant theory and Uhlmann's connection
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2610.02773
   The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system.
15. Rigidity of complex bounded domains with finite-volume quotients
   Source: math.CV (Complex Variables)
   Link: https://arxiv.org/abs/1801.00459
   We prove that any bounded convex domain in $\mathbb{C}^n$ admitting a finite-volume quotient is biholomorphic to a bounded symmetric domain.
16. Positive discrepancy of graphs far from Tur\'an graphs
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2610.02681
   We prove that, for every $\eps>0$, any $n$-vertex graph that needs at least $\eps n^2$ edge changes to become a Tur\'an graph has positive discrepancy at least $c_\eps n^{5/4}$.
17. On the endpoint estimate for discrete spherical average over sparse sequences
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2608.03004
   Let $d\ge5$, and let $A_\lambda$ denote the normalized discrete spherical average on $\mathbb{Z}^d$ with squared radius $\lambda$.
18. Embedding, compression, and the relative Hopf invariant
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2602.19266
   We establish Poincar\'e embedding results in the relative setting, generalizing previously known results in the absolute case.
19. Global existence for viscous systems of conservation laws in $L^p$ framework
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2610.03011
   In this paper, which originates from Chapter 4 of the author's PhD thesis [1, Chap.
20. Mirror symmetry for gCICY threefolds (II): $[3,-1]$ blocks, VGIT, and toric phases
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2610.03688
   The configuration matrices of generalized complete intersection Calabi-Yau (gCICY) threefolds contain negative entries, so their defining line bundles are not nef and the Batyrev-Borisov construction does not apply directly.
21. The ring of differential operators on a monomial curve is a Hopf algebroid
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2405.08490
   This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras.
22. A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2608.02405
   Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$.
23. Kodaira fibres and wrapped Floer cohomology
   Source: math.RT (Representation Theory)
   Link: https://arxiv.org/abs/2609.23884
   Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $\Omega$ be a nonvanishing holomorphic two-form near $F$.
24. Gaussian solutions to the Yang--Baxter equation and their twists
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2607.13876
   In this paper, we consider two explicit Gaussian solutions to the constant (or parameter-independent) quantum Yang--Baxter equation and produce the corresponding bialgebras using the Faddeev--Reshetikhin--Takhtajan construction (FRT).
25. Structure-Driven Methodology: An Emerging Cross-Domain Paradigm
   Source: math.HO (History and Overview)
   Link: https://arxiv.org/abs/2610.02698
   This article proposes and articulates structure-driven methodology as an emerging cross-domain paradigm.
26. Ropelength-Filtered Swept-Area Geometry
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2605.05557
   This paper studies the swept-area cost of isotopies between thick knot representatives when the isotopy is required to stay inside a ropelength window: every intermediate curve has thickness at least one and length at most $\Lambda$.
27. Linear maps preserving families of C-symmetric operators
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2608.24583
   Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$.
28. Monoidal categories graded by partial commutative monoids
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2603.16375
   Effectful categories have two classes of morphisms: pure morphisms, which form a monoidal category; and effectful morphisms, which can only be combined monoidally with central morphisms (such as the pure ones), forming a premonoidal category.
29. Small Lefschetz fibrations on the BMY line
   Source: math.SG (Symplectic Geometry)
   Link: https://arxiv.org/abs/2610.02261
   We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface.

Sources in this brief: math-ph (Mathematical Physics); math.AC (Commutative Algebra); math.AG (Algebraic Geometry); 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.GN (General Topology); math.GR (Group Theory); math.GT (Geometric Topology); math.HO (History and Overview); math.IT (Information Theory); math.LO (Logic); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.OA (Operator Algebras); math.OC (Optimization and Control); math.PR (Probability); math.QA (Quantum Algebra); math.RA (Rings and Algebras); math.RT (Representation Theory); math.SG (Symplectic Geometry); math.SP (Spectral Theory).
Selected 29 of 323 available items for this weekly brief.