Clay weekly context brief for the Maths category (ISO week 2026-W28).

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. Corner Quantization of 4D $BF$ Theory
   Source: math.RT (Representation Theory)
   Link: https://arxiv.org/abs/2605.30006
   This note studies the quantized corner structure of four-dimensional $BF$ theory, classifies the associated free and physical corner algebras and constructs possible representations.
2. A proof of Riemann's hypothesis via Hadamard-Weierstrass factorization
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2607.04338
   Using the Hadamard-Weierstrass factorization theorem for Riemann's {\xi} function, we discuss and prove Riemann's hypothesis.
3. Zeroing Diagonals, Conjugate Hollowization, and Characterizing Nondefinite Operators
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2508.00096
   We prove the conjecture by Damm and Fassbender that, for real traceless matrices $L,M$, there exists orthogonal $R$ such that $\mathrm{diag}(R^\top L R) = (0,...,0,0,0)$ and $\mathrm{diag}(R M R^\top) = (0,...,0,*,*)$.
4. Soft edge limit of the Laguerre beta-ensemble at the lower edge
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2607.08536
   We show that the lower edge of the appropriately scaled size $n$ Laguerre beta-ensemble with parameter $a=a_n$ converges to the $\operatorname{Airy}_{\beta}$ process as $n\to \infty$ when $a_n\to \infty$ and $\tfrac{a_n}{n}\to 0$.
5. Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2605.13449
   Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space.
6. The Finite Length Property of the Rado Graph and Friends
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2605.21681
   An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length.
7. Combinatorial constructions of Schubert subspace codes
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2607.07479
   We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace.
8. Minimal simplicial degree $d$ self-maps of $\mathbb{S}^{n-1}\times \mathbb{S}^1$
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2407.10128
   The degree of a map between orientable manifolds is a fundamental concept in topology, providing important information about the structure of manifolds and the behavior of maps between them.
9. Examples of non-amenable, boundary-amenable dynamical systems
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2507.19614
   Let $\Gamma$ be a discrete countable group with the (AP)-property.
10. Wold-type decomposition for doubly twisted left-invertible covariant representations
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2601.13950
   In this article, we have introduced the notion of a near-isometric covariant representation of a $C^*$-correspondence.
11. TNODEV: Toolbox for Neural ODE Verification
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2606.16567
   Neural ordinary differential equations (neural ODE) gained attention in safety critical settings such as continuous-time controllers for cyber-physical systems and classifiers integrated into automated decision pipelines, raising the question whether their behavior can be formally verified.
12. Ahlfors Currents and Symplectic Non-Hyperbolicity
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2504.10790
   Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood.
13. Braiding structures on categorical multi-Interval Jones-Wassermann subfactor
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2607.08296
   In this paper, we construct braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra associated with any unitary modular fusion category.
14. Two Conjectures on Extensions of Brouwer's Laplacian Conjecture
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2607.08452
   Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix.
15. Sobolev spaces on snowtrees
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2606.30927
   We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type.
16. Topological decoding of grid cell activity via path lifting to covering spaces
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2510.16216
   High-dimensional neural activity often reside in a low-dimensional subspace, referred to as neural manifolds.
17. Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2607.08631
   We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.
18. Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2504.01563
   We use height arguments to prove two results about the dynamical Mordell-Lang problem.
19. Testing the max-flow min-cut property and the replication conjecture
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2606.16543
   The replication conjecture [Conforti and Cornu\'{e}jols, 1993] states that every clutter with the packing property has the MFMC property.
20. On Polyhedral Formulas for Kirillov-Reshetikhin Modules
   Source: math.RT (Representation Theory)
   Link: https://arxiv.org/abs/1901.00104
   We propose a method to prove a polyhedral branching formula for Kirillov-Reshetikhin (KR) modules over an untwisted quantum affine algebra.
21. Murmurations in the depth aspect
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2603.25564
   We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N=\ell^a$, with $\ell$ a fixed odd prime and $a\to \infty$.
22. Multiphysics embedding localized orthogonal decomposition for thermomechanical coupling problems
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2507.13644
   Multiscale thermomechanical problems in highly heterogeneous media are challenging because the elastic, thermal, and coupling coefficients may vary on unresolved spatial scales.
23. Gram--Wishart--Stiefel formulation of the $N=2$, large--$d$ gauge theory in 1D
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2607.08481
   We develop in this paper the Gram/Wishart/Stiefel formulation of the \(N=2\), large--\(d\) planar endpoint theory of the BFSS/BMN matrix quantum mechanics on the lattice, obtained in our previous work.
24. Smooth Approximations of Quasispheres
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2503.09253
   We prove that every $n$-dimensional quasisphere is the Gromov-Hausdorff limit of a sequence of locally smooth uniform quasispheres.
25. Generalized Decidability via Brouwer Trees
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2602.10844
   In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable".
26. Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2604.17369
   How many black-box queries to a quantum channel are needed to learn its full classical description?
27. The Casson-Sullivan invariant for homeomorphisms of 4-manifolds
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2405.07928
   We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients.

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.DG (Differential Geometry); math.DS (Dynamical Systems); math.FA (Functional Analysis); math.GR (Group Theory); math.GT (Geometric Topology); math.IT (Information Theory); math.LO (Logic); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.NT (Number Theory); math.RT (Representation Theory).
Selected 27 of 236 available items for this weekly brief.