Clay weekly context brief for the Maths category (ISO week 2026-W45). 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. Galois module structure of Deligne--Lusztig varieties for $\mathrm{GL}_3$ Source: math.RT (Representation Theory) Link: https://arxiv.org/abs/2610.40335 Let $\overline{\mathbf{Y}}(w)\to\overline{\mathbf{X}}(w)$ be the ramified Galois cover of compactified Deligne--Lusztig varieties. 2. A further study of quandles and quandle rings Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2608.09149 We investigate core quandles and idempotents in quandle rings of core quandles. 3. Webs and finite dimensional representation theory for quantum symmetric pairs Source: math.QA (Quantum Algebra) Link: https://arxiv.org/abs/2610.08057 We develop a highest weight framework and construct explicit finite dimensional irreducible modules for the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'(\mathfrak{gl}_n^\theta)$. 4. Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang Source: math.PR (Probability) Link: https://arxiv.org/abs/2610.38893 Randomly connected networks are ubiquitous in computer science, physics or biology. 5. Convergence rates of mixed primal-dual dynamical systems with implicit Hessian damping and Tikhonov regularization Source: math.OC (Optimization and Control) Link: https://arxiv.org/abs/2610.39144 We propose a mixed-order primal--dual dynamical system with implicit Hessian damping and Tikhonov regularization for linearly constrained convex optimization, and derive a corresponding fast primal--dual algorithm. 6. Free energy and spectral edge of the SYK model Source: math.OA (Operator Algebras) Link: https://arxiv.org/abs/2607.18998 Consider the Sachdev-Ye-Kitaev model with $N$ Majorana fermions and interaction order $q$. 7. An introduction to the strategy of transcendence proofs Source: math.NT (Number Theory) Link: https://arxiv.org/abs/2610.39810 The proof by Ch. 8. Energy Stability and Error Estimates for a Globally Divergence-Free DG Method of Arbitrary Order Source: math.NA (Numerical Analysis) Link: https://arxiv.org/abs/2610.39299 We propose and analyze an arbitrary-order globally divergence-free (GDF) discontinuous Galerkin (DG) method for the two-dimensional linear magnetic induction equation on unstructured triangular meshes. 9. Solution of Laczkovich's Lipschitz mapping problem Source: math.MG (Metric Geometry) Link: https://arxiv.org/abs/2610.38834 In 1991, Laczkovich asked whether every measurable set of positive measure in $\mathbb R^d$ can be mapped onto a cube with a Lipschitz map. 10. Algebraic Quantum Field Theory Source: math-ph (Mathematical Physics) Link: https://arxiv.org/abs/2610.37904 Algebraic quantum field theory describes a quantum system through its observables, their localisation in spacetime, and the states that assign them expectation values. 11. Index and nullity of critical spherical catenoids in hyperbolic three-space Source: math.SP (Spectral Theory) Link: https://arxiv.org/abs/2610.35651 We prove that every critical spherical catenoid in a geodesic ball of hyperbolic three-space has Morse index four and nullity two. 12. Integral Points and Volume of Integral-Integral Affine Manifolds Source: math.SG (Symplectic Geometry) Link: https://arxiv.org/abs/2508.17551 Using rational Ehrhart theory and Fourier analysis, we give an elementary proof that the volume of a closed manifold equipped with an integral-integral affine structure equals the number of its integral points. 13. On End-Semidualizing Modules Source: math.AC (Commutative Algebra) Link: https://arxiv.org/abs/2610.37924 We introduce and study the notion of End-semidualizing modules which are generalizations of semidualizing modules in the classical sense, and also extend the notion of $S-R$ semidualizing bimodules studied by Holm-White and Wakamatsu. 14. Spectrahedral relaxations and tightenings of rigidly convex sets Source: math.AG (Algebraic Geometry) Link: https://arxiv.org/abs/2610.37850 We construct a natural spectrahedral relaxation of a rigidly convex set (rcs) that depends only on the cubic part and on the degree of the defining real zero polynomial. 15. Sharp $L^4$ Strichartz estimate for the Zakharov--Kuznetsov equation on $\mathbb{R}\times\mathbb{T}$ Source: math.AP (Analysis of PDEs) Link: https://arxiv.org/abs/2610.37805 We prove the local $L^4$ Strichartz estimate without derivative loss for the Zakharov--Kuznetsov flow on $\mathbb{R}\times\mathbb{T}$. 16. The equal norm Grassmannian: topology and applications to frame fields Source: math.AT (Algebraic Topology) Link: https://arxiv.org/abs/2610.38669 Given an orientable rank-$d$ vector bundle over a $d$-manifold, the Euler class is the unique obstruction to finding a nowhere-zero section. 17. Exponential decay of threshold correlations for symmetric periodic functions Source: math.CA (Classical Analysis and ODEs) Link: https://arxiv.org/abs/2610.38135 We study how often $n$ distinct integer dilates of a periodic function can all lie above a fixed threshold or all lie below its negative. 18. Torsion Theories in a non-pointed Context II: iterated torsion theories Source: math.CT (Category Theory) Link: https://arxiv.org/abs/2610.37956 Generalizing a known result in the context of homological categories, we show that, in a regular protomodular category $\mathcal{C}$ equipped with a posetal, monocoreflective subcategory whose coreflector inverts monomorphisms, every torsion theory, in a non-pointed sense, satisfying suitable properties gives rise to a torsion theory in the category $\operatorname{Arr}(\mathcal{C})$ of arrows in $\mathcal{C}$. 19. Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms Source: math.CV (Complex Variables) Link: https://arxiv.org/abs/2610.38247 Let $n\geq 2$ and $k\geq 1$ be integers. 20. The generalized Cartan-Hadamard conjecture Source: math.DG (Differential Geometry) Link: https://arxiv.org/abs/2610.37817 We establish the generalized Cartan-Hadamard conjecture, including rigidity in the equality case. 21. Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity Source: math.DS (Dynamical Systems) Link: https://arxiv.org/abs/2610.37990 Periodic attractors of spatially extended systems possess not one period but a hierarchy of periods, one for each level of observation and symmetry quotient, and this hierarchy separates how a pattern evolves from how it fits the finite domain. 22. Local Hardy and bmo Spaces on Filtered Lie Groups Source: math.FA (Functional Analysis) Link: https://arxiv.org/abs/2610.38562 We develop a theory of local Hardy and $bmo$ spaces on a connected Lie group with polynomial growth whose Lie algebra is endowed with a filtration. 23. Convexity of a ratio function involving polygamma functions and monotonicity of the sectional curvature of the Fisher--Rao manifold Source: math.GM (General Mathematics) Link: https://arxiv.org/abs/2609.10628 This paper proves that the sectional curvature of the Fisher--Rao manifold of beta distributions, restricted to the diagonal, is strictly decreasing. 24. Duality and Baire like properties in topological abelian groups Source: math.GN (General Topology) Link: https://arxiv.org/abs/2609.14074 This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. 25. Quasiisometric embeddings of right-angled Artin groups into products Source: math.GR (Group Theory) Link: https://arxiv.org/abs/2610.37824 We provide obstructions to the existence of quasiisometric embeddings of right-angled Artin groups into products of trees, hyperbolic spaces, and other right-angled Artin groups. 26. Loop Space Models and Integral Homology of Connected Quandles Source: math.GT (Geometric Topology) Link: https://arxiv.org/abs/2610.38066 We construct a marked integral delooping of a connected quandle's covering monoid, realizing its rack space as a homotopy fiber. 27. Beyond Selected Breakthroughs: Evaluating Language Models on Randomly Sampled Mathematical Conjectures Source: math.HO (History and Overview) Link: https://arxiv.org/abs/2610.40382 The last year has seen a shift in AI-assisted mathematics, with frontier language models producing proofs of and counterexamples to long-standing open conjectures. 28. The Interference-Conjugate Transform: The Tightest Interference-Linear Surrogate for Power Control, Scheduling and Beamforming in Wireless Networks Source: math.IT (Information Theory) Link: https://arxiv.org/abs/2610.37794 Closed-form fractional programming (FP), weighted MMSE (WMMSE), and the 2026 surrogate-enhanced methods SEFP and XMMSE all maximize the weighted sum rate by iteratively maximizing a surrogate linear in the interference power. 29. Analytic Kasparov product for non-separable C$^*$-algebras Source: math.KT (K-Theory and Homology) Link: https://arxiv.org/abs/2610.40351 We develop an analytic framework for the explicit construction of Kasparov products on uncountably generated Hilbert modules over non-separable C$^*$-algebras by extending the classical Kasparov bimodule picture. 30. An Arithmetic Property of Intertwining Operators for Real Reductive Groups Source: math.RT (Representation Theory) Link: https://arxiv.org/abs/2610.40261 Let $G$ be the group of complex points of a connected split reductive group defined over ${\mathbb Q}$. 31. Expander Evolution Algebras Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2605.12672 We introduce \emph{expander evolution algebras} (EEAs), a class of nonassociative algebras defined over an arbitrary field $\K$ in which the underlying undirected loopless graph of the algebra -- in the sense of Kowalski -- is an expander graph in the classical sense of Cheeger. 32. Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem Source: math.QA (Quantum Algebra) Link: https://arxiv.org/abs/2206.06000 For a split quasireductive supergroup $G$ defined over a field, we study structure and representation of Frobenius kernels $G_r$ of $G$ and we give a necessary and sufficient condition for $G_r$ to be unimodular in terms of the root system of $G$. 33. Small-time Wasserstein action for KPZ and $\Phi^4_3$ Source: math.PR (Probability) Link: https://arxiv.org/abs/2610.38785 We determine the leading small-time Wasserstein action of the full spatial laws of centered KPZ and dynamical $\Phi^4_3$, both started from zero. 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.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.IT (Information Theory); math.KT (K-Theory and Homology); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.NT (Number Theory); 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 33 of 472 available items for this weekly brief.