Clay weekly context brief for the Maths category (ISO week 2026-W31).

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. Fully Convergent Projection-based Methods with Momentum under Nonconvex Geometric
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2607.22510
   Nonlinear optimization problems with complicated, nonconvex, yet geometrically structured constraints can be tackled by projected-gradient methods: under weak regularity assumptions, these approaches were recently proved to possess convergence properties to the strongest stationarity conditions.
2. The Dirac equation in (split-)octonions: origins, variants, and modern context
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2607.09703
   Octonions and split-octonions have been used to express the Dirac equation in physics in several conceptually distinct ways.
3. Quantum Cellular Automata from Kramers-Wannier Dualities and Modular Relations
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2607.21728
   Recent work has constructed higher-dimensional analogs of non-invertible symmetries similar to 1+1d Kramers-Wannier duality.
4. Parameter-related strong convergence rates of Euler-type methods for time-changed stochastic differential equations
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2510.16405
   An Euler-type framework with equidistant step sizes is proposed for a class of time-changed stochastic differential equations. We establish the strong convergence rate of the standard Euler--Maruyama method under the global Lipschitz condition.
5. 1-Uryson width and covers
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2505.21126
   We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large?
6. An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development
   Source: math.HO (History and Overview)
   Link: https://arxiv.org/abs/2509.02758
   Euclidean geometry has historically played a central role in cultivating logical reasoning and abstract thinking within mathematics education, but has experienced waning emphasis in recent curricula.
7. Extended Weak Order for the Rank 3 Universal Coxeter Group
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2509.00871
   The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite.
8. Paper-folding models for the CAR algebra
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2508.04837
   We show that the CAR algebra admits a Cantor spectrum C*-diagonal that is not conjugate to the standard AF diagonal.
9. Double Categories of Open Systems: the Cospan Approach
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2509.22584
   This is an overview of double categories of "open systems": systems that can interact with their environment.
10. Quantitative analytic stable regularity
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2607.21762
   We prove quantitative stable regularity lemmas for binary real-valued functions, extending the work of Malliaris and Shelah for stable graphs.
11. Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2504.19864
   In this paper, we improve the $L^p$-Rellich and Hardy-Rellich inequalities in the setting of radial Baouendi-Grushin vector fields.
12. Representation stability in the (co)homology of vertical configuration spaces
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2412.01128
   In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space.
13. The Homogeneous Landau Equation with Regularised Thermal Noise
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2607.22329
   We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise.
14. Permutation twisted cohomology, remixed
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2509.00954
   For each endotrivial complex for a $p$-group arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners.
15. Dimension and topology in transserial tame pairs
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2508.16415
   Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field.
16. Langevin for Nonconvex Optimization: Exact, Inexact and Zeroth-Order
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2607.22353
   We study Langevin-based methods for non-convex optimization under smoothness and dissipativity assumptions.
17. On the Drazin Index of an Anti-Triangular Block Matrix
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2604.08137
   The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses.
18. Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2607.21698
   Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems.
19. Efficient QR-based Column Subset Selection through Randomized Sparse Embeddings
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2509.03198
   In this paper, we introduce an efficient algorithm for column subset selection that combines the column-pivoted QR factorization with sparse subspace embeddings.
20. Point-to-set principles in dynamical systems
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2607.21976
   In analogy to the point-to-set principle for Hausdorff and packing dimension of Lutz, Lutz, and Mayordomo, we develop a Kolmogorov complexity point-to-set principle framework for upper and lower topological pressure and BS dimension in dynamical systems (and by extension for upper and lower entropy).
21. An antichain condition for infinite groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2603.10759
   Let $\chi$ be a subgroup-theoretical property.
22. Mixed Poincar\'e and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2607.17315
   We prove a fixed-outer-domain content--capacity estimate for every positive codimensional gain on an unbounded complete $2$-PI space.
23. Colimits of internal categories
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2501.17769
   We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits.

Sources in this brief: 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.FA (Functional Analysis); math.GR (Group Theory); math.HO (History and Overview); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.OC (Optimization and Control); math.QA (Quantum Algebra); math.RA (Rings and Algebras).
Selected 23 of 141 available items for this weekly brief.