Clay weekly context brief for the Maths category (ISO week 2026-W30).

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. Regularity thresholds for anomalous dissipation and related phenomena in passive scalars
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2603.11466
   We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$.
2. Optimal Control of Heterogeneous Mean-Field Stochastic Differential Equations with Common Noise and Applications
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2511.18636
   We initiate the study of optimal control problems of heterogeneous mean-field stochastic differential equations with common noise.
3. A complete ultrametric on von Neumann's incomplete tensor products
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2607.09627
   We revisit von Neumann's theory of infinite tensor products of Hilbert spaces.
4. First-Order Modal Logic in HOL: Deep and Shallow Embeddings with Automated Faithfulness (Extended Preprint)
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2607.10880
   We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics.
5. The pants graph as a combinatorial model for the Teichm\"uller space of a non-orientable surface
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2607.15792
   We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover.
6. On the Howe--Moore property for automorphism groups of buildings
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2606.27993
   Let $G$ be a closed type-preserving subgroup of the automorphism group of a thick locally finite building $X$ of finite rank, and assume that $G$ acts Weyl-transitively.
7. Collatz Representations With Bounded Partial Quotients
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2504.07970
   We define Collatz representations for a subset of rational numbers and prove that each real number \( x \notin (-1,1) \) can be approximated arbitrarily well by rational numbers which have only \( 2 \)'s and \( 1 \)'s in their Collatz representation.
8. Continuation of Hamiltonian dynamics from the plane to constant-curvature surfaces
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2604.13250
   We investigate the deformation of symmetry on cotangent bundles from the Euclidean plane to two-dimensional constant-curvature surfaces and the continuation of local dynamics aspects in Hamiltonian systems.
9. Riemannian foliations on CROSSes
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2602.16491
   We classify Riemannian foliations of closed manifolds homotopy equivalent to CROSSes.
10. Continuous Hochschild Cohomology and Formality
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2512.21090
   We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories.
11. A gap theorem for non-trivial maximal intersecting families and an exact weighted asymptotic
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2607.16040
   Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$.
12. Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2604.03854
   Let $E\subset\mathbb{C}$ be a compact set with connected complement, and let $\rho_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on $E$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively.
13. On Quantum Aspects of 1-Form Symmetries II: Bordism, Invertible Phases, and Anomalies
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2606.07056
   We study quantum anomalies associated with $U(1)$ 1-form symmetries from the perspective of invertible phases and bordism.
14. Stable Determination of Coefficients in Nonlinear Dynamical Schr\"odinger Equations by Carleman Estimates
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2508.07231
   We consider the inverse problem of recovering stationary coefficients in a class of dynamical Schr\"odinger equations with locally analytic nonlinear terms.
15. Isomonodromic deformations of Higgs bundles and characterization of the non-abelian Noether--Lefschetz locus
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2606.18768
   Let $f:X\to S$ be a smooth proper family of smooth projective varieties.
16. Markov Chains Approximate Message Passing
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2512.02384
   Markov chain Monte Carlo algorithms have long been observed to obtain near-optimal performance in various Bayesian inference settings.
17. An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2507.15234
   Forward-backward stochastic differential equations have recently become a key focus in the computational field, and their role in continuous-time stochastic optimal control and reinforcement learning has grown increasingly prominent.
18. Fixing Divergence in Carleman Linearization via Analytical Continuation
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2607.05873
   Nonlinear differential equations play a crucial role in modeling a wide range of phenomena, yet their solutions remain notoriously difficult to obtain.
19. Hilbert's tenth problem for rings of holomorphic functions of bounded order
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2406.12791
   One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable.
20. Magnus submonoids and membership problems in one-relator, surface and hyperbolic groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2509.24480
   Motivated by its applications to the word problem for one-relator inverse monoids, via results of Ivanov, Margolis, and Meakin (2001), we prove several decidability and undecidability results about the submonoid membership problem in one-relator, surface and hyperbolic groups.
21. General Categorial Geometry and Algebraic Topology
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2403.19689
   In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation.
22. Ergodicity and weak mixing for group-indexed infinitely divisible stationary processes
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2601.17988
   We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing.
23. An Embarrassingly Simple Way to Optimize Orthogonal Matrices at Scale
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2602.14656
   Orthogonality constraints are ubiquitous in robust and probabilistic machine learning.
24. Wide subcategories and brick-finiteness for length categories
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2607.15991
   We extend to abelian length categories of finite rank a characterisation of brick-finiteness known for finite-dimensional algebras.
25. Generalized Nordhaus--Gaddum Inequalities for Eigenvalues
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2607.15941
   For a graph $G$, let $ \lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$.
26. A construction of Kakeya Sets in Arbitrary Dimension
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2607.14824
   We construct Kakeya sets in arbitrary dimension $d\geq 2$, generalizing the classical Perron tree construction beyond dimension $2$.

Sources in this brief: math-ph (Mathematical Physics); 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.GM (General Mathematics); math.GR (Group Theory); math.GT (Geometric Topology); math.LO (Logic); math.OC (Optimization and Control); math.PR (Probability).
Selected 26 of 196 available items for this weekly brief.