This work analyzes revises the interior penalty (IP) discontinuous Galerkin (DG) method imposed in [Chen, G., Qiu, W., \& Xu, L. (2021). Analysis of an interior penalty DG method for the quad-curl problem. IMA Journal of Numerical Analysis, 41(4), 2990-3023.] for the quad-curl problem in a nonconvex polyhedral domain, while introducing a piecewise constant coefficient matrix. We derive two main results: Under minimal regularity assumptions, we prove that the numerical solutions converge strongly to the true solution in the $ H(\text{curl}) \times H^1(Ω)$ norm. Under slightly higher regularity, we establish the optimal estimate of the convergence rate depending on the regularity of the solution. These two results, serving as a complement to the existing literature, completely answer how the concerned IP DG method performs on quad-curl problems with low regularity.

In this paper, we establish a general comparison theorem for the order statistics of the edge eigenvectors for generalized Wigner matrices. Consequently, we derive the Gumbel law for the maximal edge eigenvector component and prove the universality of the Gaussian fluctuations of the order statistics in an intermediate regime close to the maximum. In addition, our comparison result also implies a quantitative first order estimate for moderately small order statistics.

This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform $μ$-entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.

A central operator schedules each vehicle's passage time through a bottleneck to achieve a dynamic system optimum (DSO). The assignment depends on each vehicle's preferred arrival time, which is private and must be elicited from each vehicle. Mechanisms that elicit exact preferences, such as the Vickrey-Clarke-Groves (VCG) mechanism, can achieve strategyproofness but involve relatively complex rules and a computational burden on the operator. We focus instead on coarse reporting, in which each vehicle selects from a finite menu of time slots of a common width. This discrete interface already structures reservation and appointment systems in practice, including managed lanes for automated vehicles, airport slot allocation, and delivery appointment windows. We design a slot-based DSO mechanism on this coarse interface, in which the operator implements DSO assignment based on the reported slots and charges a capacity shadow price as a toll, and evaluate its performance. We prove that both the worst-case misreporting gain and the expected efficiency loss decrease quadratically in the slot width. The efficiency loss decays in this way under binding capacity, while the worst-case misreporting gain requires an additional condition on the preferred arrival time distribution and the schedule cost function. Analyzing the no-toll case, we find that the misreporting incentive persists, however finely the slots are refined, indicating that the toll also serves to elicit truthful reports. Numerical experiments support these theoretical results and show that they continue to hold in parameter regions outside the sufficient conditions.

We use perfect matchings and labelled Fano planes to construct and study the $2025$ orthogonal bases of positive roots in the $E_8$ root system. The set of these bases forms a highly structured, Bruhat-like graded poset $(Ω, \leq_Q)$ whose rank function can be computed from the cardinalities of so-called generalized Rothe diagrams. We give combinatorial characterizations of these diagrams in terms of matchings and Fano planes, and we explain how to compute the ranks of the elements of $Ω$ using suitable combinatorial statistics such as the weights of perfect matchings. We establish simple formulas for the rank generating functions of $Ω$ and of its 50 congruence classes under a natural order congruence relation. Our derivation of the generating functions contains some intermediate results on general perfect matchings and labelled Fano planes that can be stated without mentioning root systems and may be of independent interest.

In this paper we consider the friendship-bias of the vertices in an infinite rooted Galton--Watson tree. The friendship-bias of a vertex is the difference between the average degree of the neighbours of the vertex and the degree of the vertex itself. A vertex is said to be of type $χ\in S$, with $S = \{-,0,+\}$, when its friendship-bias is, respectively, strictly negative, zero or strictly positive. We consider the fractions $f_l^χ$ of vertices of type $χ\in S$ along a random downward path up to branching depth $l \in \mathbb{N}$ and derive a large deviation principle (LDP) for the triple $(f_l^χ)_{χ\in S}$ as $l\to\infty$. The branching depth of a vertex counts the number of branchings that occur along the path that connects the vertex to the root of the tree. The rate in the LDP is $l$, while the rate function in the LDP is identified in terms of a variational formula minimising a relative entropy under a linear constraint. We focus on the case of binary branching, for which the rate function is already quite involved. We identify the qualitative properties of the rate function and show how it can be computed numerically. We briefly indicate how to proceed for more general branching and for vertex types along a tree consisting of a finite number of random downward paths. Our paper is the first to consider large deviations of vertex types.

Relative symplectic cohomology is an invariant of compact subsets of a closed symplectic manifold, introduced by Varolgunes. There are many examples of computations of this invariant over the Novikov field, but the collection of computed examples over the Novikov ring is still quite limited. One reason for this is that such computations require determining the relevant Floer complexes for Hamiltonians that are not necessarily $C^2$-small Morse functions. In this work, we present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in $\mathbb{C}P^n$. Our computation relies on explicit descriptions of Floer complexes, in the Morse--Bott setting with cascades, for J-shaped Hamiltonians on $\mathbb{C}P^n$. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in $\mathbb{C}P^n$.

For datasets with unknown but stationary serial dependence, a robust long run variance estimator is essential to handle diverse scenarios. Spectral variance estimators are commonly used but tend to exhibit significant negative bias in the presence of positive correlation. To overcome this, zero lugsail estimators have been introduced, offering zero asymptotic bias regardless of the correlation structure. However, there are currently no guidelines for selecting the optimal bandwidth for lugsail estimators, a critical component in the estimation process. We propose an inference optimal bandwidth rule for lugsail estimators, based on nonstandard fixed-smoothing limiting distributions developed in our study. This approach significantly improves bias correction, accounts for variability, and provides an estimator optimized for robust inference. Our theoretical findings are supported by a simulation study.

We ask whether knotting can be recognised using persistent homology. Starting from a point-cloud representation of a curve, we compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score to these cycles. Motivated by proteins but tested more broadly, the method reveals systematic differences between knotted and unknotted structures in both protein families and synthetic examples. This suggests that knotting leaves a detectable persistent-homology-based signature.

Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $γ\in(α,1)$; the sub-linear regime is essential), not constant-$β$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.

We develop a non-overlapping domain decomposition method for the numerical solution of the Stokes-Biot model of fluid-poroelastic structure interaction in a mixed form. The model is based on a velocity-pressure formulation for the free fluid, a three-field stress-displacement-rotation formulation with weakly symmetric stress for the solid deformation, and a Darcy velocity-pressure formulation for the fluid in the poroelastic media. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman slip with friction condition are imposed on the interface. The interface conditions are incorporated through Lagrange multipliers modeling the traces of the displacement and the Darcy pressure. The system is discretized using stable mixed finite element spaces for Stokes flow, elasticity, and Darcy flow. The domain is decomposed into a union of subdomains of either Stokes or Biot type with three types of interfaces: Stokes-Stokes, Biot-Biot, and Stokes-Biot. On the Stokes-Stokes interfaces, a normal stress Lagrange multiplier is introduced to impose weakly velocity continuity, while the Biot-Biot and Stokes-Biot interfaces are equipped with displacement and pressure Lagrange multipliers to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced via Schur complement to an interface problem for the Lagrange multipliers, which is solved by GMRES. Each iteration requires the solution of local Stokes or Biot problems, which can be performed in parallel. We show that the resulting interface operator is positive definite and analyze the convergence of the GMRES iteration through fields-of-value analysis. Numerical experiments are presented to illustrate the performance of the method.

A general framework is developed for deriving sharp inequalities on simple graphs from majorization and Schur-convexity. After establishing majorization relations between the spectrum of an arbitrary graph and the spectra of the complete, complete bipartite, and matching graphs, it is shown that every positive Schur-convex spectral functional yields several sharp inequalities relating $λ_1$, $|λ_n|$, and $\|G\|_\ast$. This reduces the problem of proving graph inequalities to the choice of a suitable Schur-convex function. This optimization problem is then studied within the family of random vector norms, whose moment and cumulant expansions connect the framework to the numbers of closed walks. This yields new sharp results, recovers classical inequalities from a unified viewpoint, and produces further bounds in settings such as triangle-free and square-free graphs.

We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types $\mathbb{R}^k\times S^{n-k}$, $\mathbb{R}^k_+\times S^{n-k}$ or $\mathbb{R}^k\times S^{n-k}_+$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.

We provide several new characterizations of Property A for bounded degree graphs. In particular, we show that $(X,d)$ has Property A if and only if there is a proper gauge $ω$ such that the Lipschitz free space $\operatorname{LF}(X,ω\circ d)$ is isomorphic to $\ell_1$. As a consequence, all finitely generated groups with Property A admit proper uniformly Lipschitz affine actions on $\ell_1$. Moreover, for groups with finite Nagata dimension, we obtain actions with compression exponent 1. This result applies to higher rank lattices, such as $\operatorname{SL}(3,\mathbb{Z})$. We also show that a countable discrete group coarsely embeds into $L_1$ if and only if it admits a proper uniformly Lipschitz affine action on a subspace of $L_1$.

Mazur's knot exterior in $S^1\times S^2$ admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior through Adams' theorem on thrice-punctured spheres. The same octahedral framework applies to the family of Jester manifolds introduced by Sparks. Using hyperbolic geometry, hyperbolic Dehn filling, and recent results on systolic geodesics, we prove that the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic, regardless of orientation. Consequently, the corresponding compact, contractible $4$-manifolds are pairwise nonhomeomorphic.

We associate a profinite group to every group \(G\) acting on a set \(Ω\) with finite orbits. For each finite \(G\)-stable subset \(A\subseteqΩ\), let \(G_A\leq\operatorname{Sym}(A)\) be the induced finite permutation group. The groups \(G_A\), with the natural restriction maps, form an inverse system, and we define $Γ_Ω:=\varprojlim_A G_A$. We show that \(Γ_Ω\) acts naturally on \(Ω\) and is canonically topologically isomorphic to the closure of the image of \(G\) in \(\operatorname{Sym}(Ω)\), endowed with the topology of pointwise convergence. We introduce the finite-level exactness property \(\textup{FLEP}\), under which subgroups of \(Γ_Ω\) are recovered up to closure from their fixed-point sets, and closed subgroups are recovered exactly. We prove several equivalent formulations of \(\textup{FLEP}\). Under this condition, the fixed-point set construction gives an inclusion-reversing bijection between closed subgroups of \(Γ_Ω\) and the fixed subsets of \(Ω\) arising from closed subgroups. We apply the theory in two directions. First, every profinite group \(Γ\) is recovered from its normal finite-quotient action on $\coprod_{N}Γ/N$, where \(N\) ranges over the open normal subgroups of \(Γ\). For this action, \(\textup{FLEP}\) holds precisely when every finite quotient \(Γ/N\), with \(N\) open and normal, is a Dedekind group. Second, if \(G\leq\Aut(E)\) acts on a field \(E\) with finite orbits and \(F=E^G\), then \(E/F\) is Galois and the construction yields a canonical topological isomorphism $Γ_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where \(\operatorname{Gal}(E/F)\) has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.

We study Schubert varieties of the affine Grassmannian for the general linear supergroup $GL_{n|m}$. An explicit computational study is conducted in low-dimensional cases, namely for dimensions $n|m = 1|1$ and $2|1$. We describe the supervariety structures that arise in these settings, providing coordinate descriptions, equations, and morphisms.

In this paper we study the structure of the $\infty$-category of spectral Lie algebras. We show that this $\infty$-category admits an interesting symmetric monoidal structure, defined by an analog of the smash product of pointed spaces, and that the free Lie algebra functor $\mathrm{Sp} \to \mathrm{Lie}(\mathrm{Sp})$ is symmetric monoidal with respect to it. Moreover, this property of the free functor essentially characterizes the spectral Lie operad (among nonunital operads in spectra). This result may be thought of as Koszul dual to the more familiar fact that the free commutative algebra functor takes direct sums to tensor products. One of the key ideas is that the $\infty$-category of spectral Lie algebras behaves in many ways like the $\infty$-category of pointed spaces. More precisely, we deduce structural facts about spectral Lie algebras from familiar statements about spaces by differentiating, in the sense of Goodwillie calculus. The tool to do this is the highly structured generalization of Arone-Ching's chain rule established by Blans-Blom. Numerous other features of spectral Lie algebras follow as well, such as a version of Mather's second cube lemma, the relation between the James construction and loop-suspensions, the Hilton-Milnor splitting, and a version of the EHP sequence.

The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Apéry-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Apéry-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Padé approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Apéry-type mechanism for e+pi is found.

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

Many nonsmooth, nonconvex objectives in learning and signal recovery are $ρ$-weakly convex. We minimize such a function in deterministic and stochastic settings when the weak-convexity parameter $ρ$ is unknown. The objective is not required to be globally Lipschitz continuous or smooth. We propose the Adaptive Prox-Guided Scheme (APS), a one-trial proximal algorithm that adapts the proximal parameter online and bidirectionally through a descent test, allowing it to exploit favorable local structure. In the deterministic setting, APS obtains an $O(\varepsilon^{-2})$ iteration complexity for producing an $\varepsilon$-subgradient stationary point. In the stochastic setting, APS achieves a high-probability $O(\varepsilon^{-2})$ iteration bound for driving the Moreau-envelope gradient below $\varepsilon$. This result holds under deliberately weak oracle assumptions: the function-difference estimates may be biased and heavy-tailed, and the stochastic proximal oracle need only be sufficiently accurate with constant probability when the proximal parameter lies below $1/(2ρ)$ (unknown to the algorithm), and can be arbitrary otherwise.

Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$-nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose restricted enveloping algebras $u(L)$ and $u(H)$ are isomorphic as $F$-algebras. Such pairs exist in every dimension at least $p+5$, with $\dim L'=p$ and $\dim H'=p+1$. Thus, the restricted isomorphism problem has a negative answer over every field of positive characteristic, even for $p$-nilpotent algebras over perfect fields.

For a finite poset $\mathcal{P}$, its height $h(\mathcal{P})$ is the number of cover relations in its longest chain. When $\mathcal{P}$ is a lattice $\mathcal{L}$, we label its elements $x$ with $h(x_\downarrow) = h([\hat{0},x])$ and its cover relations $x \lessdot y$ with $h(y_\downarrow) - h(x_\downarrow)$. When a lattice $\mathcal{L}'$ extends $\mathcal{L}$, $h(x_\downarrow)_\mathcal{L} \leq h(x_\downarrow)_{\mathcal{L}'}$. We study lattices $\mathcal{L}$ and $\mathcal{L}'$ such that $h(x_\downarrow)_\mathcal{L} = h(x_\downarrow)_{\mathcal{L}'}$. Cover relations labeled $1$ in $\mathcal{L}$ induce a poset that we call the (long) skeletal poset $\mathrm{SK}(\mathcal{L})$. Its Hasse diagram is the largest spanning subgraph that the Hasse diagrams of $\mathcal{L}$ and $\mathcal{L}'$ have in common. An example of lattices $\mathcal{L}$ and $\mathcal{L}'$ is the alt-Tamari lattices introduced by Chenevière, where every alt-Tamari lattice $\mathrm{alt}\text{-}\mathrm{Tam}_n$ extends the Tamari lattice $\mathrm{Tam}_n$/refines the Dyck lattice $\mathrm{Dyck}_n$ such that $h(x_\downarrow)_{\mathrm{Tam}_n} = h(x_\downarrow)_{\mathrm{alt}\text{-}\mathrm{Tam}_n}$. We study $\mathrm{SK}(\mathrm{Tam}_n)$ with another poset we introduce. We enumerate intervals in these posets. For a well-chosen distributive lattice, we introduce its altitude lattices, which generalize the alt-Tamari lattices $\mathrm{alt}\text{-}\mathrm{Tam}_n$. Altitude lattices within a family have the same number of linear intervals. They are related to each other via extensions, refinements, and embeddings of some skeletal posets. For a poset $\mathcal{P}$ with $\hat{0}$, we define its Kneser graphs $KG(k) := (V(k),E)$, where $V(k) := \{x: h(x_\downarrow) = k, 1 \leq k \leq h(\mathcal{P})\}$ and $E := \{(x,y): x_\downarrow \cap y_\downarrow =\hat{0}\}$. We give some observations about them in a reconstruction setting.

We investigate the image of polynomials multilinear in graded variables evaluated on the algebra of upper triangular matrices endowed with a group grading. We show that, in general, such an image need not be a vector subspace. However, under the additional assumption that the identity component of the grading is commutative, we prove that the image is always a vector subspace. We further investigate the image of polynomials evaluated on inverse and direct limits of algebras. As a consequence, we prove that the image of a polynomial evaluated on a direct limit of upper triangular matrix algebras whose identity component is commutative is always a vector subspace.

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.

We prove that target-avoidance sets for Manneville--Pomeau maps are strong winning for Schmidt's game. More precisely, for the class of nonuniformly expanding interval maps considered here, there exists a single parameter $α>0$ such that for every target $p\in[0,1]$, the set of points whose forward orbit does not accumulate on $p$ is $α$-strong winning. The proof induces on the uniformly expanding region $[r_1,1]$. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu--Li--Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville--Pomeau map.

In this paper, we consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater or equal to 1, within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for plane curves.

This paper revisits the strong duality in the linear quadratic regulator (LQR) for continuous-time and discrete-time systems, and explores its interconnection with typical assumptions and the uniqueness of primal-dual solutions. Using a linear operator $Ψ$, we formulate a common nonconvex LQR problem that captures both time domains. We then derive its Lagrange dual problem and establish the strong duality via a rank-constrained tight semidefinite program (SDP) relaxation. Further, we show that the primal-dual optimal solutions to the SDP relaxation, after dropping the rank constraint, recover the classical algebraic Riccati equations and optimal feedback gains in a constructive manner. The dual derivation and strong duality analysis rely on mild standard assumptions and exploit the properties of the linear operator and its adjoint, revealing a structural symmetry between the two time domains.

Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.

The end space of an infinite graph arises naturally in many contexts as an important invariant and an interesting construction. It compactifies a locally finite graph and Diestel shows how to extend the end space to a larger space, called the tangle space, which is able to compactify any infinite graph. In both ends and tangles, it is the vertex-connectivity structure of the graph that is being studied. If we switch our attention to edge-connectivity, we can analogously define edge-ends. There is a space known as the edge-direction space which turns out to play an analogous role as the tangle space in its relationship with the end space: the edge-direction space provides a larger compact space in which the not necessarily compact space of edge-ends lives in. In this paper, we make this analogy precise, providing a natural edge analogue definition of tangles and proving they result in exactly the edge-directions. We also describe a combinatorial construction of certain Boolean algebras which give rise, via Stone duality, to the tangle and the edge-direction spaces. Finally, we pursue functorial definitions of the combinatorial constructions used in the paper, inspired by the famously functorial nature of Stone duality and by previous work by one of the authors and colleagues on trying to functorialize the end space construction. We hope our work will provide foundation and inspiration for further work on infinite graph theory that makes ample use of category theory and powerful algebraic constructions such as Boolean algebras and Stone duality.