Papers

We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.

Perhaps surprisingly, there are infinite types that can be exhaustively searched mechanically in finite time. We use ideas from topology to build plenty of them, referring to searchable types as compact types, and we use ordinals to measure their logical complexity. We consider two systems of ordinal notations under which a single notation denotes both a discrete ordinal and a compact one, with an embedding of the former into the latter whose image has empty complement. A boolean valued function decides which points in the image of the embedding are isolated and which are topological limit points. The first system consists of the traditional Brouwer codes and the second is an inductive-recursive universe generalizing them. The discrete ordinals so obtained are trichotomous, and the compact ones have the least element property for complemented subsets, but these two desirable properties cannot be fulfilled simultaneously in a constructive setting. The ordinals obtained from Brouwer codes further enjoy a boolean Leibniz principle, which has the notion of total separatedness as its topological counterpart. This extends previous work from Gödel's system T to intensional Martin-Löf type theory with univalent universes, and is formalized in Agda in the TypeTopology repository.

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke–Maass cusp forms for the group $\mathrm{SL}_n(\mathbb Z)$, where $n\geq 2$ is an integer. Our main result establishes, under the Generalised Ramanujan–Petersson Conjecture, that for all $X\leq x\leq 2X$ outside a set of measure $o(X)$ the short interval $(x,x+H]$ contains at least $H(\log X)^{1/n^2-1}$ indices with both positive and negative Hecke eigenvalues, provided that $(\log X)^{1-1/n^2+\varepsilon}\leq H\leq X/10$. For $\mathrm{GL}_2$ Hecke–Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual $\mathrm{GL}_3$ forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan–Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $\beta_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related by the quadratic bounds \begin{equation*} \hbar_{p,X}\lesssim(\beta_{p,X})^2, \qquad \beta_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\sqrt n$, and a second family with the reverse behaviour.

Let $X$ be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that $X$ admits no complete metric with uniformly positive scalar curvature when either infinite relative $K$-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset with compact complement. The proof compares twisted Dirac operators on spin covers of compact manifolds with boundary and combines a heat-kernel argument on fundamental domains with a long-neck estimate.

We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k(End_C(X))) for any ambient Frobenius algebra X; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over C has fewer than 2^(dim_C(H)) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible finite-index subfactors N < M, we improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to fewer than 2^[M:N] intermediate subfactors.

We construct a faithful functor from the category of cubespaces, arising in structured ergodic theory and nilspace theory, to the category of (condensed) simplicial sets. This functor translates the weak structure theorem of a nilspace to a Postnikov tower of the corresponding simplicial set. In particular, nilspaces are mapped to Kan complexes and the structure groups of the nilspace correspond to the simplicial homotopy groups of the Kan complex. The functor is constructed via pullback along a special cosimplicial cubical set, whose combinatorial properties may be of independent interest.

Existing airspace capacity metrics mainly quantify occupancy or flow, although the same number of aerial vehicles may result in different motion alternatives. This letter establishes mobility information capacity as an information-theoretic measure for low-altitude wireless networks. It quantifies the maximum information that trajectory observations reveal about intentional maneuver inputs under a given maneuver-resource budget and environmental uncertainty. For a common fixed feedback architecture, we formulate a lifted linear-Gaussian mobility channel and derive its finite-horizon log-determinant capacity. Cost and uncertainty whitening gives the spatiotemporal mobility eigenmodes, whose optimal maneuver-resource allocation follows water-filling. When the number of nondegenerate modes grows linearly with time and their efficiencies become asymptotically symmetric, we arrive at the Shannon-like law $R_M^{\rm G}=\frac{B_M}{2}\log_2(1+\mathrm{MNR})$, where MNR is the mobility-to-noise ratio. The proposed measure opens a motion-centric capacity perspective for the sky, while remaining a distinguishability baseline rather than a collision- or geometry-constrained airspace capacity.

Algorithmic stability refers to the property of an algorithm being insensitive to perturbations of the input data, where the type of perturbation may vary depending on the setting. In this work, we develop a general framework to quantify the extent to which any ensembling strategy defined via averaging can yield stability guarantees for any type of data perturbation. Our main theoretical result is a guarantee on the stability of this ensembled algorithm, given in terms of the norm of a certain covariance operator that describes the ensembling process. We show how our general framework yields interpretable and intuitive insights in several examples of perturbations of practical interest, and provides much sharper guarantees than those obtained from privacy considerations.

In this article, we establish $\mathrm{L}^p$-mapping properties for the generalized Stokes operator with bounded measurable coefficients on the full range of exponents that is known to be sharp for elliptic systems with rough coefficients. These include $\mathrm{L}^p$-estimates for the corresponding semigroup, its gradient and the $\mathrm{H}^\infty$-calculus, as well as $\mathrm{L}^q$-$\mathrm{L}^p$ smoothing estimates. As a key ingredient, we develop a non-local framework for $\mathrm{L}^p$-extrapolation that captures the relationship between hypercontractivity, non-local decay estimates and uniform $\L^p$-bounds.

We prove the sharp Hausdorff–Young inequality on quantum euclidean spaces. Our result implies the sharp Hausdorff–Young constants for the Weyl transform. The key ingredients are a sharp Hessian comparison inequality and a novel flow related to the mixed- norm of noncommutative Gabor transform; the Hessian comparison leads to the monotonicity of the flow. We also apply the same flow technique to prove the sharp Hausdorff–Young inequalities on Heisenberg groups and the affine group.

We determine the integer group determinant sets for $C_{24}$, $C_2\times C_{12}$ and $C_2^2\times C_6$, the three abelian groups of order $24$. Character factorization expresses each determinant as a product of cyclotomic norms, whose components must satisfy integral compatibility conditions. The classifications are constructive and reduce each nonzero signed integer to valuation conditions at $2$ and $3$ and a finite test on its remaining prime factors. The tests use conductor quotients and norm-preserving global units. Each prime contribution has a uniform bound independent of its multiplicity, and successful tests yield integer coefficients realizing the prescribed determinant. The critical cyclic strata share one compatibility group. For $C_2^2\times C_6$, a normalized rational–Eisenstein pair settles the odd values and most even strata; target stabilizers reduce the remaining tests. We also prove that the odd determinant set of $C_2^2\times C_6$ is properly contained in that of $C_2\times C_{12}$, exhibit an infinite family in the difference, and determine the signed powers of two in all three sets. Exact arithmetic certificates and coefficient constructors accompany the proofs.

We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our multi-agent, multi-model setup that chains together GPT 5.6, GPT 6, Claude Opus 5, and Claude Fable 5.1.

We study nonlinear Schrödinger equations with nonlinear Stratonovich noise \begin{equation*} \mathrm{d} u\,=\, i\bigl[ \Delta u \,+\, \lambda|u|^{p-1}u\bigr] \, \mathrm{d} t \,+\,i|u|^{(q-1)/2}u\circ \mathrm{d} {W}, \end{equation*} in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269–315] with the approach from [Ann.\ Inst.\ H.\ Poincaré Phys.\ Théor.\ 46 (1987), pp.\ 113–129] we obtain local well-posedness for all energy-subcritical nonlinearities $p,q\in [1, 1+4/(d-2)_+)$ together with a corresponding blow-up alternative. For a linear multiplicative noise $q=1$, a real-valued noise $W$ and a defocusing nonlinearity $\lambda\le 0$, we check this blow-up condition using a bound on the energy, resulting in the global well-posedness of the equation. If both nonlinearities are mass-subcritical, i.e., $p,q\in [1, 1+4/d)$, we provide an improved blow-up criterion involving the $L^2(\mathbb R^d;\mathbb C)$-norm. Using the conservation of mass for real-valued $W$, we obtain global well-posedness also in this case. Compared to previous results on stochastic nonlinear Schrödinger equations, we thereby improve the range of exponents $p$ and $q$ and the spatial regularity assumption on the noise.

Originating in ergodic structure theory, nilspace theory is concerned with the study of cubical spaces. A first key result in this theory, the weak structure theorem, is the representation of a concrete fibrant compact cubespace as a tower of its canonical truncations, exhibiting each step over the previous one as a principal bundle by a compact abelian group. We present a categorical proof of the weak structure theorem by systematically interpreting the relevant concepts in an appropriate context.

Motivated by recent work on tree independence number, we study the path independence number of a graph $G$: the minimum integer $k$ such that there is a path decomposition of $G$ where each bag induces a graph with independence number at most $k$. We show that every graph excluding both an induced forest minor and an induced star has bounded path independence number. This characterises when a graph class that excludes an induced star has bounded path independence number while also partially resolving a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel and Wiederrecht (2024). Furthermore, we show that graphs excluding both an apex-forest induced minor and an induced star have bounded tree independence number. As a consequence, for every fixed apex-forest $H$ and integer $t$, there is a polynomial-time algorithm to test whether a $K_{1,t}$-induced-subgraph-free graph contains $H$ as an induced minor. Moreover, it follows that the Maximum Weight Independent Set problem, as well as several other NP-hard problems, can be solved in polynomial-time on $K_{1,t}$-induced-subgraph-free graphs that exclude $H$ as an induced minor.

A completion theory for hyperbolic distance data is developed at the interface of matrix analysis, graph theory, and hyperbolic geometry. Krein's characterization of the metric space embeddability in Lobachevsky space leads to a natural anchoring procedure that transforms the indefinite data into a positive semidefinite kernel. In analogy with positive semidefinite and Euclidean distance matrix completion, chordality of the specification graph is shown to be the necessary and sufficient condition for local Lorentz-Gram data to admit global completion. Existence is complemented by explicit constructions. For trees, we obtain geodesic-rectification and product-distance completions; for chordal graphs, the latter extends to matrix-valued transfers along clique-trees. The resulting canonical completion is characterized by sparsity of its inverse and by a maximum-absolute-determinant principle. Its metric distortion exhibits a sharp dichotomy governed by clique separator size. Applications to exact recovery from sparse hyperbolic measurements and to hierarchical and phylogenetic data are developed.

We prove a topological generalization of the Figiel–Lindenstrauss–Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: every nonempty strongly regular CW complex with $v$ vertices, $f$ maximal cells, and a free involution that permutes its cells has $\mathbb Z/2$-index at most $4\log v\log f$. The classical inequality for centrally symmetric polytopes is the special case of polytope boundaries. As an application, we show that every finite simplicial complex homotopy equivalent to $\mathbb RP^d$ has at least $\exp(\Omega(\sqrt d))$ vertices. The previous best lower bound was $\Omega(d^2)$. We also construct triangulations with $\exp(O(\sqrt d))$ vertices. Using the same method, we prove a low-dimensional Postnikov factorization theorem, which gives similar vertex lower bounds for several classical families, such as complex and quaternionic projective spaces, the $d$-torus, closed symplectic and aspherical manifolds, Grassmannians, classical Lie groups, and spherical space forms. Further applications include results in learning theory, convex geometry, computational topology, and topological bounds for chromatic numbers and Vietoris–Rips complexes.

We say that an affine variety $X$ is (semi)rigid if all nontrivial actions of the additive group on $X$ have the same invariant ring. Then all such actions comprise the abelian subgroup denoted $\mathrm{SAut}(X)$. We prove that $X$ is (semi)rigid precisely when the neutral component $\mathrm{Aut}^\circ(X)$ of its automorphism group is nested. In this case, the neutral component is the semidirect product of any maximal algebraic torus and $\mathrm{SAut}(X)$. The proof uses Laurent expansions of algebraic curves in $\mathrm{Aut}^\circ(X)$.

Fix an integer $d\leq-2$. We characterize Hom-finite algebraic triangulated categories admitting a $(-d)$-simple-minded system as stable categories $\underline{\mathrm{CM}}(B)$ of proper $(-d)$-self-injective non-positive dg algebras; equivalently, each admits a $d$-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a $d$-Calabi–Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right $(d+1)$-Calabi–Yau structures on the finite-dimensional part with normalized right $d$-Calabi–Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav–Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left–right Calabi–Yau symmetry.

A compressible, non-isothermal dilute polymer solution is formulated as a Class-II binary mixture with separate solvent and polymer mass and momentum balances and with total energy and entropy balances for the complete mixture. The entropy exploitation is sharpened by a balance-anchoring axiom introduced here: in a fixed balance representative, each selected binary dissipative product contains a constitutive factor from an unclosed balance flux or source. Polymer configuration is resolved first by a connector distribution and then by its conformation tensor. Population kinematics fix polymer transport and, by moments, yield a two-velocity upper-convected rate; objectivity verifies covariance rather than selecting it. Within the total entropy balance, configurational transport and deformation powers cancel their chemical-potential and elastic partial-stress counterparts when deformation and stress-decomposition weights match. A Gordon–Schowalter test independently requires the affine upper-convected choice for the stated dumbbell free energy and Kramers stress unless an additional reversible channel is supplied. Coordinated stress–interaction changes shift the local entropy flux/production pair by a divergence, exposing representation dependence of local mechanism-wise production. An entropy-invariant Class-II-to-Class-I reduction selects a descendant entropy flux preserving the parent production and yields thermo-chemical, configurational-stress and partial-viscous-stress diffusion terms. The omitted quadratic relative-inertia flux is paired with relative kinetic-energy storage and transport and is a reversible truncation, not missing entropy production. The resulting compressible non-isothermal Hookean stress and temperature equations reduce to Oldroyd-B/UCM only after one-velocity, incompressible and isothermal limits.

We study factorizations of the single-particle spectrum of non-interacting quantum systems and their consequences for many-particle statistics. Representing the single-particle partition function by a spectral alphabet, tensor factorizations become multiplicative factorizations of the alphabet, which can be lifted to canonical Bose and Fermi partition functions using standard symmetric-function and $\lambda$-ring identities. We show how product spectra arise from different factorizations, how antisymmetrization can be assigned across an odd number of factors, and which tensor factorizations are compatible with a fixed spectrum. The paper therefore studies spectral factorization and provides a consolidated combinatorial framework for canonical Bose and Fermi partition functions.

We study motivic integration on varieties that are birational to smooth algebraic stacks and prove a generalization of the orbifold formula for varieties with finite quotient singularities. This leads in particular to a new way of computing stringy E-functions and we also give applications to the study of klt-singularities. Along the way we study Cluckers-Loeser motivic integration on algebraic stacks, which might be of independent interest. The proof relies furthermore on ideas from Hrushovski-Kazhdan integration and Bruhat-Tits theory.

A combination of Taylor polynomial approximation and averaging results in a powerful quasi-interpolation technique that can be used to analyze the approximation properties of finite element spaces. With this particular approach, the notion of affine equivalent families is completely bypassed in the sense that the degrees of freedom are defined locally rather than being exported from a reference element. In particular, domains with curved boundaries can be handled in a natural manner. Finally, the ease of its application to a particular finite element makes it a valuable pedagogical tool.

We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.

Mean field planning on a surface prescribes initial and terminal densities and minimizes a transport energy subject to the continuity equation. Proximal algorithms for this problem repeatedly solve a time–space Poisson equation, whose temporal derivative and surface gradient determine the density and momentum corrections. We study a gradient-enhanced approximation of this constraint projection using finite differences in time, surface finite elements in space, temporal polynomial preserving recovery, and spatial parametric polynomial preserving recovery. The same construction is incorporated into ISTA, FISTA, and Douglas–Rachford splitting. We distinguish the recovered update from an exact discrete projection and derive residual identities and conditional finite-iteration perturbation bounds that retain data, boundary, and linear-solver errors. Existing derivative-recovery estimates identify a higher-order contribution under suitable regularity and mesh assumptions; they do not by themselves establish convergence of the outer optimization iteration. Available numerical illustrations on the sphere and a more complicated algebraic surface are discussed together with the limits of the recorded refinement data.

Nonlinear complexity is a fundamental criterion in the evaluation of pseudorandom sequences. The construction of multi-sequences with high nonlinear complexity is both theoretically and practically important in cryptography. Motivated by prior constructions of multi-sequences with high nonlinear complexity in [IEEE Trans. Inf. Theory, 60(10), 2014] and [IEEE Trans. Inf. Theory, 63(12), 2017], we provide a unified framework via narrow ray class fields and the cyclic descent introduced by Guruswami and Xing in [J. Combin. Theory Ser. A 129 (2015) ]. Then we can generate new multi-sequences with high nonlinear complexity over function fields with arbitrary genera.

We characterize equality in the finite free Stam and entropy-power inequalities, proving that Hermite polynomials are the unique extremizers among simple real-rooted inputs, up to independent translations and scalings. The proof turns this classification into a rigidity problem for projective plane curves. Using hyperbolicity and the Helton-Vinnikov theorem, we express the Jacobian defect as an off-diagonal squared norm in a definite symmetric pencil. Together with score transport, this yields a matrix proof of Stam for all real-rooted inputs. For each simple base pair, the directions in which the defect vanishes are the independent translations and common dilation, forming a three-dimensional subspace in every degree whose curves split into $n$ projective lines. At a collision, the leading configurations are again finite free convolutions of normalized derivatives of the velocity polynomials of the colliding input clusters. Combined with incidence counting, this local formula shows that at most one real fiber is singular, every real singularity is an ordinary totally real multiple point, and the ordered collision multiplicities determine the real normalization covering. For $n\geq3$, every non-split curve has at least $2n-2$ non-real projective discriminant zeros, counted with multiplicity, with equality attained by irreducible curves of geometric genus zero through every simple input pair. The leading Fisher-information coefficient is determined by the colliding tangent configurations, while the finite entropy term retains the gaps between clusters. For $n\geq3$, maximal logarithmic entropy divergence along the optimally weighted score direction is equivalent to Stam equality.

We give a simple proof for the existence of weak central polynomials for the algebra of multiplications of a finite-dimensional simple (non-associative) algebra. As an example we present explicit weak central polynomials in the cases of the three-dimensional simple Lie algebra and the Jordan algebra of the two-dimensional vector space with non-degenerate symmetric bilinear form.

We study the convexity of the Bergman space Berezin range of composition operators induced by unit disk and unit ball automorphisms. We obtain several new results characterizing convexity of the Berezin range for linear fractional unit disk and unit ball automorphisms composed with unitary maps.