A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard tori, spheres, and cubes admit thresholds.

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 $Ω_+$ result for the divisor problem and his $Ω_-$ result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Voronoï summation formula. This is achieved by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the complex plane.

Recently, the authors introduced the Gamified Katětov order on filters over $ω$. This was shown to be strictly coarser than the classical Katětov order, and in fact collapses all MAD families to a single equivalence class. In the opposite direction, the present paper shows that the Gamified Katětov order also embeds $\mathcal{P}(ω)/\mathrm{Fin}$, and thus contains an antichain of size continuum. The analysis brings into focus some interesting connections with Ramsey theory. As part of a broader programme investigating the interplay between combinatorial and computable complexity, we then apply our construction to produce a large new family of non-modest degrees in the extended Weihrauch hierarchy, which arise from associated effective subtoposes.

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erdős, and Spencer showed that if $H$ is a graph on $k$ vertices whose independence number is $α$, then $K_n \to (mH)_2$, where $m\sim n/(2k-α)$, and that the $1/(2k-α)$ factor is best possible. In the 1990s, Rödl and Ruciński proved that, for all but a few graphs~$H$, the threshold for the property $\mathbb{G}(n,p) \to (H)_r$ is $n^{-1/m_2(H)}$. In this paper, generalizing the result of Burr, Erdős, and Spencer, we prove that $n^{-1/\max\{m_2(H),1\}}$ is the threshold for the property $\mathbb{G}(n,p) \to (mH)_2$, where $m\sim n/(2k-α)$. This threshold matches the one found by Rödl and Ruciński for most graphs $H$, extending their result in the case $r=2$.

We study a new class of preferential attachment trees with \emph{self-reinforcement}. At each time, each vertex is assigned a weight equal to the cumulative sum over past times of an affine function of its degree. A new vertex attaches itself via a single edge to an already present vertex with a probability proportional to the current weight of that vertex. This ``integrated popularity'' rule builds long memory directly into the attachment mechanism, thereby destroying the Markov and partial-exchangeability features that underlie the classical analysis of affine preferential attachment models. More broadly, the model connects to applied-probability work on long-memory self-interacting processes (such as the elephant random walk), emphasizing how non-Markovian reinforcement reshapes asymptotic behaviour. Despite this loss of structure, we identify an explicit exponent $ϕ=ϕ(δ)$ governing both local and global growth: typical degrees at time $n$ scale as $n^{1/ϕ}$, and the empirical degree distribution converges to a power-law with a tail exponent $ϕ+1$. We further prove Benjamini--Schramm local convergence to an infinite random rooted tree characterized via an embedded continuous-time branching process. The limiting tree is a \texttt{sin}-tree, and is \emph{not} the Pólya-type limiting tree arising in the non-reinforced setting. Our results provide a tractable probabilistic description of a natural ``memoryful'' network-growth mechanism, and quantify precisely how reinforcement renormalizes the classical preferential-attachment exponents.

The paper is concerned with the error analysis of a numerical scheme for the approximation of parametric mean curvature flow. The scheme we study is based on a reparametrization using the DeTurck trick and was proposed by Elliott and Fritz in [15]. In the semidiscrete case, for a spatial discretization by finite elements of order $k \geq 2$ we prove an optimal $H^1$-error estimate for the position vector. We present numerical experiments that confirm this error bound and demonstrate that the scheme has good properties with respect to the distribution of mesh points as already observed in [15].

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

When discretizing symmetric stress tensors in variational problems arising in continuum mechanics, one has to choose how to enforce the symmetry of the stress tensor: (i) strongly by requiring the discrete tensors to be pointwise symmetric or (ii) weakly by introducing a Lagrange multiplier. For $H(\mathrm{div})$-conforming finite element discretizations of Hellinger--Reissner elasticity and velocity--stress formulations of incompressible flow, where symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum, we show that this choice may substantially impact the accuracy of the numerical scheme. Through a series of benchmark problems featuring anisotropic constitutive laws inspired by fiber reinforced material, liquid crystal polymer networks, and polar fluids, we show that schemes enforcing symmetry weakly can yield arbitrarily poor stress approximations -- even for zero-stress configurations. However, schemes enforcing symmetry strongly deliver accurate stress approximations independently of the constitutive law, a property we term material robustness. We present a unifying theory that rigorously explains this behavior.

We consider a simple dice game, which leads to an intriguing study of multinomial walks, with surprising and seemingly paradoxical properties. The winning and losing probabilities of a general version of the game are investigated via conjugacy relations between Gamma and Poisson distributions, as well as between negative multinomial and inverted Dirichlet distributions. We show a monotonicity property of the regularized beta function, which implies a monotonicity property of the winning probability. Furthermore, the asymptotic behavior of the game for one or several parameters of the game tending to infinity is analyzed, as well as the probability of being last in the game.

We introduce the data driven extreme value distribution (DDEVD) estimator, a kernel-based method for estimating extreme value distributions from data. We derive its mean integrated squared error (MISE) in detail, use it to compute the optimal bandwidth and establish stability conditions for the bandwidth optimization procedure.

We prove a Serre type vanishing property for the twisted primitive cohomology of a symplectic manifold. It is based on Tseng and Zhou's vanishing property under the symplectic flatness. These vanishing properties emphasizes the necessity of the symplectic flatness when generalizing certain results from the sheaf cohomology in complex geometry to the primitive cohomology in symplectic geometry.

We investigate a one dimensional flux limited Keller Segel system (FLKS) in which the chemical decay rate is allowed to vary explicitly in time, a feature motivated by enzymatic regulation and environmental variability in chemotactic signalling. Treating the decay rate as an arbitrary function, we carry out a systematic Lie symmetry analysis of the resulting PDE system and employ equivalence transformations to perform a complete group classification, we identify the kernel symmetry algebra admitted for arbitrary decay functions and determine three distinguished cases that extend the symmetry algebra constant decay rates, inverse time (power law) decay, and exponential decay. For each case, we construct an optimal system of subalgebras and derive the corresponding similarity reductions. Finally, we find some explicit solutions for our FLKS model. Our results provide a rigorous mathematical foundation for understanding which temporal decay patterns admit similarity reductions, thereby enabling analytical progress on flux limited chemotaxis models with realistic time varying degradation mechanisms.

Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.

This paper connects two seemingly different ways of studying knots: quantum group invariants and the dynamics of Morse flows. For fibered knots, we define a two-variable series invariant by counting Morse flow loops in the complement. This dynamical series is conjectured to agree with the BPS $q$-series of the knot complement, which arises from Verma modules for quantum groups and encodes all colored Jones polynomials. We prove this correspondence for all braid-homogeneous knots.

Relativistic hydrodynamics successfully provides an effective field theory description for the low energy regime of many out-of-equilibrium systems. On the other hand, in this paper we proof that any stand-alone hydrodynamic EFT is inherently acausal and therefore requires the addition of transient UV modes in order to restore causality. This is made possible by the exponential decay of dissipative hydrodynamics in a majority of the lightcone, allowing the possibility of a causal description that still reduces to the hydrodynamic one at late timescales. We then investigate the emergence and possible restrictions of the non-hydrodynamic modes in these causal UV completions.

We develop a finiteness notion for unbounded chain complexes over a commutative noetherian integral domain $R$ employing the Abel summation method. The algebraic K-theory of such complexes is defined, and shown to be non-trivial. We also exhibit a natural map from the (usual) algebraic K-theory of $R$ into the new K-theory and show that its image contains a canonical infinite cyclic subgroup.

In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the extrinsic distance, and the extrinsic volume growth or the pinching curvature. In particular, we prove that the essential spectrum of a complete non-compact submanifold $M^n$ in a Euclidean space is $[0, +\infty)$ provided the second fundamental form $A$ of $M^n$ satisfies $\|A\|_{L^p} < \infty$, $p>n$.

In the directed landscape, the Martin boundary coincides with the horofunction boundary. We show that functions in this boundary are precisely the eternal solutions possessing a spatial growth rate, and that the minimal Martin boundary is given by the Busemann functions. Moreover, every eternal solution can be expressed as a max-plus convex combination of countably many Busemann functions. Horofunctions are exactly those eternal solutions that admit a representation in terms of at most two Busemann functions with a common growth rate. As a consequence of instability, not all horofunctions are Busemann functions, and the Martin boundary is strictly larger than its minimal part.

Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In applications, however, the measures of interest are almost never known precisely, calling for a theory of optimal transport that accounts for statistical uncertainty. We construct such a framework, lifting the classical theory to the setting of random probability measures. We introduce the $L^2$ over Wasserstein space establishing that it inherits the formal Riemannian structure of the Wasserstein space by characterising distances and geodesic geometry. The structure induces random flows with Wasserstein gradient flow sample paths, making it the natural extension of the Wasserstein space which allows for random gradient flow dynamics. We ensemble statistical convergence results of the optimal transport machinery using the empirical measure within the $L^2$ over Wasserstein framework. Moreover, in the setting of Bayesian non-parametrics, we refine Schwartz's consistency theorem to the Wasserstein topology and deduce posterior convergence of the same machinery in the $L^2$ over Wasserstein space. We demonstrate that the growing theory of random token sampling for transformer models using self-attention flow paths can be embedded into the our framework. The results provide a unified treatment of random optimal transport and its consequences for principled inference and generative modelling under the statistical uncertainty of random sampling.

We study the problem of testing $H_0: ξ^\topβ=t_0$ in high-dimensional sparse linear regression with Gaussian random design and unknown design covariance. The loading vector $ξ$ is arbitrary, and the exact sparsity level $k$ is unknown but bounded by a known value $k_u$. Tests are required to control Type I error uniformly over the $k_u$-sparse null, while power is evaluated against $k$-sparse alternatives. We construct a computationally efficient mixed test that gives an upper bound on the adaptive separation distance and establish an information-theoretic lower bound calibrated to the magnitude profile of $ξ$. In the ultra-sparse regime $k_u\lesssim \sqrt n/\log p$, these bounds characterize the adaptive separation rate up to logarithmic factors for arbitrary $ξ$. In the moderately sparse regime $\sqrt n/\log p\ll k_u\lesssim n/\log p$, these bounds match for several classes of loading vectors but may differ in general. In this regime, we further prove a low-degree lower bound that matches the upper bound up to logarithmic factors. This provides evidence that improving on the rate of the mixed test, if statistically possible, may be computationally hard. For flat sparse loadings, we complement this evidence with a polynomial-time reduction from sparse CCA. Finally, we examine how information about the design covariance affects the adaptive separation rate in two settings. Under a sparse signed-spiked covariance model, the information-theoretic lower bound is attainable up to logarithmic factors by a computationally inefficient procedure, while the low-degree lower bound and sparse-CCA reduction continue to apply, providing evidence for a statistical-computational gap. When the design covariance is known and diagonal, the adaptive separation rate takes the same form as in the ultra-sparse regime.

We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $λ\geq0$. Under suitable conditions, the operator is self-adjoint for every $λ>0$, while the formal limit at $λ=0$ is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-$λ$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $λ\to0$ in the strong resolvent sense, proving that for every sequence $λ_j\to0$ one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.

Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.

Eduardo Sontag and coauthors studied Input-to-Output Stability (IOS) and the output asymptotic gain property. These notions changed control theory and recently had an impact on robust adaptive control through the Deadzone-Adapted Disturbance Suppression (DADS) control scheme. Moreover, recently the notion of IOS was extended to systems described by Partial Differential Equations (PDEs). In this work, we celebrate Eduardo Sontag by combining DADS and IOS for PDEs: we study the partial-state regulation problem for a scalar Ordinary Differential Equation (ODE) which is interconnected with a possibly infinite-dimensional system. In such a case the DADS control scheme can allow an escape from the requirements of the small-gain theorem that is mainly used for partial-state feedback. We show the design procedure of partial-state DADS controllers and we prove robust regulation even in the presence of external inputs (disturbances) without assuming knowledge of any disturbance/parameter bounds. The DADS controller is applied to three different cases of the interconnection of an ODE with an almost completely unknown: (a) heat PDE, (b) transport PDE, and (c) wave PDE with viscous damping. We show that the same DADS controller can achieve robust regulation in all three cases.

Completeness proofs in categorical semantics usually proceed by building a syntactic category whose composition is given by substitution. For untyped effectful Call-by-Value languages, this runs into a basic obstacle: there is no canonical notion of simultaneous substitution of computations, since evaluation order is semantically meaningful. We address this by taking single computation substitutions, that is, binding steps, as primitive, and representing computation substitution by finite sequential lists composed by concatenation. We formalize this idea in a one-object Freyd-multicategorical setting. We introduce Freyd operads, separating a cartesian operad of values from a symmetric Ren-cartesian preoperad of computations, connected by a Freyd functor, and from any Freyd operad we construct a corresponding Freyd PROP of substitutions. We prove that this construction is representable and, in the strict one-object setting, left adjoint to restriction to codomain 1. Using the induced term model, we interpret untyped computational lambda-calculus with procedures and higher-order functions in weakly closed Freyd operads, and prove soundness, initiality, and completeness. This yields a categorical semantics tailored to untyped effectful computation and broad enough to encompass realizability-oriented models such as monadic combinatory algebras.

In future large-scale deployments of 6G and beyond networks, collecting timely information, as measured by the Age of Information (AoI) metric, is becoming increasingly important. At the same time, the environmental impact, often characterized by the resulting Carbon Footprint (CF), depends on both the amount of consumed energy and the Carbon Intensity (CI), i.e., the amount of CO$_2$-equivalent emissions produced per unit of consumed energy. Since CI varies over time, minimizing energy is not equivalent to minimizing CF, as a status update with the same energy demand may result in a different carbon cost depending on when it is transmitted. This makes timely status updating a nontrivial scheduling problem. To address this challenge, we formulate carbon-aware status updating as a constrained Markov Decision Process (MDP) that minimizes AoI subject to CF budget, transmission duty-cycle, and channel-capacity constraints. We then propose Sustainable Age-of-Information-Based Timely Status Updating for Hardware-constrained Edge networks (SAOITHE), a Whittle-index-based scheduling solution that enables scalable real-time scheduling. Using real-world CI traces across low-, medium-, and high-CI regions, the results show that SAOITHE remains within the allocated CF budget while achieving lower AoI than baseline policies. Moreover, the gains are around 25% and 20% in low- and medium-CI regions, respectively, and up to 75% in high-CI settings, while preserving scalability.

We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.

We give a complete calculation of the cobordism ring of stably almost complex $C_p$-manifolds in terms of generators and relations. We also compare these generators with the geometrically-defined generators obtained by Kosniowski.

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo powers of $2$ satisfied by $\bar{a}_s(n)$ for infinitely many $s$. Our approach builds upon generating function manipulations, Hecke eigenform theory and results of Newman.

We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional.

A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay between nonlinear diffusion and density saturation leads to a rich variety of behaviors across different parameter regimes. We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits. The analysis relies on a priori estimates derived from suitable entropy functionals. Pattern formation is demonstrated by reducing the system to a first-order equation and conducting a detailed analysis of the resulting nonlinearity. Numerical simulations from a one-dimensional finite-volume scheme illustrate the asymptotic regimes.