Papers

Given a fixed past-directed timelike vector field, does there exist a time function whose gradient is optimally aligned with it? We address this question by introducing a functional that, on the one hand, captures the misalignment between the timelike vector field and the gradients of suitable Sobolev functions, and, on the other hand, penalizes null gradients. Our analysis focuses on compact subsets of smooth stably causal spacetimes. More precisely, we prove that, under suitable assumptions on the Sobolev index and the strength of the null gradient penalization, there exists a unique smooth temporal function which minimizes the considered functional. We refer to this minimizer as the alignment time function. Furthermore, several useful properties of the alignment time function are established: there exists a canonical procedure to improve its steepness, it is stable under $C^{p}$ convergence of the underlying metrics and vector fields and it inherits the symmetries shared by the metric and the given vector field.

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$–exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le \beta\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

In this paper, a braid is regarded as a dynamical system of points in the plane. The states of this dynamical system are given by Delaunay triangulations. This construction makes it possible to define an abstract groupoid $\overset{{abc}}{\mathcal{G}^{4}_{n+3}}$, which gives a representation of the colored braid groupoid $\text{ColB}(n)$. We define homomorphisms ${f}_{n+3}:\overset{{abc}}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{Q})$ and ${f}'_{n+3}:\overset{{abc}}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{C})$, and describe an algorithm for computing the resulting invariants.

We introduce the notion of a coarsely minimal Reeb flow, generalizing the notion of minimal geodesic flow, and prove the following rigidity theorem: That a coarsely minimal Reeb flow satisfying a divergence property is orbitally equivalent to the geodesic flow of a Riemannian metric of negative sectional curvature. Without the divergence assumption, we obtain an orbital semi-equivalence. This extends a rigidity result for geodesic flows of negatively curved Riemannian metrics which is due to Gromov. We use Floer homology and Morse's hyperbolic `stability' Lemma.

Numerical methods and neural networks have transformed our understanding of nonlinear partial differential equations (PDEs), yet discovering mathematical representations that explain their dynamics remains largely an expert task. We introduce Agentic Symbolic Search (ASYS), in which coding agents construct and refine explicit approximations from a known PDE and supplied mathematical knowledge. ASYS couples evolutionary search over differentiable symbolic programs with gradient-based fitting, using the governing equation to evaluate candidates and guide their revision. For a singular generalized Constantin-Lax-Majda profile, ASYS infers a missing decay term that sets how fast the field approaches its singular-time limit, and builds a basis that fits the profile to a median error of $3.90\times10^{-5}\%$. For Fujita blow-up, separating diffusion from reaction and deriving a local response lowers the supplied formula's error from 55.5% to 0.0879%. For Allen-Cahn interfaces, a 14-parameter construction yields a mode-coupling law confirmed by independent simulations. These studies show how agentic search develops mathematical starting points into accurate, analyzable approximations.

We prove that a lacunary hyperbolic group $G = \varinjlim G_i$ with sufficient generics is selfless in the sense of Amrutam–Gao–Kunnawalkam Elayavalli–Patchell, provided the hyperbolicity constants $\delta_i$ and injectivity radii $r_i$ satisfy $\delta_i(\log r_i)^{7} = o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $\delta$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^*$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson's mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^*$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^*$-algebra with strict comparison.

We prove that every graph admits a linked, componental, rooted tree-cut decomposition of finite adhesion that displays all undominated edge-ends. As a first application, we deduce that this tree-cut decomposition also displays the edge-degrees of all undominated edge-ends. For locally finite graphs $-$ where every end is an undominated edge-end $-$ this yields a linked tree-cut decomposition of finite adhesion into $\textit{finite}$ parts that displays all ends and their edge-degrees. As a second application, this latter tree-cut decomposition yields short, unified deductions of Thomassen's theorem on boundary-linked finite partitions, and of Bruhn and Stein's characterisation of Eulerian locally finite graphs in terms of even ends.

We report some work in progress on the relationship between étale and quasicoherent cohomological dimensions.

Torsion pairs, and in particular t-structures, play a central role in the study of triangulated categories. Specifically, t-structures induced by silting (or tilting) objects often admit desirable properties with strong connections to derived equivalences. In this paper, using the correspondence of Saorín-Šťovíček between cohereditary cotorsion pairs in Frobenius exact categories and t-structures in their stable categories, we construct a family of t-structures in the $Q$-shaped derived category of Holm and Jorgensen, arising from admissible partitions of $Q$. We give an explicit description of the associated cotorsion pairs inside the Frobenius exact category of the bifibrant objects, and we identify the corresponding co-aisles by certain homological vanishing conditions. Such t-structures are proved to be induced by a silting object, that can be completely determined by the combinatorics of $Q$. Finally, we illustrate our results by recovering well-known equivalences in the $Q$-shaped setting, while also providing examples where the combinatorial conditions fail (e.g. cyclic quivers), showing that such categories may admit no non-trivial t-structures, revealing phenomena analogous to those observed by Linckelmann in stable module categories.

We establish a direct high-probability last-iterate guarantee for the standard same-sample two-point Gaussian zeroth-order SGD method in smooth, strongly convex stochastic optimization. At each iteration, the method draws a fresh Gaussian direction, evaluates two symmetric perturbations with the same stochastic sample, and takes a norm-normalized stochastic approximation step. Assuming unbiased stochastic gradients and a conditional exponential-moment bound on the squared norm of the stochastic gradient noise, we prove a finite-horizon bound, valid for dimension \(d\ge2\), with an explicit product-weight factor. When the offset in the stepsize schedule is large enough relative to the logarithmic confidence terms, this factor is bounded and the result gives \[ f(\bx_T)-f(\bx^*) = \widetilde{\mathcal O}\!\left(\frac{d}{T}\right) \] with probability at least \(1-\delta\), up to fixed problem parameters and logarithmic factors. Thus the confidence dependence is logarithmic rather than polynomial in \(1/\delta\), and the proof neither invokes Markov's inequality nor truncates the noise. To the best of our knowledge, this is the first direct high-probability last-iterate result at this zeroth-order scale for the same-sample Gaussian recursion under conditional sub-Gaussian stochastic-gradient noise. The proof combines uniform weighted lower and upper scans for Gaussian angles, a product-martingale boundary for the signed suffix-product term, and terminal nonnegative concentration estimates. We also formulate the resulting general pathwise framework for stochastic recursions with random contraction and signed perturbations, identifying the scan, filtration, variance, and terminal-control conditions under which the same argument applies.

Real-time process monitoring requires methods that extract actionable information from high-dimensional time-series data. In this work, we present a new approach for process monitoring that combines tools of topological data analysis (TDA) and machine learning. In the proposed approach, we represent multivariate time-series data as manifolds and use topological descriptors to summarize the structure of such data; we then use a neural ordinary differential equation to learn the dynamic evolution of the topological structure of the system. Using real data from an industrial process, we show that this trajectory-based event detection approach is effective at detecting diverse types of events. We contrast this approach against reconstruction-based approaches such as principal component analysis and autoencoders and against a trajectory-based approach that uses Koopman autoencoders.

Physics-Informed Neural Networks (PINNs) solve Partial Differential Equations (PDEs) by embedding physical laws into neural network training. However, their performance suffers from unstable convergence, training plateaus, and strong sensitivity to architectural and optimization hyperparameters due to the highly non-convex and multi-term structure of the physics-informed loss. In this setting, the outer-loop hyperparameter search is a noisy and black-box optimization problem over heterogeneous parameters, where classical local or gradient-based strategies are easily trapped in suboptimal regions. Evolutionary algorithms, with their population-based exploration and ability to handle mixed, non-differentiable search spaces, provide a more robust mechanism for discovering promising configurations. We propose and investigate a two-stage approach based on evolutionary algorithms that combines exploration and exploitation parts of PINNs training to improve solution accuracy and robustness under fixed computational budgets. In the first stage, we perform low-fidelity training runs with truncated epochs to rapidly screen candidate configurations, treating hyperparameter selection as a black-box outer-loop problem. In the second stage, only the most promising candidates are fully trained with standard gradient-based optimizers to refine the solution. Evaluated on three popular problems, namely Advection, Klein-Gordon and Helmholtz equations, our method consistently outperforms standard training and achieves significantly lower mean error within constrained computational resources.

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

We say that a convex polytope has the clique-face property if every clique in its 1-skeleton is the vertex set of a face. We establish this property as a geometric necessary condition for quadratic generation of toric ideals. More precisely, we prove that every lattice polytope with primitive edges and a quadratic toric ideal has the clique-face property; in particular, this holds for every $(0,1)$-polytope with a quadratic toric ideal. For $(0,1)$-polytopes satisfying condition (E), we characterize the clique-face property in terms of divisibility by monomials occurring in quadratic binomials, and show that, under the clique-face property, such toric ideals have no indispensable monomials of degree at least three. For edge polytopes and cut polytopes, we prove that the clique-face property is equivalent to quadratic generation. This yields new geometric characterizations of quadratic generation for these classes. We also prove that all simple polytopes, matroid independence polytopes, and matroid base polytopes have the clique-face property, and discuss the case of stable set polytopes in connection with conjectures on quadratic toric ideals.

We introduce the $G$-Daugavet property ($G$-DPr, for short) for Banach spaces endowed with an action of a group $G$ by surjective linear isometries. This notion provides a common framework for the classical Daugavet property and the alternative Daugavet property, which correspond respectively to the trivial action and to the scalar action of $S_{\mathbb{K}}$. We establish several characterizations of the $G$-DPr in terms of $G$-slices and closed convex $G$-invariant hulls, recovering the usual slice descriptions of the DPr and the aDPr as particular cases. We show that the presence of a group action leads to new behavior in Daugavet theory. In particular, the $G$-DPr may hold on classical reflexive spaces in sharp contrast with the classical Daugavet property. We relate this phenomenon to convex transitivity, almost transitivity and finite-dimensional rotation problems. We also prove group-action versions of the classical characterizations for $L^1(\mu, X)$- and $C(K,X)$-spaces. The paper also studies group separable determination, $G$-versions of numerical radius and numerical index, and connections between the $G$-DPr and strong Radon-Nikodým and SCD operators. Finally, we introduce a parameter which measures how far the $G$-DPr is from the classical DPr in a quantitative manner. As a consequence of these results, we obtain conditions under which the $G$-DPr recovers several classical implications, including the failure of the RNP for both $X$ and $X^*$, the presence of copies of $\ell_1$ and the failure of the unit ball to be an SCD set.

We reconcile privacy protection and rate-double-robust inference. The privacy of individuals is protected by a local privacy mechanism: injecting noise into their sensitive data, revealing only the noisy data for inference. Hence, privacy protection hinders inference. In contrast, the inference of a target parameter is rate-double-robust when the large-sample bias of an estimator of the parameter is characterised by a trade-off between the estimation errors of two other, nuisance, parameters. Hence, rate-double-robustness facilitates inference. Our starting point of reconciliation is a class of rate-double-robust target parameters indexed linearly by an infinite-dimensional and nonlinearly by a low-dimensional regression. Among others, this includes causal parameters. To infer these targets privately, we show how suitable privacy mechanisms transfer the semiparametric properties of the sensitive-data model to the private setting. Rate-double-robustness is transferred, enabling locally-private, unbiased and semiparametrically efficient inference of our target parameters. Finally, we transform general nonparametric nuisance estimators into private ones, which inherit convergence properties of their nonprivate counterparts. For parametric nuisance models, we develop a private method-of-moments estimator and its large-sample inference theory.

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. We distinguish classical Ziegler pairs, for which the minimal degree of a Jacobian relation denoted by ${\rm mdr}$ differs, from the more general notion detected by distinct graded Betti data. First, we show that for arrangements of $d<9$ lines the intersection lattice determines ${\rm mdr}$, and hence classical Ziegler pairs do not occur in this range. Then we list several distinct Ziegler pairs with $d=10$ in the broader sense. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions. Another rather surprising consequence is that being maximal Tjurina for a line arrangement is not determined by the combinatorics.

Integrated Sensing and Communication (ISAC) refers to the capability for the network to provide communications services whilst also being able to sense the environment in a scalable manner. One of the key functions of ISAC is the accurate localization of passive and mobile sensing targets. This paper introduces a novel hybrid TRP-UE sensing mechanism that improves network-based sensing performance. Evaluation results are provided using 3GPP-compliant ISAC channel models. The results demonstrate the significant benefit in complimenting TRP-based sensing with UE-assisted sensing in challenging propagation environments such as indoor factory.

A c-direct category is a small category equipped with an ordinal degree function such that every morphism is level or degree-raising. Every c-direct category is c-Reedy. The c-Reedy model structure on any functor category from a c-direct category to a model category coincides with the projective model structure. In this framework, a realization functor is a colimit-preserving functor satisfying some mild homotopical conditions from the category of presheaves on a c-direct category with cofibrant representables to a model category. We prove that any two such realization functors are weakly equivalent on cofibrant presheaves. For categories of cubes, we prove that thick categories have cofibrant representables. As an application, we introduce the $\varepsilon$-branching space of an $\mathcal A$-set for any thick category of cubes $\mathcal A$. It is obtained as a coend over a c-direct category with cofibrant representables constructed from $\mathcal A$. We prove that, on free $\mathcal A$-sets generated by precubical sets, this new definition coincides with the earlier one. We prove that, for cofibrant $\mathcal A$-sets, the resulting space is independent of $\varepsilon$ up to homotopy.

We define a combinatorial property of a projective Fraisse category which we call the approximate Ramsey property. Let $F$ be a continuum, $G$ a closed subgroup of the homeomorphism group of $F$, and $\mathbb{F}$ the limit of projective Fraisse category $\mathcal{F}$ such that $\textrm{Aut}(\mathbb{F})$ is dense in $G$. We prove that $\mathcal{F}$ has the approximate Ramsey property if and only if $G$ is extremely amenable. We prove that the group of homeomorphisms of the universal pseudo-solenoid has non-metrizable universal minimal flow.

We develop a new theory of thermodynamic formalism for non-compact flows, providing criteria for existence and uniqueness of equilibrium states, as well as ergodicity and a Gibbs property. These criteria include new versions of the expansivity and specification properties that hold for geodesic flows on non-compact negatively curved manifolds. Previous results on thermodynamic formalism for these geodesic flows were restricted by a pinched curvature condition, which we remove, allowing curvature to approach both $0$ and $-\infty$. Our results also apply to geodesic flows beyond the smooth setting, including locally CAT(-1) spaces.

We prove that every $K_5$-free $n$-vertex graph with sublinear independence number can be made bipartite by removing at most $n^2(1/18+o(1))$ edges, where the constant $1/18$ is best possible. The proof method is related to extensions of Turán Theorem in edge-weighted settings, and part of the proof uses flag algebra.

Spacecraft guidance, navigation, and control functions are increasingly realized as learned policies distilled from expert solvers. Developing such a policy is itself a research process: an investigator selects an architecture and hyperparameters, runs experiments, and must determine whether an apparent improvement is genuine or merely seed noise. This paper presents AutoResearch, a framework in which a large language model autonomously drives that loop for aerospace control problems, coupled with a credibility layer, built into the loop, that certifies each reported result against the problem's own measured seed noise. The language model serves only as the offline research agent that develops the control policy; the trained policy it produces is then deployed onboard the spacecraft, while the model itself never operates the vehicle. At each iteration the agent reads a plain-language problem description and the run history, proposes a single edit to the training script, executes it, and logs the outcome. No reported result is credited until it passes the same three checks: measured per-problem seed noise, reseeded verification of the best configuration, and leave-one-out pruning of the agent's edits. The same loop is applied, unchanged, to two aerospace control problems: a Clohessy-Wiltshire relative rendezvous and a safety-constrained collision-avoidance docking past a keep-out zone, each calibrated against a known optimal control benchmark. In both, the audited policy clears the measured seed noise by many standard deviations; an undirected search over the same parameters does not. On the docking problem the gap becomes categorical: undirected search yields no feasible policy, while the learned policy stays outside the keep-out zone on every seed.

This paper investigates the vanishing and non-vanishing of Betti and Bass numbers for non-finitely generated modules. We prove that for $d$-dimensional Cohen–Macaulay local rings, every non-zero $\mathfrak{m}$-torsion module satisfies $\beta_d(M)\neq 0$, and we establish the Betti number behavior of the injective hull $E_R(k)$. We study tor-rigidity for $H^d_{\mathfrak{m}}(R)$. We also provide positive answer to Schoutens' question on whether the vanishing of some Betti number of a big Cohen–Macaulay algebra forces the Cohen–Macaulay property of $R$. For the absolute integral closure $R^+$, we establish both Tor and Ext results. On the Tor side, we prove that $\beta_i(R^+)=0$ for some $i>0$ implies regularity in a series cases. On the Ext side, we prove that $\mu_i(R^+)=0$ for some $i> d$ forces regularity for Gorenstein domains of prime characteristic, and we obtain analogous results for graded normal domains of dimension $2$ and also for quotient and isolated singularities in any dimension. Also $\mu_i(R^\infty)=0$ forces regularity for isolated singularity.

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathrm{g}$ and the classical cone of dominant integral weights $P^+\subset \mathfrak{h}^*$. The non-integrable simple highest weight $\mathfrak{g}$-modules $L(\mu)$'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for $\mu$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}\alpha_j$ of mutually orthogonal and imaginary simple roots $\alpha_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s (over all BKM LA's), and character of $L(\rho)$ for Weyl vector $\rho$. These needed a family of “integrable” $L(\mu)$'s for $\mu$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $\mu(\alpha_i^{\vee})\leq 0$ for $\mu\in P^{\pm}$ are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu)$ contains “Chevalley-Serre relations” $f_i^{\frac{2}{A_{ii}}{\mu(\alpha_i^{\vee})}+1}L(\mu)_{\mu}=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(\mu)$'s of $L(\mu)$'s for $\mu\in P^{\pm}$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.

In this article we revisit regularity results for eigenfunctions in the discrete spectrum of the electronic Schrödinger equation and study their consequences for approximation complexity. In particular, for the convergence to the complete basis set limit, it can be shown that the curse of dimensionality in the leading algebraic exponent can be mitigated. That is, for general sparse grid constructions, the main term of the convergence rate with respect to the number of degrees of freedom is independent of the number of electrons. These insights indicate potential benefits for classical numerical solvers of the electronic Schrödinger equation and also for quantum-computing approaches through new qubit-efficient wavefunction encodings.

Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain–displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr–Coulomb model and the Duncan–Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

In this paper, we study multi-stage stochastic programming (MSP) problems with nonsmooth composite objectives. Tailored to their intrinsic stage-wise and scenario-wise structure, we develop a single-loop minorized dual decomposition method, in which each iteration constructs a minorized problem and its restricted Wolfe dual, and then performs one iteration of the symmetric Gauss–Seidel based inexact alternating direction method of multipliers on the resulting dual problem to generate the next iterate. A key feature of the proposed optimization framework is that the resulting updates preserve the stage-wise and scenario-wise decomposable structure of the MSP problem and are suitable for parallel implementation. We establish global convergence of the generated iterates for the three-stage case and further establish the corresponding global convergence theorem for the general multi-stage setting. Numerical experiments illustrate the computational viability of the proposed framework and its favorable scaling behavior with respect to the stage-wise and scenario-wise structure.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We report on recent results concerning minimal $G$-invariant characters of $N$ (which are the sums of the characters on each orbit of the action of $G$ by conjugation on $\text{Irr}(N)$) and their influence on the structure of $N$, as well as their relationship to the $G$-conjugacy classes of $N$.