Papers

We study a class of content delivery network design problems in which service performance is explicitly modeled through queueing-based response times. In contrast to standard formulations relying on distance-based approximations, the resulting models incorporate congestion effects and give rise to nonlinear expressions involving ratios of affine mappings, reciprocal terms, and interactions across multiple traffic classes. To address these challenges, we develop a systematic framework based on mixed-integer conic liftings that enables exact reformulations of such expressions within a tractable optimization paradigm. The proposed approach combines McCormick-type envelopes and second-order cone representations to derive mixed-integer conic programming formulations that jointly capture geometric design decisions and stochastic service dynamics within the same system. The framework is first introduced in a single-server setting, where the optimal solution exhibits a connection with the classical Fermat–Weber problem, and is then extended to the multi-server case, leading to a general formulation that simultaneously optimizes server locations, capacity allocation, and routing decisions. Computational experiments on a realistic case study illustrate the impact of congestion-aware modeling on network design and demonstrate the effectiveness of the proposed formulations when solved using modern conic optimization solvers.

We present a theory of point estimation with e-statistics (e-values and e-processes) by introducing the "ME-estimator": the parameter that minimizes the corresponding e-statistic, or the evidence against it. Our approach is based on the intuitive idea of e-statistics as a measure of evidence and betting pay-off, and naturally generalizes the classical method of maximum likelihood estimation. First, we establish the consistency as well as the almost sure convergence rate for ME-estimators relating to the high-probability bounds on the size of the confidence set derived from thresholding the e-statistics, an approach that sets ME-estimators apart from traditional M-estimators. Second, we conduct classical M-estimator-style analysis on the consistency and asymptotic normality of ME-estimators in the bounded mean estimation setting, discussing the notion of efficiency (or lack thereof) from various choices of betting strategy. Our work brings e-statistics, a fundamental tool for inference and uncertainty quantification, to the space of estimation.

We study privately estimating the sum of $n$ user-held values in the presence of an honest-but-curious server. This motivates requiring privacy not only at data release but also throughout server-side computation. We therefore adopt the local (pure) differential privacy model, in which each user transmits a noise-perturbed value. It is well known that independent local noise typically incurs a substantial utility loss compared to the centralized model, where noise is added only after aggregation. We show that this gap is not fundamental. By carefully designing correlations among the locally added noise variables, we construct $\varepsilon$-DP mechanisms whose estimation cost matches the optimal cost achievable in the centralized setting, up to an arbitrarily small error.

In this article, we consider a particular Hayashi $R$-matrix that satisfies the Yang-Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$. Using the FRT-bialgebra technique and Woronowicz's method of construction, we construct a concrete compact quantum group $U_{q,\Theta}(3)$ for non zero real $q$ and modulus one complex deformation parameters $\theta_{ij}$. We then study in detail the irreducible $*$-representations of the underlying $C^*$-algebra $C(U_{q,\Theta}(3))$, using the representations of the three dimensional noncommutative torus. Also, a monomial basis for the dense Hopf *-algebra $\mathbb{C}[U_{q,\Theta}(3)]$ is obtained.

We consider polynomial rings over finite fields and rings of integers of imaginary quadratic fields $\mathbb{Q}(\sqrt{-D})$. In this paper, we formulate the density of squarefree elements divisible by all elements of $T$ but by none of $P$, where $T$ and $P$ are subsets of squarefree elements and $T$ is finite. We also define Mersenne irreducibles in order to estimate the density of squarefree elements divisible by none of $P$.

Many learning tasks in science and engineering are characterized by sparse datasets, which limits the effectiveness of purely data-driven approaches. At the same time, these problems are often accompanied by rich domain knowledge derived from physical laws, operational requirements, and expert heuristics. Such knowledge is frequently expressed as rules involving logical propositions and linear inequalities. Existing neuro-symbolic methods typically enforce these rules approximately through soft penalties, assume input-independent rules when designing specialized architectures, or rely on non-differentiable post-processing at inference time to achieve hard constraint satisfaction. While recent advances in differentiable optimization layers enable end-to-end feasibility enforcement within neural networks, extending these approaches to logical or mixed-integer rules remains challenging due to inherent nonconvexity. In this work, we propose a unified end-to-end framework for enforcing hard, input-dependent mixed integer linear constraints within neural networks. Our approach represents rules as disjunctive constraints and applies hierarchical convex relaxations to obtain convex hull formulations. These relaxations yield tractable linear constraints that can be embedded as differentiable optimization layers while enabling exact rule satisfaction. We demonstrate the effectiveness of the proposed framework on real-world datasets, achieving perfect rule satisfaction and strong predictive performance.

Many machine learning problems involve multiple inherent trade-offs that are best addressed by gradient-based multi-objective optimization (MOO) algorithms. Existing methods are often proposed with various motivations, analyzed case by case, and differ algorithmically in how the component gradients are aggregated at each step. In this work, we develop a unifying framework for gradient aggregation in MOO, establishing (optimal) rates of convergence to Pareto stationarity, the standard measure of performance in MOO. Central to our analysis is a sufficient alignment condition, from which we derive a theorem showing that non-conflicting directions, when chosen within the convex hull of gradients, form a fundamental sufficient condition for convergence. We further show that feasibility can be ensured through projection onto the dual cone, broadening the scope of methods that admit convergence guarantees. In parallel, we present a primal optimization perspective of gradient aggregation that encompasses established algorithms, clarifies their theoretical relationships, and enables the design of new variants. As an illustration, we introduce capped MGDA, derived from a CVaR-based formulation, and demonstrate its robustness in adversarial federated learning. Finally, we validate our theory through experiments on synthetic problems and practical benchmarks.

We compute the small quantum cohomology ring of a Verra fourfold. Using the theory of atoms recently developped by Katzarkov–Kontsevich–Pantev–Yu, and building on recent papers of the authors, we deduce that a Verra fourfold is never birational to a very general cubic fourfold, nor to a very general Gushel–Mukai fourfold, whereas it was previously known that a general Verra fourfold is birational to a general nodal Gushel–Mukai fourfold. More precisely, we show that for every smooth cubic fourfold or smooth Gushel–Mukai fourfold that is birational to some Verra fourfold, the primitive cohomology is isomorphic, as a rational Hodge structure, to the middle cohomology of some projective K3 surface.

Given two Blaschke products $A$, $B$ of the same degree, a welding homeomorphism of $A$, $B$ is a circle homeomorphism $k: \mathbb{T} \rightarrow \mathbb{T}$ satisfying $A\circ k=B$ on $\mathbb{T}$. We show that any two pairs of Blaschke products sharing a welding homeomorphism must factor through a common pair of Blaschke products, and we give a criterion for detecting whether two Blaschke products admit such a factorization.

We prove that a very general complex symmetric Verra fourfold is not \(\ZZ/2\)-birational to \(\PP^4\), using the theory of atoms introduced by Katzarkov–Kontsevich–Pantev–Yu~\citep{Katzarkov2025BirationalIF} in the equivariant setting as in Cavenaghi–Katzarkov–Kontsevich~\citep{cavenaghi2026atomsmeetsymbols}.

We investigate the structural and operator-theoretic foundations of indirect signal-generation mechanisms in Keller–Segel-type chemotaxis models. By analyzing a physically motivated multi-stage signaling cascade, we establish a precise mathematical dichotomy between instantaneous equilibration and transient kinetic memory. Specifically, we prove that the fully equilibrating parabolic–elliptic–elliptic (PES) cascade reduces to a static fourth-order elliptic interaction. In dimension four, an exact algebraic cancellation of the leading Newtonian singularity yields a purely logarithmic kernel, shifting the mass-critical dimension from $N=2$ to $N=4$. Through an $L^2$-gradient flow formulation, we identify the corresponding concentration-scaling candidate critical mass $M_* = 64\pi^2\tau/\chi$. In sharp contrast, we demonstrate that the mixed elliptic–parabolic (MEP) cascade retains a genuine Volterra memory effect that defies static reduction. Its interaction drift acts as a singular perturbation in time-exhibiting classical two-dimensional Keller–Segel principal order near the time diagonal, yet providing fourth-order smoothing in its frozen-time average, necessitating a mixed space-time threshold theory. These results isolate the physical origin of mass-critical dimensional shifts in multiscale biological systems and formulate the specific adapted Adams/logarithmic Hardy–Littlewood–Sobolev (log-HLS) inequalities and mixed-norm criteria required to close the threshold problems.

Social systems consist of networks of individuals who influence one another through social interactions. Studying how processes evolve on these networks can help us better understand patterns of social behavior. We study a system that couples online and offline social activity and investigate how to learn effective models directly from data using Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy), a method for discovering governing equations. We assess learning performance using data generated by a mean-field approximation model of a stochastic interaction process on networks and test how accurately the system can be recovered under different noise levels. Our results show that using more trajectories improves accuracy when noise is high, but only a small number of additional trajectories is needed to gain most of the benefit, with little improvement beyond that. We also learn effective ODE models from averaged stochastic data on networks. When traditional mean-field approximations fail, identifying continuum ODEs directly from stochastic processes yields efficient models that better match the data and provide deeper insight into the underlying dynamics.

Killing forms on finite groups arise as special cases of braided Killing forms on braided Lie algebras. If $\mathcal{C}$ is a conjugation-stable subset of a finite group $G$, the Killing form on $\mathbb{C}\mathcal{C}$ is given by $K_\mathcal{C}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b \in \mathcal{C}$. It is conjectured in previous work by López Peña, Majid and Rietsch that $K_\mathcal{C}$ is non-degenerate for any real conjugacy class $\mathcal{C}$ in a finite simple group. In this article, we reformulate the conjecture and introduce combinatorial conditions - the $\textit{1-element condition}$ and the $\textit{2-element condition}$ - that are sufficient for non-degeneracy to hold. This allows us to prove the conjecture for simple groups of the form $\mathrm{PSL}_2(q)$ and certain conjugacy classes in the alternating and symmetric groups. Moreover, we verify computationally that every real conjugacy class in a simple group of order $\leq 10^9$ fulfills at least one of these two conditions, thereby significantly extending the computational evidence for the conjecture. This raises the question whether these conditions are satisfied by all conjugacy classes in finite simple groups.

For a simple graph $\Gamma$ and a real number $\lambda$, the general reduced second Zagreb index is defined by the formula $$GRM_\lambda(\Gamma)=\sum_{ab\in E(\Gamma)}[(\deg_{\Gamma}(a)+\lambda)(\deg_{\Gamma}(b)+\lambda)]\,.$$ A sharp lower bound for $GRM_\lambda$ over all trees of given order and maximum degree under the condition that $\lambda\ge -\frac{1}{2}$ is established. A parallel result is proved for unicyclic graphs under the condition $\lambda\ge -\frac{1}{2}$. The corresponding minimal trees and unicyclic graphs are identified. These findings improve upon the lower bounds previously established by Buyantogtokh, Horoldagva, and Das concerning $GRM_\lambda$ of trees and unicyclic graphs of given order.

Brualdi, Friedland, and Pothen studied sparse bases of row spaces and proposed a rank-intersection criterion for elementary row-space vectors of sparse generic matrices. We give a \(4\times 8\) sparse generic counterexample. The example has four elementary vectors in its row space whose zero sets satisfy all the proposed rank-intersection inequalities, but the four vectors satisfy a nontrivial linear relation and hence do not form a basis. The construction also explains the obstruction: the proposed inequalities see only ranks of column sets indexed by intersections of zero sets, while linear independence of the corresponding elementary vectors is governed by the relative position of the hyperplanes annihilated by their coefficient vectors.

Modular Invariant Theory is a branch of mathematics that explores the behavior of polynomial functions invariant under group actions, particularly over fields with positive characteristic. Overall, modular invariant theory serves as a vital link connecting algebraic methods with combinatorial and topological applications. Based on the existing literature, this undergraduate thesis aims to investigate conjectures and problems emerging from the work of Macdonald (1992), and the recent work of Lewis, Reiner, and Stanton (2017), as well as subsequent developments by L. M. Ha, N. D. H. Hai, and N. V. Nghia (2024). In particular, we state and prove a generalization of Conjecture (7.25) by Macdonald (1992), leading to an extension of the Stong-Tamagawa formula, which is a basis-free characterization of Schur functions over finite fields. Besides, we examine conjectures by Lewis-Reiner-Stanton (2017) about invariant spaces of truncated polynomial algebras under the action of parabolic subgroups, and the proof for the Borel subgroups by Ha, Hai, and Nghia (2024), from which we extend the investigation to the invariant rings under the unipotent group's action. Additionally, we consider the delta operators-a pivotal family of operators used in the proof of Ha, Hai, and Nghia (2024)-with particular attention to their polynomiality properties.

Regulatory institutions (from content moderation platforms to financial supervisors) observe, deliberate, and intervene only after a characteristic delay. We ask whether this processing lag alone can destabilize a multi-agent system that would otherwise remain stable, without exogenous shocks, coordination among agents, or malicious actors. We study this in two stages. First, we analyze a delayed replicator equation in which autonomous agents benefit from radical behavior but face punishment based on a lagged institutional alarm signal. We derive a closed-form critical delay beyond which the unique interior equilibrium loses stability through a Hopf bifurcation, and prove via center manifold reduction that the bifurcation is supercritical (bounded oscillations, not explosive growth) for the entire sigmoid response family. Second, we embed N=240 agents on a network with reinforcement learning (tabular Q-learning) and cross institutional delay with three decision architectures: fixed-policy, reactive (a memoryless threshold heuristic), and Q-learning. The hierarchy is opposite to the naive expectation that learning amplifies instability. Reactive agents are perfectly stable without delay yet collapse once delay is introduced (96% runaway by delay >= 8); fixed-policy agents are immune (0% at all delays); Q-learning agents are only partially resilient (66% at delay 20). The destabilizing ingredient is reactivity to delayed signals, not learning: agents that immediately exploit low-alarm windows trigger oscillatory feedback loops, while learning buffers this through punishment memory encoded in value functions. Throughout, "runaway" denotes bounded large-amplitude oscillation crossing a radical-fraction threshold, consistent with the supercritical bifurcation, not unbounded growth.

We introduce a boundary–residue incidence coalgebra associated with the face poset of a positive geometry and apply it to associahedral scattering forms. The construction is motivated by the analogy between the Connes–Kreimer coproduct on Feynman graphs and the recursive residue structure of canonical forms. For the Stasheff associahedron \(K_n\), whose faces are indexed by non-crossing dissections of an \((n+1)\)-gon, we prove that the incidence coproduct records all intermediate nested planar factorisation channels of the corresponding tree-level scalar amplitude. The residue of the canonical form on a face labelled by a dissection factorises as the exterior product of canonical forms on the lower associahedra associated with the resulting subpolygons. We illustrate the construction explicitly for the pentagon associahedron \(K_4\), corresponding to the five-point planar scalar amplitude. We then formulate a loop-level extension: whenever a planar loop integrand is represented by a positive geometry, the associahedral face poset is replaced by the boundary poset of the corresponding loop geometry. The one-loop halohedron gives a concrete scalar example, while in the non-planar case we define the associated incidence coalgebra at the level of logarithmic singularity strata. Finally, we compare the boundary–residue coalgebra with the cellular incidence coalgebra of a triangulated or regular CW spacetime. The face poset of a finite regular CW complex reconstructs its barycentric subdivision, and hence its underlying polyhedron, while in positive geometry the same incidence mechanism organises canonical-form residues. This yields an incidence-first bridge between cellular spacetime topology and positive-geometric amplitude factorisation, without assuming that metric or causal data are determined by topology.

Global symmetry anomalies of a quantum field theory (QFT) can be packaged as specific couplings of a higher-dimensional symmetry theory (SymTh). In this work we show that for 5D superconformal field theories (SCFTs) engineered from M-theory backgrounds $X$ a Calabi-Yau cone, this data can be extracted from the path algebra of branes probing $X$. This provides a complementary algebraic approach compared with more geometric computations based on the explicit calculation of triple intersection numbers in a resolved geometry and / or $\eta$-invariants extracted from the boundary geometry $\partial X$. Our method applies in situations where the counterpart geometric computation is either unknown or combinatorially unwieldy. We illustrate with several toric threefold examples, including orbifolds $\mathbb{C}^{3} / \Gamma$ and more general non-orbifold Calabi-Yau cones of Sasaki-Einstein five-manifolds.

We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $\alpha$) and Rényi entropies (for all $\alpha > 1$), also recovering these properties for the Shannon entropy in the process. We further strengthen these inequalities, showing that: (i) all these entropic functionals are strictly subadditive on the majorization lattice; (ii) Tsallis entropies (and therefore the Shannon entropy as well) are strictly supermodular on the majorization lattice.

Frontier reasoning models are produced by post-training base language models with reinforcement learning. Recent work has challenged this by showing that sampling from a sharpened version of the base model's distribution, a so-called power distribution, elicits comparable reasoning without additional training, curated datasets, or verifiers. However, making this method practical requires efficiently sampling from the power distribution. A sampler needs to "mix" to the power distribution, which necessitates moving between modes of the target distribution; intuitively, e.g., trying different reasoning strategies. The samplers proposed in prior works repeatedly select a "cut" position in the current reasoning trace uniformly at random and resample the suffix from that position onward. However, reasoning traces typically contain a few consequential decisions (e.g., the choice of proof strategy or algorithm), and we observe that a uniformly chosen cut tends to rewrite local details rather than revisit decision points. We introduce an algorithm (Entropy-Cut Metropolis-Hastings) that uses the base model's next-token entropy as a proxy to identify key decision points and resample from those positions. We empirically verify that entropy jumps are a useful proxy for decision points and, in a stylized model of reasoning, prove that our method's mixing time scales with the number of decisions in a trace rather than with the number of tokens, which can be much larger. Across MATH500, HumanEval, GPQA Diamond, and AIME26, our method consistently improves over baselines and RL-trained models, including best-of-N at the same token budget.

In this paper, we give a short Bayesian proof of Talagrand's celebrated majorizing-measure theorem (MMT). While the upper-bound direction of MMT follows relatively directly from standard arguments, the lower-bound direction is widely regarded as the more difficult part and has received several distinct proofs. Unlike previous approaches, our proof does not rely on existing Gaussian processes lower bounds techniques, nor on combinatorial, geometric, or coding-theoretic constructions. Instead, we derive the lower bound from two area identities for Gaussian additive models. We show that the Gaussian width of a finite set is the integrated mean-squared error of the maximum-likelihood estimator (MLE), while the integrated minimum mean-squared error (MMSE) is larger than the Fernique-Talagrand functional, up to a universal constant. Simply then comparing the MLE with Bayes-optimal estimation, combined with a recent duality minimax argument by Liu, gives a direct proof of the hard direction of MMT.

A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like "how does an intervention affect each unit," rather than only on average. We study this problem with panel-data where we observe $n$ units across $m$ times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit–time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$. Technically, our analysis establishes the first sharp row-wise $\ell_2$-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.

It is shown how some deviations of ergodic averages can be structured. The deviations tend to zero almost everywhere. They are asymptotically almost invariant with respect to the action due to averaging. In this situation, the question of the distribution of the values of such deviations is meaningful. It turns out that for any free ergodic $Z^d$-action these distributions can be weakly close to any given distribution if we change the scale on the value line. This article shortly provides related proofs of the ergodic theorems of von Neumann, Birkhoff, Wiener, and Rokhlin's lemma for $Z^d$-actions with an invariant measure.

We study invariant statistical connections on the space $\mathcal{N}_0^n$ of zero-mean multivariate normal distributions (the multivariate centered Gaussian model) equipped with the Fisher metric $g^F$. We introduce moduli spaces of invariant statistical connections on homogeneous Riemannian manifolds via two natural equivalence relations arising from a categorical viewpoint, and apply this framework to $(\mathcal{N}_0^n, g^F)$. We explicitly determine the $GL(n,\mathbb{R})$-invariant and $\mathrm{Isom}(\mathcal{N}_0^n, g^F)$-invariant statistical connections, with particular emphasis on the dually flat case, and describe the corresponding moduli spaces.

We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a partial reversal. As a main application, we show that the quantity $\varphi_{p_c}(S)$, introduced by Duminil-Copin and Tassion (Comm.\ Math.\ Phys., 2016), is uniformly bounded over all $S\subset \mathbb Z^d$. This partial reversal further yields a short and self-contained route to several key results, including near-critical estimates on the two-point function and sharp bounds on the critical one-arm probability.

We study the moduli stacks of real vector bundles of fixed rank and degree on a type I real algebraic curve and determine its mod $2$ cohomology algebra in terms of characteristic classes induced from the complex Atiyah-Bott classes.

Conformal prediction methods enjoy strong theoretical and empirical predictive inference performance, provided the data is exchangeable and is treated symmetrically during training. However, these assumptions are impractical in many settings, such as time series, where temporal dependence violates exchangeability and it is preferable to use predictors that leverage dependence by treating data asymmetrically. Recent work shows that split conformal prediction is robust to these issues, but sample splitting can reduce accuracy, motivating the study of methods that do not rely on data splitting in the time series setting. In this work, we show that the vanilla leave-one-out jackknife can suffer arbitrary loss of coverage even in canonical time series models with mild temporal dependence. As a remedy, we propose a modification tailored to such settings, which we term the leave-a-window-out (LWO) method, and show that it can achieve valid coverage provided that the model-fitting procedure satisfies mild stability properties. Our proofs are based on quantifying the degree to which the data departs from cyclic exchangeability, which we introduce new coefficients to measure. Experiments on time series demonstrate that our method often enjoys valid coverage when the vanilla jackknife fails to cover, while producing much narrower intervals than split conformal prediction.

In this paper, we are dealing with constrained vector optimisation problems where the objective function acts between real linear-topological spaces. Our aim is to study the relationships between the sets of properly efficient solutions to constrained and unconstrained vector optimisation problems under certain cone convexity assumptions on the objective function using a vectorial penalisation approach.

We study $KU_G$-localization for finite groups. For a finite nilpotent group $G$, we identify the $KU_G/p$-local sphere as the fiber of an Adams operation, and reduce the computation of its homotopy Mackey functors to the corresponding computation for the Sylow $p$-subgroup of $G$. At the prime $2$, we compute $\underline{\pi}_* L_{KU_G/2}S_G$ for abelian $2$-groups, resolve the resulting extension problems, and determine the degree-zero Hurewicz image. As a consequence, we completely determine the $\mathbb Z$-graded homotopy Mackey functors of $L_{KU_G}S_G$ for all finite abelian groups. Finally, for an arbitrary finite group $G$, we prove that the $KU_G/p$-localization of $G$-spectra splits as a wedge of height-one equivariant Morava $K$-localizations, indexed by conjugacy classes of cyclic subgroups of order prime to $p$.