Papers

We prove that every simply connected nilpotent Lie group endowed with a left-invariant nilpotent complex structure is biholomorphic to $\mathbb{C}^n$. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this consequence as a holomorphic analogue of polynomial crystallographic actions introduced by Dekimpe, Igodt, and Lee.

We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

We give an overview of a recent conjecture of the authors on the leading constant in Malle's conjecture on number fields of bounded discriminant. This comes from applying the philosophy from Manin's conjecture on rational points of bounded height on Fano varieties to classifying stacks. To make these ideas more accessible we assume no background in algebraic geometry, which requires some new perspectives and alternative approaches to the theory. We also give some new conjectures on multi-heights and Bhargava's heuristics on counting with local conditions imposed.

This paper analyzes the convergence of parallel overlapping domain-decomposition methods with impedance boundary conditions for the time-harmonic Maxwell equations in heterogeneous media. We prove that the parallel iterative method is well-posed in an appropriate function space, and characterize the error propagation operator through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, we derive explicit convergence estimates in terms of the norms of the impedance-to-impedance maps. At the discrete level, we develop the finite-element counterpart of these results based on Nédélec-element discretisations. Under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined, we show that the discrete method inherits the convergence behavior of the continuous method. We illustrate this theory with numerical experiments for strip domain decompositions, and also present numerical experiments for checkerboard domain decompositions that go beyond our theory.

We establish a homological mirror theorem for the 4-manifolds arising as moduli of (irregular) rank two local systems on the projective line. Specifically, we prove that the Fukaya category of a moduli of such local systems with generic microlocal monodromy at punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli of local systems with trivial microlocal monodromy.

In this paper, we extend localized Turán theorems to signed graphs and study the corresponding spectral Turán problems. Firstly, we establish a vertex-localized Turán-type inequality for signed graphs and characterize the extremal graphs reaching this bound. Secondly, we derive a sharp upper bound for the largest eigenvalue of a signed graph via localized Turán parameters, and identify the extremal graphs attaining equality. Finally, we generalize a walk-based local spectral Turán inequality to signed graphs. Our results generalize and improve several existing theorems for unsigned and signed graphs.

We study the stationary and dynamical Vlasov-Poisson-Boltzmann system in a bounded, convex domain subject to a confining external potential field. For the stationary problem, we construct a unique stationary solution with an inflow boundary condition. A key difficulty is to obtain pointwise regularity for stationary solutions due to the intricate coupling between the self-consistent electric field and the Boltzmann collision operator. To overcome this issue, we establish a $W^{1,p}_{x,v}$–$\alpha C^1_{x,v}$ bootstrap framework and derive an unweighted $C^1_v$ estimate by exploiting the structure of the external potential field. We then investigate the dynamical Vlasov-Poisson-Boltzmann system near the stationary solution. We prove the global existence and uniqueness of solutions for small perturbations and establish exponential convergence toward the stationary state in weighted $L^\infty$ norms. Our results reveal the stabilizing effect of the external potential field and provide a framework for the stationary and dynamical theories of the Vlasov-Poisson-Boltzmann system in bounded domains.

We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's bound for the Dirichlet Laplacian. The operators considered include the Baouendi-Grushin operator and operators with ellipticity degenerating on the boundary.

We study the Cauchy problem for the Tricomi equation perturbed by space-time Gaussian White Noise. To prove existence and uniqueness of the solution, we employ a Fourier transform approach that allows to obtain its representation in terms of certain integrals of the Airy functions. Then, via a careful analysis of the asymptotic behaviour of those integrals, we obtain all the desired properties of the solution, such as square integrability, continuity of its sample paths and stationarity with respect to the space variable. In relation to that stationarity, we also provide the precise description of how the correlation function behaves for small values of the space-lag. We also remark that, in contrast to the findings of the recent paper [11], the properties of the solution to our stochastic Tricomi equation are equivalent to those derived in studying the corresponding problem for the wave operator.

The escalating accumulation of orbital debris poses a critical threat to future space operations. Space-based lasers leveraging laser ablation have emerged as a promising approach for mitigating debris proliferation and preserving the orbital environment. Current literature, however, treats space-based laser debris remediation as a deterministic problem, assuming that momentum transfer and the resulting debris perturbations are precisely known. In reality, laser-to-debris engagement outcomes are inherently stochastic due to partially known debris characteristics. Compounding this challenge, estimating critical laser-matter parameters in situ, such as the momentum coupling coefficient, requires ablation that consequently perturbs the debris trajectory. This establishes a coupled ablation-and-estimation problem in which the laser platform and target debris encounter geometry influence remediation effectiveness and estimation accuracy. To address this problem, we present a joint ablation-and-estimation methodology that provides insights into the driving factors that make different encounter geometries improve or degrade overall remediation and estimation performance. Results across multiple coplanar and out-of-plane encounter geometries demonstrate how periapsis-lowering capacity, linear system observability, and nonlinear estimation performance evolve as laser parameters and relative orbit geometry vary. By identifying the key drivers behind these metrics, this study highlights critical considerations for the safe and effective operation of space-based lasers under uncertainty.

We propose a novel linearized mixed finite element method for the Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system. Specifically, the method combines a staggered time discretization that eliminates the need for expensive nonlinear solvers by carefully treating nonlinear terms in a time-staggered manner, with a mimetic spatial discretization that preserves the exact structure of the discrete de Rham complex. Both semi-discrete scheme and its fully discrete counterpart are developed, which preserve key physical properties, including conservation of the total mass and energy stability. Under appropriate assumptions on the initial data, a rigorous theoretical analysis is carried out for the fully discrete scheme. Numerical experiments using mimetic spectral elements are presented to demonstrate the properties and to verify the accuracy and effectiveness of the proposed decoupled approach.

Moment/Sum-of-Hermitian-Squares relaxations for noncommutative polynomial optimization problems have become an important tool for analyzing problems within quantum theory. Despite their widespread success, little is known about their rate of convergence and, consequently, their accuracy. In this work, we develop explicit convergence rates for relaxations of noncommutative polynomial optimization problems generated from the Pauli algebra – covering applications to the ground state energy problem for n-qubit systems. In particular, we show that the rate of convergence can be bounded in terms of the smallest roots of a family of orthogonal polynomials known as Krawtchouk polynomials. Our result represents the first quantitative analysis of the rate of convergence for relaxations of noncommutative polynomial optimization problems.

We study the short-time asymptotic behavior of the heat trace associated with the Neumann Laplacian on polygonal domains in the plane. By establishing locality principles for the heat kernel near corners, edges, and the interior, we approximate the heat kernel on the polygon by model heat kernels defined on infinite sectors, half-planes, and the full plane, respectively. Although it is known that the Neumann heat trace of polygons admits a three-term asymptotic expansion followed by an exponentially small remainder, an explicit estimate for the exponent in this remainder term is not known. In this article, we provide such an estimate. We also discuss whether the exponent is sharp, and how it relates to known results. Finally, we discuss issues that arise when trying to extend the results to Robin boundary conditions.

We classify the cocommutative Hopf algebras which coact inner-faithfully on (one-parameter) skew polynomial rings $A_q(n) = \Bbbk \langle x_1,\dots,x_n \rangle/(x_j x_i - q x_i x_j \mid i < j)$ for $n = 2$ and $3$. As a direct corollary, we obtain a classification of group gradings on two- and three-variable skew polynomial rings, recovering a result of Crawford in the two-variable case. Our results are achieved via Manin's universal coacting Hopf algebra construction, often denoted $\underline{\operatorname{aut}}(A_q(n))$, by classifying all its cocommutative quotients. We therefore also give an explicit presentation of $\underline{\operatorname{aut}}(A_q(n))$ for arbitrary $q \in \Bbbk^*$ and $n \in \mathbb{N}$.

We introduce a novel notion of Ollivier-Ricci curvature for causal sets using Lorentzian optimal transport. The construction is motivated by a new Lorentzian asymptotic formula of independent interest, which recovers timelike Ricci curvature, up to higher-order terms, from the transport distance between probability measures on nearby causal diamonds. Passing to the discrete setting, this leads to a mesoscopic notion of Ricci curvature defined along maximal chains and built from probability measures on causal diamonds. We study several variants, including idle and Lin-Lu-Yau type curvatures, prove local-to-global propagation results and timelike Bonnet-Myers theorems, and compute the curvature for a range of explicit causal sets. We design high-density Poisson sprinkling numerical experiments recovering the expected constant-curvature signatures of Minkowski, de Sitter, and anti-de Sitter space. These results provide evidence that the construction captures timelike Ricci curvature from order-theoretic data.

We study the weak Brill–Noether problem for vector bundles on nodal Enriques surfaces by exploiting the classical realization of a nodal Enriques surface $X$ as a Reye congruence in the Grassmannian $\operatorname{Gr}(2,4)$. We consider restrictions of irreducible homogeneous bundles from the ambient Grassmannian and study their behavior with respect to the weak Brill–Noether property and slope stability.

Finite Difference Schemes are widely used in the approximation of different hyperbolic (wave-like) differential equations, and are particularly important for seismic wave modelling and its applications. Classical methods based on Taylor Series are dominant in the literature; however, it is known that these methods can suffer from excessive numerical dispersion. In this paper, we review and extend existing high-order in space finite difference schemes for acoustic wave propagation, featuring different stencil geometries ranging from classical cross stencils to stencils with rhombus or square-like shapes, and propose a general mathematical framework for their derivation. The numerical implementation is performed in a symbolic, high-level framework (Devito), which compiles and runs highly optimized, stencil-based computations, allowing for a low-level interpretation of the methods efficiency. We demonstrate that non-cross stencil shapes, such as rhombus and square-based stencils, do not necessarily provide added accuracy or dispersion reduction in general, despite their increased computational cost. However, results on both idealized and realistic velocity models confirm the benefits of using dispersion-optimized cross-stencils, indicating adequate accuracy with reduced computational cost on more compact stencils compared to classic approaches. Finally, our implementation of the methods provides ease of use for full-scale acoustic seismic inversion problems using Devito.

We present a two-grid algebraic multiscale preconditioner for large sparse symmetric positive definite systems arising from elliptic problems with highly heterogeneous coefficients. The coarse space is constructed directly from the system matrix by graph partitioning and local generalized eigenvalue solvers, yielding basis functions that capture the low-energy modes responsible for slow convergence. The method requires no geometric information, making it suitable for unstructured and matrix-only settings, and its construction is naturally parallelizable. Numerical results for heterogeneous Darcy flow problems show robustness with respect to coefficient contrast and problem size, better performance than standard algebraic multigrid on challenging large-scale cases, and good parallel scalability.

In this work, we develop Stein's method for the Wishart distribution on the cone of positive definite matrices. We establish the basic ingredients of a Wishart Stein framework: we derive an extended-generator-based Stein characterization from the Wishart diffusion process, identify the corresponding transition semigroup through the noncentral Wishart law, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is demonstrated in four applications: (i) an order $n^{-1}$ bound, for smooth test functions, for the Wishart approximation of uncentered group-mean scatter matrices in MANOVA; (ii) a quantitative multivariate Satterthwaite approximation; (iii) local/integrated De Bruijn identities and logarithmic Sobolev inequalities for the Wishart measure; and (iv) Stein's method of moments for the shape and scale parameters, including structured scale estimation.

In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.

Using methods from spectral graph theory and semidefinite programming, we obtain sharp spectral bounds for several graph parameters related to distance colorings and packing, including the injective chromatic number, the open packing number, the injective chromatic index, and the strong chromatic index. The new spectral bounds improve several existing combinatorial bounds. Furthermore, we apply the obtained eigenvalue bounds on the first two parameters to estimate the code covering number and the open packing number of hypercubes, obtaining new exact values and strengthened bounds regarding the existing literature. The obtained results illustrate the power of combining spectral and semidefinite programming tools for tackling coloring and packing problems in graph theory and coding theory.

This paper contains a proof of a recent conjecture of Artemev that connected the resonance transformations for the $(2,2p+1)$ minimal string to the $x-y$ swap in the theory of topological recursion.

We study coKähler structures on real Lie algebras by using the Fino-Vezzoni correspondence, which relates them to Kähler Lie algebras endowed with compatible skew-symmetric derivations. Our first result completes the flat case: every odd-dimensional flat Lie algebra admits a coKähler structure, giving a converse to the theorem of Fino and Vezzoni asserting that unimodular coKähler Lie algebras are flat and hence solvable. We then characterize almost abelian Lie algebras carrying coKähler structures and we determine the possible Reeb directions. As a consequence, every non-unimodular almost abelian coKähler Lie algebra splits as the direct product of a non-unimodular Kähler Lie algebra and a line. Finally, we apply these results in low dimensions: we classify the almost abelian cases in dimensions five and seven, and we give the classification, up to Lie algebra isomorphism, of five-dimensional coKähler Lie algebras by reducing the corresponding extensions arising from four-dimensional Kähler Lie algebras. The resulting examples show that the direct product splitting above is a special feature of the almost abelian setting.

Designing a neural network processor is an end-to-end co-design problem: network architecture and training budget determine the inference workload; hardware mapping decisions determine chip area, latency, and energy; and these characteristics govern fabrication yield and manufacturing cost. In practice, these decisions are made in separate stages, and existing co-design methodologies are tightly coupled to specific algorithms, making it difficult to improve one component without reworking the entire pipeline. This paper presents a unified framework, grounded in monotone co-design theory, that composes four interoperable design blocks spanning network training, chip mapping, wafer-level fabrication, and compute resource allocation. Each block exposes only a functionality-resource interface to the rest of the system, so any block can be refined without structural changes elsewhere. A central contribution is the treatment of uncertainty: rather than collapsing stochastic outcomes into point estimates, the framework introduces Confidence, the inverse of success probability, as an explicit and optimizable resource alongside cost, time, and power. Three case studies validate the approach. The first recovers Pareto-optimal implementations across heterogeneous application scenarios. The second confirms that Confidence functions as a continuously tunable design knob rather than a post-hoc diagnostic. The third demonstrates that improving a single block's implementation set automatically propagates to the global Pareto front, without modifying the co-design diagram.

Future 6G networks are envisioned to integrate low Earth orbit satellite mega-constellations to enable seamless global connectivity, particularly in underserved and remote areas. However, the deployment of dense mega-constellations introduces interference among satellites operating over shared frequency bands. This represents a rather new setup for studying spectrum sharing, which exacerbates the limited flexibility of conventional FDD systems based on fixed bands for downlink and uplink transmissions. We address this spectrum-sharing problem and propose dynamic re-assignment of FDD bands for improved interference management in dense deployments, as well as evaluate the performance gain of this approach. To this end, we formulate a joint optimization problem that incorporates dynamic band assignment, user scheduling, and power allocation in both directions. This non-convex mixed integer problem is solved using a combination of equivalence transforms, alternating optimization, and state-of-the-art industrial-grade mixed integer solvers. Numerical results demonstrate that the proposed approach of dynamic FDD band assignment significantly enhances system performance over conventional FDD, achieving up to 30% improvement in throughput in dense deployments.

Sequential decision-making problems are often modelled as a Markov decision process (MDP). We focus on the stochastic shortest path (SSP) problem, which is an infinite-horizon undiscounted MDP with absorbing terminal states. We develop a Bayesian framework to learn the optimal decision strategy through interactions with the decision-making task. Specifically, we learn the optimal action-value function $Q^*$, but unlike many existing Bayesian approaches, we do not rely on unrealistic modelling assumptions and ad-hoc approximations. Our approach is to directly construct the posterior beliefs for $Q^*$ through Bellman's optimality equations. For deterministic rewards, we characterise the posterior as a distribution with a manifold density. To facilitate simpler inference, we relax the likelihood so that a Lebesgue density exists. The flip side is to create unidentifiability issues. Specifically, the relaxed posterior can have significant mass on improper decision rules, while the exact posterior will not. We also calculate the exact posterior probabilities for optimal action selections for the tabular parametrisation of $Q^*$, a Gaussian likelihood relaxation and a Gaussian prior, which is useful in benchmarking studies. Numerical studies on variants of the Deep Sea benchmark verify our findings. We demonstrate that our framework faithfully quantifies uncertainty and, compared to other temporal-difference-based Bayesian methodologies, is more data efficient. We conclude with recommendations for future work.

From $N+1$ random points on a line we wish to select $M+1$ points so as to maximize the minimal spacing between them. We consider an initial configuration with independent and identically distributed spacings. Equivalently, the points are arrival times of a generic renewal process. For general spacing distributions, and for all $M\leq N$, we derive exact distributional identities for the maximal minimal spacing and obtain its asymptotic behavior. The problem admits a reformulation in terms of a threshold-resetting random walk. The walk advances by successive random increments and is reset to the origin upon exceeding a fixed threshold. The probability that the optimal spacing exceeds a given value coincides with the probability that the walk completes at least $M$ reset cycles within $N$ steps. This yields an exact representation in terms of first-passage functionals of the walk. The same mapping suggests a numerical scheme for the max-min spacing problem in the regime of large $N$ and $M$, whose accuracy is tested against the exact results obtained here.

The Maker Breaker Domination Game is a two player game played on a graph $G$ in which the players take turns to claim a vertex from the graph. The aim of the Dominator is to claim the vertices of a dominating set, and the aim of the Staller is to prevent this. In this paper, we consider the following problem: for a given integer $d$, what is the size of the smallest (with respect to the number of vertices) graph with minimum degree $d$ such that the Dominator loses going first? We write $\beta(d)$ to denote the answer to this question. We determine the precise value of $\beta(d)$ for $d\leq 3$. For general $d$ it was known that $2^{d+1} \leq \beta(d) \leq 2^{d+1}+2d$; the upper bound is due to a construction communicated to us by Valentin Gledel, while the lower bound follows from a simple application of the Erdős-Selfridge Theorem. We improve the lower bound to $\beta(d) \geq 2^{d+1}+2$.

We study the Cauchy problem for a time-fractional Airy-type equation on an infinite metric star graph with edge-dependent Caputo derivatives. By constructing layer potentials generated by fractional Airy fundamental solutions, the problem is reduced to a generalized Abel integral equation. Using Pskhu's solvability results, we establish existence, uniqueness, and an explicit integral representation of the solution. The obtained framework extends previous results where identical fractional orders were assumed on all edges.

We study the existence theory for steady compressible Navier-Stokes-Fourier system in a three dimensional bounded domain for the case of viscosities depending on the temperature in the form $(1+\vartheta)^\alpha$ for $0\leq \alpha\leq 1$ and the hard sphere pressure $p(\varrho,\vartheta) = \vartheta\varrho h(\varrho)$, $h(\varrho)$ is increasing and singular at $\varrho = a > 0$. This paper considers both the heat-flux and Dirichlet boundary conditions for the temperature and Dirichlet boundary conditions for the velocity. The key point to controlling the pressure is based on some estimates of the Bogovskii operator.