Papers

We take a sum $C_1+r C_2,\ r\in\Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside the octic Fermat surface and with no intersection points. We gather strong evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+r C_2$ are set theoretically distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a possible counterexample to a conjecture of J. Harris.

Linear structural equation models on directed mixed graphs encode causal relationships among variables subject to latent confounding. The half-trek criterion (HTC) provides a graphical sufficient condition for the structural coefficients to be rationally identifiable from the observable covariance matrix, and yields a corresponding closed-form rational estimator. Despite this, the asymptotic distribution of the HTC estimator, and hence valid standard errors and confidence regions, have not been derived. We derive the semiparametric influence function of this estimator for all HTC-identified directed mixed graphs, including cyclic ones. The influence function combines the structural residual at the target node with the identification instruments, recursively corrected for uncertainty from earlier estimation stages. The HTC estimator is asymptotically normal with variance computable in closed form, yielding confidence regions, marginal intervals, and Wald tests for individual structural coefficients. Applied to the Fulton Fish Market dataset, our theory delivers a complete inferential summary for the causal effect of supply on demand.

The Onsager–Machlup action functional provides a variational framework for characterizing the most probable transition paths of stochastic systems and plays an important role in the study of nonequilibrium fluctuations. Its extension to McKean–Vlasov stochastic differential equations is complicated by the intrinsic distribution dependence of the coefficients. In this paper, we address this difficulty by introducing an Euler-type approximation scheme based on classical, distribution-free stochastic differential equations. Combining the classical Onsager–Machlup theory with a convergence argument for the approximation sequence, we derive an explicit expression for the Onsager–Machlup functional associated with the McKean–Vlasov SDE. The proposed approach is constructive and extends to a broad class of distribution-dependent stochastic systems.

Phylogenetic reconstruction is one of the major challenges in computational biology. Among existing reconstruction methods for phylogenetic networks, an important subtask emerges in extending a leaf-labelling on a phylogenetic network to determine a most parsimonious tree inside the network. There exist different variants of this subtask depending on the biological model assumptions for which distinct evolutionary phenomena are captured by the network. In this article we assume that next to hybridization or recombination events, also allopolyploidy or incomplete lineage sorting are present. Then, finding the most parsimonious tree inside the network is called the parental parsimony score problem (PPS), a NP-hard combinatorial optimization problem. We provide the first constant-factor approximation for the PPS on arbitrary but fixed leaf labels and a class of networks on which the PPS remains NP-hard, namely binary, semi-simplex, tree-child phylogenetic networks. Furthermore, we introduce a novel exact solution algorithm for the PPS on binary, tree-child phylogenetic networks and analyze its performance on simulated data.

This paper presents a wave-particle decomposition (WPD) for kinetic relaxation equations, formulated around a local evolution timescale and its associated kinetic horizon. By leveraging the characteristic integral solution, we decompose the distribution function into an analytically accumulated wave component and a purely kinetic particle component. The latter is defined by the collisionless transport that survives beyond a prescribed local domain of influence, termed the horizon. This continuous formulation yields a unified wave-particle system valid across the entire Knudsen spectrum, comprising a source-free total conservation law, a wave equation, and a particle equation. The wave operator admits a Chapman–Enskog expansion, whose moments yield Euler and Navier–Stokes fluxes with horizon-dependent coefficients, while the particle equation governs the remaining non-equilibrium kinetic transport. At the algorithmic level, this system is discretized by a conservative macro-micro method. The total conservative variables are advanced by a finite-volume update using the sum of a Navier–Stokes gas-kinetic wave flux and a particle flux computed by either a deterministic discrete-ordinate $(S_N)$ method or a Monte Carlo representation. Unlike the global time-step splitting in the unified gas-kinetic wave-particle (UGKWP) method, the present partition is defined at the PDE level and governed by the local ratio of evolution timescale to relaxation time. The particle component is therefore a fractional kinetic population generated by the collisionless factor of the integral solution. Formal analysis establishes the asymptotic-preserving continuum limit, rarefied-regime consistency, and regime-adaptive scaling of active kinetic degrees of freedom. Numerical tests in one, two, and three dimensions validate the accuracy, multiscale capability, and efficiency of the framework.

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with step-like initial data. Using the nonlinear steepest descent method, we derive the long-time asymptotic expansion of the solution to the Cauchy problem in three distinct transition regions. In the first two transition regions, the leading-order asymptotics are characterized by Painlevé \uppercase\expandafter{\romannumeral34\relax}-type formula, while in the third one is a collisionless shock region, the leading-order asymptotics is describedin terms of Riemann theta functions. Our analysis is based on the Riemann-Hilbert formulation associated with the Cauchy problem of the defocusing NLS equation.

The radius-one limited permutation channel $\operatorname{LPC}_{\infty}(1)$ maps a transmitted word to any word obtained by an arbitrary set of pairwise disjoint adjacent transpositions. This is the $r=1$ case of the $\ell_\infty$-limited permutation channel of Langberg et al., and is also the zero-error version of simultaneous adjacent-swap errors. We study zero-error block-concatenation codes for this channel. Our first contribution is a $q$-ary two-stage local criterion for certifying free block-concatenation codes. The criterion replaces the global all-length confusability problem by finitely many local checks between blocks: a same-length truncated-ball test and a second-stage prefix test for unequal lengths. In the binary case, it yields explicit block-concatenation codes of rates $0.649872$, $0.652018$, and $0.653618$. The best construction improves the previous string-concatenation rate $0.642805$ and comes within $0.013049$ of the known upper bound $2/3$. Although the criterion is only sufficient, we prove that it is rate-complete: for every alphabet size $q$, the supremum of $q$-ary block rates certified by the criterion is exactly the $q$-ary zero-error capacity $C_0^{(q)}$. Thus it imposes no asymptotic rate loss. We also give an exact product-automaton verifier which decides, for a fixed prefix-free binary block set, whether the induced finite-length codes are correcting for all lengths. Finally, motivated by feedback settings, we study error detection. We prove a $q$-ary pairing upper bound and give a $q$-ary local detecting criterion. In the binary case, we construct a detecting block-concatenation code of rate $0.756707$, compared with the upper bound $\frac12\log_2 3\approx0.792481$.

To mitigate the computational burden due to the high dimensionality of linear kinetic equations, low-rank solvers and data-driven reduced order models (ROMs) have been developed. Low-rank solvers based on matrix or tensor compression exploit phase-space low-rank structures within individual solutions, while data-driven ROMs exploit low-rank structures of the solution manifold across parameters of parametric multi-query applications such as sensitivity analysis and uncertainty quantification. It is attractive to combine them to exploit their complementary strengths. A key challenge is to extract a reduced basis without reconstructing full solution or prescribing the ranks. To address this challenge, we develop a deterministic incremental tensor-train (TT) compression algorithm that operates directly on streaming TT data and adapts the ranks to a prescribed approximation tolerance. Given a new TT tensor, the proposed method updates an accumulated TT representation via core-wise projection, residual orthogonalization, and adaptive enrichment, retaining only the complementary information that cannot be represented within a prescribed tolerance. By operating entirely at the core level, the algorithm avoids reconstructing either the incoming tensor or the accumulated full tensor. We establish approximation error bounds for the proposed approach. Moreover, we show that the accumulated TT representation provides a compressed analogue of proper orthogonal decomposition for full-order snapshot data. Using this connection, we construct reduced order models for linear kinetic equations directly from streaming low-rank solution data. Numerical experiments on parametric radiative transfer equations demonstrate that the proposed method achieves comparable reconstruction accuracy with substantially reduced wall time and yields efficient and accurate ROMs directly from compressed low-rank data.

The outer approximation algorithm is a widely used method for solving convex mixed-integer nonlinear programs. While the algorithm is well established in theory and practice, certain assumptions underlying its convergence are rarely discussed in the literature. In this paper, we examine two such assumptions: that a constraint qualification holds at the optimal solution of each nonlinear programming subproblem, and that these subproblems can be solved exactly. We argue that both assumptions are connected to the issue of cycling, by which we mean that the same integer assignment reappears in successive iterations of the algorithm. When a constraint qualification fails, separation of the current iterate from the feasible set is not guaranteed, which can cause the algorithm to stall. When the nonlinear programming subproblem is solved only approximately, separation may likewise fail, and we show that high precision can be required in certain cases. To formalize the connection between these issues, we prove that when Slater's condition is nearly violated, a point close to the exact solution can be found at which separation fails. Furthermore, within the outer approximation algorithm, we propose to use extended cutting planes as a fallback strategy when cycling is detected. We prove that this approach yields a finitely convergent algorithm under relaxed assumptions regarding constraint qualifications, thereby generalizing the convergence theory of the outer approximation algorithm.

We establish a generalized Zariski cancellation theorem for Brieskorn–Pham varieties over the field of complex numbers. More precisely, we show that if two complex Brieskorn–Pham varieties become isomorphic after taking a product with an arbitrary separated complex scheme having a smooth point, then they are already isomorphic not merely as complex algebraic varieties but, in fact, as $\mathbf{C}^*$-varieties. The proof combines our general cancellation theorem for complex algebraic varieties with a unique singularity, whose proof relies on the analytic cancellation theorem of Hauser–Müller, with an exponent rigidity theorem for Brieskorn–Pham varieties. The latter asserts that, over any field of characteristic zero, the exponent tuple appearing in the defining equation completely determines the isomorphism class of the corresponding Brieskorn–Pham variety.

Newton polygons are powerful tools for computing the decomposition of prime ideals in extension rings. Methods based on Newton polygons have been developed through the work of Ore, Montes, and Nart. Kölle and Schmid obtained a method for determining decomposition groups from Newton polygons and associated data under assumptions introduced by Ore. In this paper, we extend their approach and show how to determine decomposition groups under the weaker assumptions introduced by Montes and Nart, formulated in terms of indices.

We study the local well-posedness of the Cauchy problem for a class of hyperbolic quasilinear evolution systems with fractional dissipation and commutators of the fractional Laplacian $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$. The analysis is motivated by the compressible isentropic Navier-Stokes equations with fractional diffusion and the compressible Euler alignment system with singular communication weights. We show that the contribution of the commutators to the energy is absorbed by the dissipation. Our approach does not rely on the cancellation condition previously used to recover the coercivity of the highest-order non-local terms. We further show that the commutator term induces a loss of regularity in the stability estimates, preventing the standard contraction argument from being applied. Therefore, we instead establish the relative compactness of the solution operator and apply Schauder's fixed point theorem. The resulting theory provides local existence for arbitrary orders $\alpha\in(0,1)$ and identifies the obstacles to local uniqueness in the presence of fractional commutators.

We introduce an accelerated Langevin-based sampling method that is based on two complementary devices: SamAdams adaptive timestepping, which automatically shrinks the effective integration step in stiff regions of phase space using a relaxed stiffness monitor, and position-adaptive Langevin (PAL) dynamics, which concentrates friction along the local force direction while preserving the canonical distribution as the exact invariant measure. The resulting combined scheme (SA-PAL) is implemented in a palindromic integrator which requires only one force evaluation per iteration through suitable organisation of the integration steps and by exploiting the rank-one-plus-scalar structure of the PAL friction tensor. We test the method on various model problems: the Rosenbrock function, a thin entropic channel, the Mueller-Brown potential, and a Bayesian parameterisation problem with a sparsity-inducing shrinkage prior. On the Rosenbrock and Mueller-Brown potentials mixing rates are improved by 1.5-3 times compared to fixed stepsize integration. Efficiency gains of more than an order of magnitude are documented in the other examples.

We motivate two new approaches to the study of Lefschetz point-free periodicity. The first focusses on spaces satisfying the Wecken property. As an example, we study the bubble spaces. The second is a relative study of the Lefschetz point-free periodicity. This becomes important, for example, in the study of repellers of dynamical systems.

Local Cohomology, since its introduction, has served as an important invariant for commutative rings and their modules. They furthermore provide the local model for relative cohomology groups for schemes. As with all cohomology theories, vanishing theorems are widely sought after, and for local cohomology a classical theorem of Grothendieck states that all local cohomology vanishes past the dimension of the ring. Hartshorne-Lichtenbaum vanishing tells us when local cohomology vanishes at the dimension of the ring, and for vanishing one below the dimension of the ring we arrive at the Second Vanishing Theorem. This paper proves the Second Vanishing Theorem in the final unknown case for regular local rings, that being the case of ramified mixed characteristic rings, and gives a few applications of this result. The method of this paper works in equicharacteristic, and we show how we can reduce the unramified case to the ramified case as well, yielding a unified proof of the Second Vanishing Theorem in all characteristics.

Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative $L^{2}(\mu)$ universal approximation theorem for this architecture, achieving a convergence rate of $\mathcal{O}(n^{-1/2})$. This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.

We investigate the geometry of plane curves in $\mathbb{R}^2$ endowed with a canonical linear connection. By introducing the notions of the tangential and geodesic curvatures, we prove the existence and uniqueness theorems for curves with prescribed curvatures. We find all curves with constant curvature and we generalize the curves whose curvatures are linear functions of the arc length parameter.

We introduce a particle method for the numerical approximation of time-dependent first-order Mean Field Games (MFGs) systems with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary time-horizons and (possibly) singular initial player distributions in $\mathcal{P}_2(\mathbb{R}^d)$. The numerical scheme is based on an implicit Euler discretization in time and sampling in space of the characteristic Hamiltonian/Pontryagin system associated with the continuous MFGs system. We prove convergence of the approximations of the player distribution in the $L^{\infty}(\mathcal{W}_2)$-metric and the approximations for the gradient of the value function along optimal trajectories in the $L^{\infty}{(L^2)}$-norm as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. The error bound that we establish for this convergence further implies rates of convergence of the scheme for a range of spatial sampling techniques. Provided that the Lagrangian and terminal costs are additionally locally Lipschitz continuous, we also establish an asymptotic error bound in the $L^{\infty}(L^1)$-norm for the approximations of the value function along optimal trajectories. This is the first work in the literature on rigorous numerical approximation and analysis of first-order MFG systems that handles non-separable Hamiltonians and potentially singular initial agent distributions for arbitrary long time horizons. We illustrate the performance of the scheme in numerical experiments for a range of initial agent distributions, time horizons and space dimensions.

Let ${\mathbf A}$ be the affine coordinate ring of $E\setminus\{\infty\}$, where $E$ is an elliptic curve over $\mathbb F_q$ and $\infty$ is rational. We solve the rank-two case of the open problem of explicitly classifying sign-normalized Drinfeld ${\mathbf A}$-modules in generic characteristic with a fixed determinant motive; the classification for rank greater than two remains open. Outside an explicit exceptional locus, the rank-two Drinfeld modules are parametrized by the open modular curve $Y^{q+1}=\pi(X)$ with $Y\neq0$. The polynomial $\pi$ is squarefree, and the smooth projective completion of this curve has genus $q^2$. The roots of $\pi$ split into two types. Above roots of the first type, all coefficient functions extend regularly, and their extensions yield exactly the exceptional modules. Roots of the second type correspond to cusps of the modular curve; above these roots, the coefficient family has poles.

We present a semidefinite programming framework for constructing time-varying Lyapunov densities for nonautonomous dynamical systems on a hypertorus. The formulation leverages Gram matrix representations of hybrid (real-trigonometric) polynomials. In addition, we introduce a novel block decomposition of these Gram representations to confine the blow-up of the resulting density to a prescribed set. The results are then applied to establish the almost global synchronization of a time-varying Kuramoto model and the robust almost-global stability of a parameter-varying nonautonomous system. These examples demonstrate the applicability of the proposed method and validate the theoretical results. All computational results are obtained using an open-source MATLAB implementation, as referenced in the text, thereby facilitating reproducibility of the reported examples.

With the explosive growth of network scale and data volume, wireless communication is facing an increasingly severe limitation of spectrum resources. Semantic communication has emerged as a promising paradigm to break the bandwidth bottleneck by transmitting significant task-oriented semantic information rather than raw data. In practical real-time wireless applications, semantics of information exhibit diverse spatial and temporal correlations depending on intrinsic dynamics of source and extrinsic dynamics of environment. Motivated by this observation, this paper develops a novel information-theoretic metric to quantify the semantic value of spatiotemporal information. Specifically, a semantic value of information (SVoI) framework is proposed based on the mutual information, which characterises the reduction in uncertainty when predicting an unknown system state using past semantic spatiotemporal correlated observations. Focusing on general Gaussian Markov models, closed-form expressions of the SVoI are derived. Effects of both separable and coupled spatiotemporal correlations on SVoI are further investigated analytically. Numerical simulations are conducted to validate the theoretical analysis of SVoI and its bounds. The proposed SVoI metric jointly captures the impact of semantic spatiotemporal correlation of source, timeliness of information, and channel conditions, which could serve as an effective optimisation objective for the design of next-generation semantic-aware communication systems.

A central program in infinite-dimensional Riemannian geometry is to understand which classical finite-dimensional principles remain valid beyond finite dimensions. In this article, we study the existence of periodic geodesics on Hilbert half-Lie groups equipped with strong right-invariant Riemannian metrics. Building on recently established completeness results for this class of infinite-dimensional manifolds, we prove that every nontrivial free homotopy class contains a periodic geodesic whenever the fundamental group is nontrivial. We further establish a Lyusternik-Fet type theorem in this setting. Assuming a Palais-Smale condition modulo right translations and the existence of a nontrivial higher homotopy group, we prove the existence of a nonconstant contractible periodic geodesic. Thus, as in the finite-dimensional setting, both first and higher homotopy information continue to force the existence of periodic geodesics in infinite dimensions. In addition, we describe a reduction principle based on compact finite-dimensional totally geodesic submanifolds and apply our results to groups of Sobolev diffeomorphisms equipped with right-invariant Sobolev metrics.

The set of stably trivial complex vector bundles over complex projective spaces and spheres has a natural group structure when the corank is small enough. With respect to this group structure, the operations of taking the underlying real vector bundle (realification) and of adding a trivial line bundle (stabilisation), are group homomorphisms. Building on Hu's recent enumerations of stably trivial complex bundles, we compute these homomorphisms in a range by using Weiss calculus to translate the problem to stable homotopy theory.

We derive quantitative uniform resolvent estimates for Schrödinger operators on the half-line with inverse-square potentials, which provide a sharp behaviour in the limit of large coupling. Our approach is based on a matrix representation of the boundary value of a weighted resolvent. The partial wave decomposition then turns these one-dimensional channel estimates into explicit weighted resolvent estimates for the Laplacian, its inverse-square potential perturbations and for the magnetic Laplacian with an Aharonov–Bohm potential. We also obtain exact Simon-type identities for the imaginary parts of the weighted resolvents of these operators.

The notion of a quasi-holomorphic homotopy of two immersions of a $3$-manifold into a $5$-manifold extends the notion of their regular homotopy by also allowing the homotopy to pass through instances of maps with an isolated singularity around which the path of the homotopy forms a cross-cap (complex Whitney umbrella). We describe the local form of such homotopies and explain connections with the theory of holomorphic map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$. Our main result is a complete a description of how the fundamental group of the complement of the image of an immersion of a $3$-manifold into a $5$-manifold (i.e. its knot group) changes under a quasi-holomorphic homotopy. As a corollary we will see that certain quasi-holomorphic homotopies of the standard embedding of the $3$-sphere into the $5$-sphere do not change its knot group.

Instrumental variable regression quantifies causal effects between a possibly confounded treatment variable $ X_2 $ and a response variable $ X_3 $ by leveraging an instrument $ X_1 $. Our work considers the setting where some prior information of the joint distribution of $ X_{123} $ is given, potentially through an initial dataset. However, further samples must be gathered to improve the accuracy of the estimation. We show that under specific parameter configurations in a Gaussian graphical model, taking partial samples from, e.g., $ X_{12} $ can reduce the asymptotic variance of a consistent estimator. This idea is developed by adding a budget constraint over the cost per (partial) sample. The optimization problem is analytically solvable over the real numbers and gives the optimal number of requested partial and complete samples. We provide significance level, power, and sample-size calculations for detecting a non-zero causal effect under optimal budget allocation. Our method can considerably reduce the necessary budget and the number of complete samples. Finally, we showcase the advantages and applicability of adaptive causal effect estimation for automotive analytics and pharmaceutical research.

We consider a scattering problem for a wave equation $\partial_t^2 u = \frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j)u$ in a multilayer domain $\Omega \subset {\bf R}^{n+1}_x = {\bf R}^n_y \times {\bf R}^1_{x^{n+1}}$ of the form $\Omega = \mathcal K \cup \Omega_1 \cup \cdots \cup \Omega_N$, where $\mathcal K$ is a bounded open set and $\Omega_k$ is asymptotically equal to a slab domain ${\bf R}^n \times (c_k,c_k + d_k)$ as $|y| \to \infty$. Assuming that $\partial_x^{\alpha}\big(g_{ij}(x) - \delta_{ij}\big) = O(|x|^{-|\alpha| - \delta_0}), \ \delta_0 > 1, \forall \alpha$, we show that $\Omega$ and $g^{ij}$ are determined by one diagonal component $S_{11}(\lambda)$, for all energies, of the S-matrix associated with the slab $\Omega_1$, provided $\Omega_1$ is flat: $\Omega_1 \cap \{|y| > R\} = \{|y| > R\} \times (c_1, c_1+d_1)$ for some constants $c_1, d_1, R > 0$, and the metric is Euclidean on $\Omega_1\cap \{|y| > R\}$.

We study the use of an educational proof assistant in an introductory proof course through a quasi-experiment in a varied setting: multiple teachers, students with different study programs, and a mixed Dutch-English language environment. First-year university students are known to struggle with writing proofs. Waterproof is a proof assistant that is designed to support the transfer of skills to paper proofs by working with controlled natural language. We focus on the students' ability to construct valid mathematical proofs, and on their learning process. We study this through in-class observation, surveys, and analysis of student performance and proof structure. We present evidence that effects of using an educational proof assistant carry over to the pen-and-paper context,even when the assistant is English and the proof is given in Dutch. We also present evidence that suggests students in the Mathematics-Computer Science program achieve higher grades when using Waterproof. Our most important conclusion is that an educational proof assistant can help students be more explicit in their proofs. As students self-selected into using Waterproof rather than being randomly assigned, these results are suggestive rather than causal.

The first goal of this paper is to provide concrete chain complexes computing the homology of (unordered) configuration spaces of manifolds in positive characteristic, lifting a theorem by Knudsen to the model category level. We make them fully explicit and provide a computer program to compute their homology. Our methods also allow us to construct several new spectral sequences converging to these homology groups. Finally, we conjecture that this equivalence of chain complexes can be promoted to an equivalence of twisted $\EE_\infty$-coalgebras in right $\EE_d$-modules, and we explain how this conjecture would imply the homotopy invariance of the $\EE_d$-homotopy type of configuration spaces in positive characteristic via new “twist” and “detwist” functors.

Multiobjective composite optimization problems arise in sparse regularization, constrained multiobjective models, and multi-task learning, but their numerical solution remains challenging when the smooth components are ill-conditioned. Proximal gradient methods are inexpensive per iteration but may converge slowly, while proximal Newton and quasi-Newton methods exploit curvature information at the cost of evaluating expensive metric proximal mappings. To address these issues, we propose a preconditioned proximal Barzilai–Borwein method for multiobjective composite optimization. The method combines objective-wise Barzilai–Borwein scaling, which reduces imbalance among objectives, with a common preconditioner that captures shared curvature information. To avoid non-diagonal metric proximal mappings, we develop a subspace variant in which the search direction is computed in a two-dimensional subspace generated by a proximal-gradient-type direction and a projected historical direction. By constructing a conjugate basis with respect to the preconditioning metric, the subspace model decomposes into tractable one-dimensional subproblems. The framework is further extended to nonsmooth terms of the form $g_i(Ax)$ through a linear-operator-aware preconditioner, yielding explicit proximal evaluations via dual subproblems. We also analyze an inexact version based on relaxed descent conditions. We establish the global convergence of the inexact algorithm in the nonconvex setting and prove a linear convergence rate under an error-bound condition. Numerical experiments on ill-conditioned $\ell_1$-regularized, structured $\ell_1$-regularized, and linearly constrained problems demonstrate the effectiveness of the proposed method.