数学前沿

In this paper we give an overview of several developments in spray geometry and provide some further interpretations of the geometric quantities that arise naturally in this subject. We discuss, in particular, the $S$-curvature, the $\chi$-curvature, and several notions of Ricci curvature for sprays. We rederive a number of identities relating these quantities and explain their applications to locally projectively flat sprays and to projectively related sprays. We also discuss applications to Finsler geometry, with emphasis on the $\chi$-curvature of a Finsler metric and on the inverse problem of determining whether a given spray is induced by a Finsler metric.

We present an efficient method for image segmentation in the presence of strong intensity inhomogeneities. The approach combines two levels of clustering: pixels are first grouped into superpixels using Power-SLIC, which admits a discrete optimal transport (OT) interpretation, and these superpixels are subsequently merged using squared 2-Wasserstein distances between their empirical distributions. We formulate the merging process as an adjacency-constrained hierarchical optimization problem with a regularized objective that accounts for the heterogeneities of previously formed regions. We establish NP-completeness of the associated global optimization problem, even for two final regions and two-bin histograms. For the proposed greedy algorithm, we prove exact recovery under suitable histogram separation conditions and derive an additive approximation bound for the regularized energy. Numerical experiments demonstrate competitive segmentation accuracy on images exhibiting substantial intensity inhomogeneity, while retaining computational efficiency.

We develop a uniform framework for Stevens' approach to Kloosterman sums on split groups over $p$-adic fields. The main result reduces the bounding of Kloosterman sums to that for the so-called Pl\"ucker-constrained sets, which are certain subsets of affine spaces constrained by conditions giving by the Pl\"ucker coordinates. We then give a nontrivial bound for the Kloosterman sums on the exceptional group $G_2$. As an application, we obtain the regularity of Bessel distributions associated with generic representations on $G_2$.

The approximation theory by finite elements is usually described on the simplified geometry of polygonal or polyhedral domains. While avoiding the question of boundary approximation makes the analysis simpler, real applications may involve piecewise smooth domains and in some cases the extension of classical results to general boundaries may be tricky. In this paper we discuss in a systematic way the approximation properties and the error analysis of standard non-isoparametric finite element spaces on piecewise curvilinear domains in two and three dimensions. We show in detail how the geometric error introduces a barrier for the $H^1$ and $L^2$ error of the solution of both Dirichlet and Neumann elliptic problems independently of the regularity of the problem and of the degree of the finite element spaces used.

Developing general-purpose algorithms with global asymptotic convergence guarantees for Nash equilibria of general-sum games remains a longstanding challenge. This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from simplex domains to product-of-simplex domains, enabling its direct application to Nash equilibrium computation. The resulting fixed-point bundle exhibits a composite fiber bundle structure. The key innovation is to construct an equivalent VI on a simplex and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, we show that the path-following algorithm for VIs on product-of-simplex domains inherits the global linear convergence guarantees of the simplex-domain framework. In numerical experiments, the algorithm attained a precision of $10^{-5}$ on all 5600 randomly generated game instances, with dimension ranging from 2-player 128-action to 128-player 2-action. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.

We study whether a transformer network can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence $\mathbb{N}\mathscr{T}$ defines an arithmetic text with measurable statistical structure. A transformer network (the GPT-2 architecture) is trained from scratch on the first $10^{11}$ elements and evaluated on next-token prediction and masked-token prediction tasks, with a Hidden Markov Model as baseline and a scaling analysis over context window, dataset size, vocabulary size and model size. The model reaches a word accuracy of about $0.4$, well above the baseline, and its performance remains stable on test blocks located at $10^{13}$-$10^{15}$, far beyond the training interval. Moreover, the likelihood assigned by the trained model separates the arithmetic text from two controls: synthetic token sequences that preserve the marginal token frequencies but not their order, with a separation that widens with the context, and sequences containing more than three consecutive square-free integers, a configuration that arithmetic forbids. These results indicate that the transformer captures regularities of the arithmetic text that go beyond its frequency profile.

Accurate simulation of atmospheric flows is challenging. Low-Mach-number dynamics often need to be resolved in the presence of a dominant, nearly hydrostatic background state. Conventional upwind numerical fluxes can introduce excessive acoustic-speed-based dissipation in the low-Mach regime. Discretization errors in the balance law can also generate spurious motion. This work investigates these two issues within a high-order Direct Flux Reconstruction (DFR) framework coupled with a third-order exponential time integrator for the 3D compressible Euler equations with gravity. Low-Mach-number accuracy is addressed using a modified AUSM$^+$-up flux, denoted AUSM$^+$-upH. Preservation of the background equilibrium is improved through a deviation-based formulation. The resulting approach is assessed using a hierarchy of idealized atmospheric test cases. The results show that AUSM$^+$-upH preserves low-Mach flow structures more effectively than the more dissipative Rusanov flux. However, reduced numerical dissipation also makes AUSM$^+$-upH more sensitive to residual errors in the discrete hydrostatic balance. The deviation formulation substantially suppresses these errors by several orders of magnitude, with the strongest improvement observed for flows over orography. Overall, AUSM$^+$-upH exhibits greater robustness when combined with the deviation formulation across the atmospheric test cases considered.

We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interest in their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide a complete characterization of a compactness problem posed by Ozawa for graph products of finite groups and right-angled Hecke von Neumann algebras.

We propose a variant of the approximate Bregman proximal gradient (ABPG) algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. ABPG is known to converge globally to a stationary point even when the smooth part of the objective function does not have a globally Lipschitz continuous gradient, and its iterates can often be expressed in closed form. However, ABPG relies on an Armijo line search to guarantee global convergence, which can slow down its practical performance. To address this issue, we propose a variant of ABPG with a variable metric Armijo line search, which relaxes the Armijo condition. We prove that the line search terminates in finitely many steps for a general nonsmooth convex function. Under the variable metric Armijo condition, we establish global subsequential convergence of the algorithm. Moreover, assuming the Kurdyka--\L{}ojasiewicz property, we also prove that the algorithm globally converges to a stationary point. Numerical experiments on $\ell_p$-regularized least squares problems, nonnegative linear inverse problems and phase retrieval demonstrate that the proposed algorithm outperforms existing algorithms.

We study the additive energy and the Fourier approximation of graph adjacency matrices labeled by a cyclic group. For a graph with $N$ vertices, $s$ edges and adjacency matrix $M$, the additive energy of its directed edge set is at most $\min\{(2s)^3,N^2\operatorname{tr}(M^4)\}$, and equality in the spectral bound holds exactly when the labeling attains the graph-energy lower bound for the Fourier ratio. Such a labeling also minimizes every best $k$-term Fourier approximation error, whose square equals the tail of the squared adjacency singular values. We prove that graphs of odd prime-power order admit an equality-attaining labeling exactly when they are cyclic Cayley graphs, and a connected example on $15$ vertices shows that equality need not imply vertex transitivity at other orders. We construct regular graph families with asymptotically maximal normalized additive edge energy and unbounded minimum Fourier ratio, one of them with density tending to one half. For dense graph sequences under uniformly random labelings, the collision dimension differs from the reciprocal squared edge density by a quantity tending to zero in probability. We also prove a quantitative stability estimate for the graph-energy bound and show why its dependence on a spectral scale cannot be omitted, even for regular graphs. For a rank-$m$ orthogonal projector we prove $m\leq D_4\leq\operatorname{FR}^2$, characterize the equality $D_4=m$ by cyclic translation invariance, and obtain a quantitative approximation by a circulant projector.

We develop energy-stable parametric finite element methods for surface diffusion of solid films with moving contact lines on curved substrates. Discretizing both the film surface and the substrate patches allows the introduction of linear schemes that are unconditionally energy stable. A nonlinear variant of the schemes ensures that, in addition, the vertices of the substrate approximations remain attached to the true substrate throughout. Moreover, our framework can incorporate anisotropic surface energy densities, for both the film surface and the substrate, as well as different tangential motions. Numerical experiments in two and three space dimensions illustrate the practicality of the introduced methods.

For $d,n\in\mathbb{Z}_{\geq1}$ and $p\in[1,\infty)\setminus\{2\}$, we characterize all linear maps $s\colon\ell_d^p\to\ell_n^p$ for which there is $E\subseteq X_d$ such that $\|s(\xi)\|_p=\|\xi|_E\|_p$ for every $\xi\in\ell_d^p$. We call these maps \emph{restricted isometries}. Our Lamperti-type theorem gives a unique description of each such operator using a set transformation, a positive amplitude function, and a unimodular phase function. In particular, it includes noninjective and nonsurjective operators, even when the dimensions differ. We further characterize restricted isometries between finite $\ell^p$-spaces with arbitrary positive atomic weights. Applications include formulas counting set transformations and uniform restricted isometries, as well as a geometric description of all restricted isometries with fixed column supports.

In this article we give an algorithm to produce infinitely many explicit quadratic twists of a given elliptic curve over $\mathbb{Q}$ with a rational point and fixed root number. Moreover, under the Parity Conjecture (or other technical hypotheses), for each $n\geq 1$, we produce explicit multi-quadratic extensions $L_n$ of $\mathbb{Q}$, of degree $2^n$, such that the growth in rank from $\mathbb{Q}$ to $L_n$ is at least $\max\{2n+2^{n-1}-1,\,2^n-1\}$ when the root number of $E$ is $-1$, and at least $\max\{2n,\,2^{n-1}\}$ when it is $+1$. As a result, assuming Parity, we exhibit elliptic curves over $\mathbb{Q}$ and extensions of degree $2^n$ where the rank after base-extension is greater than or equal to $34$ or $30+\max\{2n+2^{n-1},\,2^n\}$.

We classify two-species mass-action networks with at most three internal directed reactions on $\{X,Y,2X,X+Y,2Y\}$, positive inflows, and linear removal of both species. Among the 681 networks up to species exchange, exactly 27 admit two distinct positive hyperbolic attracting equilibria. The same 27 admit bistability with equal removal rates and with independently chosen positive removal rates. Each is minimal under reaction deletion. We give rational equal-removal witnesses for all positive cases and all-parameter exclusions for the remaining 654. The proof combines nullcline geometry, polynomial sign tests, resultant root bounds, and exact flux-cone decompositions. A separate rational-arithmetic implementation verifies the complete classification and its equilibrium certificates.

Continuous-time multiparametric model predictive control solves the optimal control problem once, offline, for every initial state, and divides the initial-state space into critical regions, each with its own control law and sequence of active constraints. A boundary between two regions is either a hyperplane, given by one linear equation, or curved. The curved boundaries have not been computed to date; a classifier has been trained instead to distinguish the regions, so the partition has been known only approximately. In this paper we show that a curved boundary can be computed exactly, without solving any optimal control problem. The method rests on one property. Once the sequence of active constraints is fixed, together with the instants inside it at which a constraint becomes active or inactive (the switching times) and the instant at which the change of the neighboring region first appears (the event time), every condition on a boundary is a linear equation in the initial state, so the computation of a boundary, curved or not, is exact. We fix one of these instants, a switching time or the event time, and solve the linear equations that correspond to it for the corresponding initial state on the boundary; stepping the fixed instant along the horizon yields the whole boundary. A hyperplane is the case in which no instant is left to fix. We illustrate the method on the Newell-Lee evaporator, which has seven critical regions and eight boundaries, four hyperplanes and four curved, computing one boundary of each type in full with the adjacent regions at its ends.

Max-pressure (MP) traffic signal control selects phases using pressure scores computed from estimated traffic states, but these states are uncertain and their consequences depend on whether they alter phase ranking and how much pressure is lost. This paper develops a decision-oriented Regret-Based Max-Pressure framework that represents uncertainty through phase-selection regret, defined as the pressure loss of a candidate phase relative to the pressure-maximizing phase under each plausible latent state. We propose Stochastic Regret MP, which minimizes expected regret under a calibrated residual distribution, and Distributionally Robust (DR) Regret MP, which minimizes worst-case expected regret over a Wasserstein ambiguity set. Robustifying regret rather than absolute pressure makes the formulation invariant to state-dependent pressure components shared by all candidate phases, thereby focusing robustness on relative phase quality. We further derive an explicit queue-linear pressure-loss bound and establish strong queue stability when the net-service reserve exceeds the resulting loss coefficient. Numerical experiments show that both methods reduce normalized true-state pressure regret by about 70% relative to Point-estimate MP, while DR Regret MP yields lower network queues than direct robustification of absolute pressure. In microscopic SUMO experiments on a 28-intersection real-world network, Stochastic Regret MP and DR Regret MP reduce mean delay by 18.9% and 14.9%, respectively, and retain benefits under detector undercount and failure. The results show that representing uncertainty through its decision consequences can improve reliable traffic control under imperfect state information.

We prove that the 2-d cubic nonlinear Schr\"odinger equation with nonlinearity supported on a line or a sphere is globally well-posed in $H^1(\mathbb{R}^2)$. For higher dimensions, we prove that the equation with hyperplane-supported nonlinearity is locally well-posed in suitable anisotropic Sobolev spaces. We also prove that the equation with sphere-supported nonlinearity is globally well-posed in $H^1(\mathbb{R}^n)$ for radial data.

Developing general-purpose algorithms with global asymptotic convergence guarantees for variational inequalities (VIs) remains a longstanding challenge. This paper proposes a path-following framework for finite-dimensional VIs with arbitrary continuous mappings and polytope domains. The approach first transforms a general VI into a smooth VI on a simplex. Its key innovation is to formulate the resulting problem on a fiber bundle, called the fixed-point bundle, whose geometric structure enables the systematic integration of initialization, path-following, and singularity avoidance. Without imposing monotonicity or similar assumptions, the framework guarantees global linear convergence for nonsingular solutions and global linear convergence up to any prescribed precision for singular solutions. In numerical experiments, the algorithm attained a precision of $10^{-5}$ on all 6400 randomly generated instances, with dimensions up to 1024. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.

We give a probabilistic approach to the ultraviolet renormalization of the two-dimensional Sine-Gordon model. The model is formally described by the density $\exp(\alpha\int \mathopen{:} \cos(\beta X) \mathclose{:} )$, where $X$ is a log-correlated Gaussian field. Our construction applies on bounded convex domains for every particle activity $\alpha\in\mathbb R$ and throughout the subcritical range $\beta^2\in[0,4)$, below the Berezinskii-Kosterlitz-Thouless threshold. We apply the martingale method introduced by Lacoin, Rhodes and Vargas to two dimensions by using refined estimates based on charge conjugation symmetries. Our result extends existing constructions by treating the full subcritical regime for a broad class of covariances.

Let $f$ be an isolated hypersurface singularity and let $J(f)$ be its Jacobian ideal. We study the higher Milnor numbers $\mu_k(f)=\dim_{\mathbb C}\mathcal O_n/\mathfrak m^kJ(f).$ Our main result concerns an ordinary semi-homogeneous singularity $f=f_r+f_{r+1}+\cdots$, where the projective hypersurface defined by $f_r$ is smooth. We prove that $\mu_k(f)=\mu_k(f_r) \qquad\text{for every }k\geq0.$ This gives the equal-weight case of a conjecture of Shi, Yau, and Zuo. The main algebraic input is the common-order case of the Valabrega--Valla criterion. From it, we obtain the graded Jacobian module and the exact intersection formula. As a consequence, we obtain a closed formula for $\mu_k$ and determine the sharp stabilization index of $\mathfrak m^kJ(f)$. We also determine the optimal uniform range in which the Tjurina formula holds for all ordinary semi-homogeneous germs with fixed $n$ and $r$. We next consider the log-concavity property. For ordinary semi-homogeneous singularities, the higher Milnor sequence has an initial log-convex part and a log-concave tail. For an arbitrary isolated hypersurface singularity in $n\geq2$ variables, both the higher Milnor and higher Tjurina sequences are eventually strictly log-concave. In the plane curve case, we prove that the log-concavity defects change sign at most once, and we characterize the sequences which are log-concave at every positive index. Finally, we give examples showing that the higher Milnor sequence does not determine the contact-equivalence class.