数学前沿

We prove that the polynomials counting locally free, absolutely indecomposable, rank 1 representations of quivers over rings of truncated power series have non-negative coefficients. This generalizes the positivity of toric Kac polynomials to higher depth. The proof goes by inductively contracting/deleting arrows of the quiver and is inspired by a previous work of Abdelgadir, Mellit and Rodriguez-Villegas on toric Kac polynomials. We also relate counts of absolutely indecomposable quiver representations in higher depth and counts of jets over fibres of quiver moment maps. This is expressed as a plethystic identity involving generating series of these counts. In rank 1, we prove a cohomological upgrade of this identity, by computing the compactly supported cohomology of jet spaces over preprojective stacks. This is reminiscent of PBW isomorphisms for preprojective cohomological Hall algebras. Finally, our plethystic identity allows us to prove two conjectures by Wyss on the asymptotic behavior of both counts, when depth goes to infinity.

We study a mathematical model of in situ leaching of rare metals, in which the joint filtration of two liquids is governed by the microscopic model $\mathbb{A}^{\varepsilon}$. A key difficulty is the unknown (free) boundary $\Gamma(r)$ between solid and liquid components, determined by an additional condition on $\Gamma(r)$; no standard methods exist for this nonlinear problem. To resolve it, we apply the fixed point theorem. For a given function $r(\boldsymbol{x},t)$ from a set $\mathfrak{M}_{(0,T)}$ of sufficiently smooth functions describing the skeleton structure, we consider the auxiliary problem $\mathbb{B}^{\varepsilon}(r)$: an elliptic system for displacements of the liquid and solid components coupled with parabolic equations for the acid concentration. Selecting the weak solution of minimal smoothness, we apply the homogenization method to pass from the microscopic to the macroscopic description. The resulting macroscopic model $\mathbb{H}(r)$ contains a homogenized boundary condition that expresses the normal boundary velocity $V_{N}=\partial r/\partial t$ as a linear function of the acid concentration $c$. Since $c$ depends on $r$ via an operator $\mathbb{F}\colon\mathfrak{M}_{(0,T)}\to\mathfrak{M}_{(0,T)}$, we prove that $\mathbb{F}$ is Lipschitz continuous and, by Banach's theorem, possesses a unique fixed point $r^{*}$, which yields the unique solution $\mathbb{H}=\mathbb{H}(r^{*})$.

We study divisorial ideals and the $v$-class semigroup of a Noetherian domain satisfying Serre's condition $(S_2)$. We prove that strict closedness is equivalent to Clifford $v$-regularity and describe the structure of the associated $v$-class semigroup. We use these results to characterize finite generation of the class semigroup and Boole $v$-regularity. We also construct prime ideal decompositions of integrally closed divisorial ideals and clarify the relationship between these decompositions and local Boole $v$-regularity. Finally, we study polynomial extensions and use them to construct Boole $v$-regular and locally Boole $v$-regular domains in arbitrary dimension.

It is an open question whether the Lie algebra $\mathrm{HH}^1(B)$ of a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ is simple only if the block is nilpotent with an elementary abelian defect group of order at least $3$. We show this to be the case if $B$ has a normal abelian defect group. Another open question is whether $\mathrm{HH}^1(kS_n)$ is a solvable Lie algebra, where $S_n$ denotes a symmetric group on $n$ letters. We give some general solvability criteria for $\mathrm{HH}^1(B)$ of blocks $B$ with a normal defect group, and, making use of results of Chuang-Kessar and Chuang-Rouquier, we deduce that if $B$ is a block of a symmetric group algebra with an abelian defect group, then $\mathrm{HH}^1(B)$ is solvable.

A path or cycle in a $2$-edge-coloured graph is alternating if consecutive edges have different colours. These objects are natural analogues of paths and cycles in graphs, and even more so in digraphs. A well-studied class is that of acyclic $2$-edge-coloured graphs, that is, those forbidding alternating cycles. In particular, (acyclic) $2$-edge-coloured graphs generalize directed (acyclic) graphs (DAGs), but the complexity of problems on DAGs is much better understood than that of the corresponding problems on acyclic $2$-edge-coloured graphs. We investigate the complexity of two such problems, whose directed counterparts are well-known. Our first result is an $n^{O(k)}$-time algorithm for the alternating $k$-linkage problem on acyclic $2$-edge-coloured graphs. This problem asks for $k$ vertex-disjoint alternating paths joining $k$ prescribed pairs of vertices. This generalizes the directed $k$-linkage algorithm for DAGs. For every fixed $k$, this also yields a polynomial-time algorithm for finding $k$ internally vertex-disjoint alternating $(s,t)$-paths, solving an open problem by Abouelaoualim et al. from 2008. We then consider the alternating chromatic number of $2$-edge-coloured graphs, defined as the minimum number of parts in a vertex partition such that each part induces an acyclic subgraph. This notion is analogous to the dichromatic number of digraphs, where it is known that deciding dichromatic number $2$ is NP-complete even on tournaments. Our second result is that deciding alternating chromatic number $2$ is NP-complete on complete $2$-edge-coloured graphs. Equivalently, deciding whether a graph can be partitioned into two induced threshold graphs is NP-complete.

We establish a new dynamical convergence-divergence criterion for the existence of Lipschitz functions with continuous sub-actions, over general dynamical systems. This unifies and generalises various versions of the Ma\~n\'e cohomology lemma, in the setting of uniformly expanding maps, and systems with indifferent fixed points. One consequence is that for any compact metric space $(X,\rho)$, and Lipschitz Ruelle-expanding map $T:X\to X$, a metric transform $\omega$ satisfies Dini's criterion $\int_0 \omega(s)s^{-1}\, ds <\infty$ if and only if every function that is Lipschitz for $\omega\circ\rho$ has a continuous sub-action. Another consequence is that if $T$ is an expanding circle map of Manneville-Pomeau type $\alpha\in(0,1)$, then $\omega$ satisfies the convergence criterion $\int_0 \omega(s)s^{-1-\alpha}\, ds <\infty$ if and only if every function that is Lipschitz for $\omega\circ\rho$ has a continuous sub-action. For the latter class of maps $T$, with $\omega$ satisfying the convergence criterion, we prove a Liv\v{s}ic theorem for functions that are Lipschitz for $\omega\circ\rho$: if $\int f\, d\mu=0$ for all invariant measures $\mu$, then there exists a continuous function $\varphi$ such that $f=\varphi\circ T-\varphi$, where $\varphi$ is Lipschitz with respect to $D_\alpha(\omega)\circ\rho$, and $D_\alpha(\omega)(t)= \int_0^t \omega(s)s^{-1-\alpha} ds$.

We construct complete smooth K\"ahler metrics on $\mathbb{C}^n$ , $n\ge2$, with nonnegative Ricci curvature, Euclidean volume growth, and nonunique tangent cones at infinity. More generally, we prescribe the entire tangent cone set within a class of smooth Hopf cones, using a compact connected family of positive transverse Ricci forms on $\mathbb{C}\mathbb{P}^{n-1}$. We identify all metric blow-downs and their Ricci currents, compute the exact asymptotic volume ratio, and classify isometries between the model cones. In particular, the K\"ahler condition does not ensure uniqueness at infinity under nonnegative Ricci curvature and Euclidean volume growth, even in complex dimension two.

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 $30+\max\{2n+2^{n-1},\,2^n\}$.