数学前沿

We define tropical band varieties as $\mathbf{T}$-reductions of separated finitely presented $\mathbf{T}$-band schemes. We construct a faithful functor from the category of tropical band varieties to the category of extended polyhedral spaces in the sense of Gross--Shokrieh. As a byproduct, we show that tropical cohomology can be defined functorially on tropical band varieties.

We study a proximal-gradient method for the sum of a convex smooth function $g$ and a possibly nonconvex, nonsmooth, strongly quasar-convex regularizer. The gradient step is weighted by the quasar-convexity parameter, so, for step sizes in the descent range, the iteration decreases a weighted objective, while the unweighted sum serves for comparison. We establish a geometric envelope around every global minimizer of the unweighted objective, with an explicit radius independent of the step size; a common minimizer yields global linear convergence. We also prove that every critical point $u$ of the weighted objective lies within an explicit multiple of $\|\nabla g(u)\|$ from every global minimizer of the unweighted objective. Together with an optimized composite comparison, this yields asymptotic distance bounds. An example shows that the two objectives can have different critical points. For power-regularized least squares with powers between zero and one, descent provides a confinement ball for the sharp local modulus, and support identification combined with a KL argument yields whole-sequence convergence.

We study a construction for producing Finsler metrics on the sphere from convex bodies in $\mathbb{R}^4$ using the quaternions. Among several applications, we prove the weak Viterbo conjecture for convex bodies in $\mathbb{R}^4$ that are invariant under the natural action of the quaternion group $Q$. This is the first instance where Viterbo's systolic inequality is established for convex bodies with a finite group symmetry, addressing a question of Haim-Kislev-Ostrover. Moreover, a smooth $Q$-symmetric convex domain that achieves equality is symplectomorphic to the ball, without any assumption of strict or uniform convexity. Finally, we show that any link of minimal Reeb orbits on the boundary of a uniformly convex $Q$-symmetric convex body is transversely isotopic to a union of Hopf fibers, strengthening a result of Abbondandolo-Edtmair-Kang in the $Q$-symmetric setting.

This paper addresses the Target-Attacker-Defender (TAD) game within a planar environment, focusing on the protection of a stationary target against incoming threats. To compensate for potential speed disadvantages of the defenders, we leverage geometric pre-positioning within an annular defense perimeter. The problem is formulated as an area interception problem, for which we propose a distributed multi-agent algorithm consisting of two primary modules: target allocation and interception control. For target allocation, an equitable region partitioning strategy is employed to divide the defense area into annular sectors and assign defenders to attackers. Under specified angular-separation and parameter conditions, the proposed partitioning mechanism ensures that each defender is assigned at most one attacker, thereby reducing the complexity of multi-agent assignment. For the control layer, movement protocols based on zero-sum differential games and coverage optimization are designed, achieving Nash equilibrium strategies while maintaining effective barrier coverage. Numerical simulations show that the proposed method enables slower defenders to intercept faster attackers through proactive deployment and maintains nearly constant local computation time as the swarm size increases.

We prove exact dimensionality of ergodic stationary measures for random compositions of $C^1$ diffeomorphisms of a closed manifold $M$ that contract at a single Lyapunov scale. More precisely, let $\nu$ be a Borel probability measure on $\mathrm{Diff}^1(M)$ such that $\log\sup_{x}\|{D_xf}\|$ and $\log\sup_{x}\|{D_xf^{-1}}\|$ are $\nu$-integrable, and let $\mu$ be an ergodic $\nu$-stationary probability measure. If the top and bottom Lyapunov exponents of the derivative cocycle coincide and equal $\lambda<0$, then $\mu$ is exact dimensional and $\mathrm{dim}(\mu)=h_\mu^{\mathrm{F}}(\nu)/(-\lambda)$, where $h_\mu^{\mathrm{F}}(\nu)$ is the Furstenberg entropy.

Conner and Raymond proved that if a connected, compact Lie group $G$ acts effectively on a closed aspherical manifold $M$, then $G$ is a torus and the fundamental group of $G$ injects into the center of $\pi_1(M)$. We prove the corresponding result for effective isometric actions on closed, aspherical Alexandrov spaces.

We construct the first examples of uncountable families of residually finite duality groups of type $FP$, all of which have isomorphic profinite completions, while being pairwise not quasi-isometric. To establish these finiteness properties, we prove conjectures of Leary regarding groups of deck transformations of certain cubical branched covers being residually finite and virtually special. The main new tool is a strong separability result for Bestvina-Brady groups, allowing the extension of any finite quotient of many Bestvina-Brady subgroups to a finite quotient of the whole group. This gives a finite version of the congruence extension property, which we also establish for the same group pairs.

In this study, we analyze the rapid stabilization of Stokes and the two- and three-dimensional Navier-Stokes systems in the presence of an unknown spatially localized disturbance. A multivalued feedback control strategy is proposed which combines the Lyapunov frequency method with the multivalued sign operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator provides robustness against disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach guarantees global rapid exponential stabilization for the Stokes system and local rapid exponential stabilization for the two- and three-dimensional Navier-Stokes systems, with arbitrarily large disturbance bounds. In three dimensions, we obtain global strong solutions and rapid exponential stabilization for initial data sufficiently small in the $L^2$-norm with a prescribed, arbitrarily large bound in the $H^1$-norm.

We study optimal control problems for incompressible fluids in two-dimensional domains with a cold boundary and internal heat sources and sinks. Given a kinetic energy budget, measured by the square of a nondimensional P\'eclet number $\mathrm{Pe}$, the goal is to maximize a cooling efficiency inversely proportional to the mean square gradient of the fluid's temperature. Using Lagrange duality, we formulate a well-posed dual problem whose solution yields an upper bound on the maximum cooling efficiency $\mathcal{E}(\mathrm{Pe})$. We then numerically approximate the dual problem using a convergent hierarchy of semidefinite programs obtained via discretization. We illustrate this computational approach on optimal cooling problems in a square and in an annulus, explaining also how problem symmetries can be exploited to reduce computational complexity. Finally, we construct admissible points for the dual problem to obtain new analytical upper bounds on the optimal cooling efficiency $\mathcal{E}(\mathrm{Pe})$. Specifically, we prove that $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^{2}$ for arbitrary domains and heat distributions, and that $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^{2}/ \ln^2\mathrm{Pe}$ for cooling flows in disks and annuli with heat source/sink distributions with a uniformly positive azimuthal average. These results generalize and improve known efficiency bounds for energy-constrained cooling flows in a disk.

In this paper, we establish the existence of a local-in-time strong solution to the compressible Navier--Stokes--Poisson system in a three-dimensional bounded smooth domain, subject to the general Navier--slip boundary condition. The solution is shown to be uniformly bounded in conormal Sobolev spaces, with uniform regularity estimates that are independent of the viscosity coefficients. As a consequence, we obtain the inviscid limit to the corresponding compressible Euler--Poisson system via a strong compactness argument.

Modern mesh generation pipelines whether learning-based or classical often produce outputs requiring post-processing to achieve production-quality geometry. This work introduces MeshCone, a convex optimization framework for guided mesh refinement that leverages reference geometry to correct deformed or degraded meshes. We formulate the problem as a second-order cone program where vertex positions are optimized to align with target geometry while enforcing smoothness through convex edge-length regularization. MeshCone performs geometry-aware optimization that preserves fine details while correcting structural defects. We demonstrate robust performance across 56 diverse object categories from ShapeNet and ThreeDScans, achieving superior refinement quality compared to Laplacian smoothing and unoptimized baselines while maintaining sub-second inference times. MeshCone is particularly suited for applications where reference geometry is available, such as mesh-from-template workflows, scan-to-CAD alignment, and quality assurance in asset production pipelines.

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,\rho)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_\beta(x)=d_M(x,x_0)^{-\beta}$, with suitable $x_0$ satisfying $\rho(B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $\alpha$-stable laws for $\alpha=d/\beta\in(0,2)$ and, at the boundary value $\alpha=2$, a Gaussian limit with the non-standard normalisation $\sqrt{n\log n}$. We also establish a conditional central limit theorem for $L^2$ observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets. The results and remarks have been formalised in Lean~4.

We offer a classification of a broad and practically relevant class of symmetric double saddle-point system. At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms. We describe relevant applications, invertibility conditions, spectral properties, and block preconditioners. Our discussion is kept within a general framework rather than tailored to specific applications.

We prove that if a $(k+2)$-Ahlfors--David regular measure $\mu$ in parabolic space $\mathbb P^n$ has support equal to a parabolic Lipschitz graph with sufficiently small Lipschitz constant, and the parabolic Riesz transform is bounded on $L^2(\mu)$, then $\mu$ is parabolic uniformly rectifiable and the graph is regular parabolic Lipschitz. The result holds for every integer $0\leq k\leq n-1$.

Mean-field control barrier functions (MF-CBFs) enforce safety of large-scale multi-agent systems at the level of densities, but existing formulations require the control to act directly on the population to be kept safe. We consider populations that are not directly actuated and are influenced only through other, controlled populations, possibly in the presence of diffusion. We show that if the constrained population lies at directed graph distance d from the actuated one, then, under a nondegeneracy condition, the safety functional has relative degree d+1 and the control enters its (d+1)-th time derivative affinely. Higher-order safety constraints can therefore still be enforced through a convex quadratic program. We apply the framework to leader-follower density control among dangerous regions, where the follower safety functional has relative degree two and the leader safety functional has relative degree one. We characterize exactly when the two constraints are jointly infeasible and bound the tracking errors induced by the safety correction. Simulations in cluttered environments show that the proposed filter preserves follower safety, whereas enforcing leader safety alone does not.

This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar--Gross--Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.

Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.

We establish an $\Omega(d)$ lower bound on the competitive ratio of convex body chasing in $\mathbb{R}^d$ with $\ell_p$ movement costs for any $p \in [1, \infty)$. Previously, the best known lower bound for general $p$ was $\Omega(\max\{\sqrt{d}, d^{1-1/p}\})$, established by Bubeck et al. (2019); a linear lower bound was known only for $p = \infty$, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior bound for every fixed $p \in [1, \infty)$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every $\ell_p$ norm. Our lower bound is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $\Omega(r^2)$ for requests of size $O(r)$ in that setting. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional vector space while approximately preserving movement costs up to a factor of $O(r)$, which establishes the lower bound.

We investigate the operator distance between the Identity and the Copson operator on the cone of nonnegative, nonincreasing sequences within the Lebesgue sequence spaces $\ell^p(\mathbb{N})$. By proving the exact value of this distance, we confirm a conjecture originally formulated by A. Ben Said, S. Boza, and J. Soria in ``The norm of Hardy-type oscillation operators in the discrete and continuous settings'' (\emph{Analysis and Applications}, 2025).

Distributed Space Systems offer unique capabilities but have significant hardware limitations on computational processing and fuel when employing small satellites. These constraints drive interest in deterministic maneuver planning that incorporates orbital perturbations to improve final state accuracy and decrease reconfiguration delta-v cost. This paper proposes a novel closed-form, provably optimal framework to generate complete roto-translational solutions for propulsive-differential drag control of multiple satellites. The approach leverages relative orbit elements and reachable set theory to characterize all control with a single methodology that can be readily generalized to generic impulsive-continuous control. The algorithm is robust to large reconfigurations and extended control windows where differential drag is most effective.