数学前沿

We obtain bounds for the $F$-pure threshold of isolated hypersurface singularities over an algebraically closed field of positive characteristic $p$ in terms of classical singularity invariants. For an irreducible plane curve whose value semigroup has first two generators $\beta_0,\beta_1$, we show that the $F$-pure threshold is at most $\frac{1}{\beta_0}+\frac{1}{\beta_1}$ and differs from it by less than $\frac{1}{\gcd(\beta_0,\beta_1)p}$, with equality when certain base-$p$ digits add without carrying, and we give an example showing that higher weighted-order terms can change the $F$-pure threshold. We also conjecture a lower bound in terms of the Milnor number, and prove it for semi-quasi-homogeneous singularities, for all plane curves, and whenever the number of variables exceeds $p$. In particular, it holds in characteristic $2$. As applications, we reprove Igusa's formula for the log canonical threshold of a complex plane branch (with integer coefficients), prove the conjecture of Musta\c{t}\u{a}, Takagi and Watanabe for such branches, and bound the log canonical threshold of complex isolated hypersurface singularities.

We study bounded realizations of Caputo differentiation on coefficient Banach spaces of fractional power series. The main contribution is an explicit quantitative analysis of local approximation and periodic reconstruction. We determine the least possible simultaneous error in the initial displacement and the prescribed target. The optimal value is attained, and a minimizing perturbation is given explicitly. We also solve the exact-arrival problem and establish uniqueness of its least-norm solution. These formulas provide computable approximation times and identify the sharp threshold for hypercyclicity, mixing and Devaney chaos. Every periodic point is described by a finite block whose geometric repetition yields exact norm formulas. The resulting approximations attain any prescribed sufficiently large minimal period near any given vector. For distinct positive powers, a simultaneous block construction gives quantitative error estimates and the same sharp threshold for disjoint mixing. Compact embeddings relate the Banach realizations to the Mittag-Leffler-Caputo Fr\'echet space. They also exhibit dense stable invariant subspaces inside chaotic realizations. The local approximation condition is a norm formulation of classical transitivity. Its quantitative evaluation, together with explicit periodic and simultaneous reconstruction, provides the central results of the study.

We present a general framework to derive augmented Lagrangian preconditioners for saddle point systems arising from mixed finite element discretizations. Our approach is based on augmenting the variational formulation at the continuous level prior to discretization, providing a unified strategy for designing efficient preconditioners that are robust under mesh refinement for a broad class of problems leading to symmetric saddle point systems. The robustness of the proposed preconditioners is established through a detailed spectral analysis. This framework is applied to different model problems, focusing on mixed-dimensional and fictitious domain formulations involving Lagrange multipliers, with some configurations featuring large coefficient jumps. Extensive numerical experiments on different test cases and geometries are presented, validating the effectiveness and robustness of our preconditioners when used in conjunction with flexible GMRES.

We study rational and integral points lying on finite collections of conics on non-ruled cubic surfaces over a number field. We extend the global determinant method to hypersurfaces in linear subvarieties, retaining dependence on the height of their Cayley forms. The universal Deligne pairing identifies the Chow line bundle on a residual conic pencil with three times of the universal bundle. Uniform bounds for cubic curves then control the number of conics of bounded height and give uniform estimates for their contribution to auxiliary coverings.

Let $k$ denote an algebraically closed field of characteristic zero and let $X$ denote a smooth elliptic curve over $k$. Given a three-periodic elliptic helix $\underline{\mathcal{E}}$ of vector bundles over $X$ with endomorphism $\mathbb{Z}$-algebra $\operatorname{End} \underline{\mathcal{E}}$ and quadratic cover $\mathbb{S}^{nc}(\underline{\mathcal{E}})$, we prove that $\operatorname{End} \underline{\mathcal{E}}$ is the quotient of $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ by a degree three family of normal elements, generalizing a result of the authors to the case in which $\operatorname{dim }(\operatorname{End} \underline{\mathcal{E}})_{i, i+1}$ isn't a constant function of $i$. We then show that $\operatorname{End} \underline{\mathcal{E}}$ is noetherian if and only if it has polynomial growth, and in this case, the ranks of any three consecutive bundles in the helix are a Markov triple. Furthermore, in this case $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ is a noetherian GK-three $\mathbb{Z}$-algebra which is ${\sf Proj }$-equivalent to an elliptic algebra. We conclude the paper by constructing several new families of elliptic helices with exponential growth.

We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, $$\begin{equation*} T(x,y)=(\ell x, A y+\phi(x)), \end{equation*}$$ where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$, with $r>(d+3)/2. We allow the fibers to have any dimension and$A$to be non-conformal. We show that: when$|\det(A)|\ell<1$, for almost every$\phi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equal 1+ the affinity dimension; when$|\det(A)|\ell>1$, for almost every$\phi$, the SRB measure is absolutely continuous. In the latter case we investigate the Sobolev regularity of the density of the SRB measure. We introduce a derivative dispersion condition, replacing the usual transversality condition.

In this paper, we lift the instanton TQFT from ordinary categories to the $\infty$-categorical setting by constructing a functor $CI:\mathrm{BI}^{\partial}\longrightarrow\mathsf{D}$ from an $\infty$-cobordism category $\mathrm{BI}^{\partial}$ for instantons to an $\infty$-derived category $\mathsf{D}$ of $2$-periodic chain complexes. The construction of $\mathrm{BI}^{\partial}$ is a modification of the $\infty$-cobordism category $\mathrm{Bord}_4$ introduced by Lurie and by Calaque--Scheimbauer using complete Segal spaces. The target $\infty$-category $\mathsf{D}$ is obtained via the dg nerve of the dg category of $2$-periodic chain complexes of finitely presented projective $\mathbb{Z}$-modules. The information encoded by the functor $CI$ was already developed by Kronheimer--Mrowka using families of metrics on cobordisms. Our reinterpretation in terms of $\infty$-categories conceptually simplifies the construction of the hypercube complexes underlying the link spectral sequence. This viewpoint also clarifies hyperbox complexes and compression in Manolescu--Ozsv\'{a}th's work, and possible $A_\infty$-structures, especially their signs and higher coherence. In addition, we lift the generalized cap product $\mu$-operators in instanton Floer homology to the chain level and construct explicit homotopies and higher homotopies establishing the commutativity of multiple $\mu$-operators in even degrees.

Assuming $V=L$, we construct proper coanalytic subfields of $\mathbb{R}$ and coanalytic Hamel and transcendence bases of every possible Hausdorff dimension, uniformly in the relevant parameters. There is a single lightface $\Pi^1_1$ set $G\subseteq[0,1]\times\mathbb{R}$ whose sections $A_\alpha$ are proper subfields with $\dim_{\mathrm H}A_\alpha=\alpha$, form a strictly increasing chain, and satisfy $A_\alpha=\bigcup_{\beta<\alpha}A_\beta$ for $\alpha>0$. For every $n\ge1$, every countable subfield $K\subseteq\mathbb{R}$ and every $s\in[0,n]$, we construct a coanalytic Hamel basis of $\mathbb{R}^n$ over $K$ of Hausdorff dimension $s$. Likewise, for every countable subfield $K\subseteq\mathbb{C}$ and every $s\in[0,2]$, we construct a coanalytic transcendence basis of $\mathbb{C}$ over $K$ of Hausdorff dimension $s$. In each case, the bases are sections of a single lightface $\Pi^1_1$ relation, uniformly in codes for the coefficient field and in the prescribed dimension, as well as in $n$ for Hamel bases. More generally, we prove a uniform theorem for suitably coded families of pregeometries on Euclidean spaces. The constructions combine Cantor sets of controlled dimension with hyperarithmetic coding and coanalytic transfinite recursion. The complexity is optimal: proper subfields of $\mathbb{R}$ of positive Hausdorff dimension and bases of the stated kinds cannot be analytic.

Quaternion tensor singular value decomposition (T-SVD) under the T-product provides a framework for analyzing multidimensional quaternion data. However, the T-SVD construction of Pan and Ng (SIAM J. Matrix Anal. Appl., 2024) \cite{PanNg2024} contains a structural gap: an unrestricted quaternion matrix SVD need not preserve the paired Fourier block structure required for tensor factor recovery, potentially leading to loss of T-unitarity and inaccurate reconstruction. To address this issue, we develop a structure-preserving construction that transforms each paired quaternion block into a complex matrix and maps its complex SVD factors back to quaternion factors with the required structure. This construction corrects the proof of \cite[Theorem 4]{PanNg2024} and leads to a corrected version of \cite[Algorithm 2]{PanNg2024}, hereafter referred to as the ``corrected algorithm'', ensuring T-unitary tensor factors and an f-diagonal core for arbitrary third-order quaternion tensors. A numerical counterexample demonstrates the failure of \cite[Algorithm 2]{PanNg2024} and verifies the corrected algorithm at machine precision. Notably, we propose a fast structure-preserving algorithm, Algorithm~\ref{alg:t-fast}, and compare it numerically with the corrected algorithm. The experimental results show that Algorithm~\ref{alg:t-fast} reduces the decomposition time while maintaining near-machine-precision accuracy. Furthermore, its application to color video encryption yields high reconstruction quality, favorable ciphertext statistics, and stability under input perturbations.

This paper is about the derivation of "integrability results" (i.e., for example, explicit formulas for the probability of events involving a few marked points) for Conformal Loop Ensembles, that are the (conjectured) scaling limits of discrete crititical planar lattice models from Statistical Physics. We discuss the explicit solutions to the PDEs provided by the original SLE/CLE martingale approach that AI now makes it possible to uncover. This provides short and direct new proofs of a number of formulas that have recently been derived based on Liouville quantum gravity, conformal field theory or Coulomb gas methods, and also makes it possible to derive some new ones. We briefly illustrate this with the examples of the backbone exponent and annulus crossing probabilities for critical percolation, but this direct approach seems to apply to a wide class of observables of two-dimensional systems related to conformal field theories.

We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $\Lambda_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. More precisely, for every fixed $d\ge2$ and $q\ge3$, there exists $\beta_0=\beta_0(d,q)>0$ such that, for $0<\beta<\beta_0$, the cutoff occurs in an optimal $O(1)$ window centered at $$t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |\Lambda_n|,$$ where $\mathfrak{r}\in(0,1)$ is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive $O(1)$. It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.

For the $H^{2|2}$ model on $\mathbb Z^2$, we show that, at energy zero, the fractional moments of Green functions, $\mathbb E[ G(x,y)^s]$ decay exponentially in the distance between $x$ and $y$, for every constant edge weight $W>0$ and every $0<s<1/2$. Here $G$ is the infinite-volume Green function of the associated random Schr\"odinger operator. The main ingredient is a quantitative lower bound on the derivative of the mean effective conductance with respect to the edge weights.

Let $\Omega\subset\mathbb{R}^n$ be a domain, with $n\ge2$. We characterize the real-valued harmonic functions that can occur as prescribed real components of monogenic functions. For any fixed nonzero real Clifford module $M$ and any fixed unit vector $s\in M$, a smooth real-valued function $h$ can be written as $h=\langle F,s\rangle$ with $F$ monogenic if and only if the $(n-1)$-form $*dh$ is exact. Consequently, every real-valued harmonic function on $\Omega$ admits such a representation if and only if $H_{\mathrm{dR}}^{n-1}(\Omega)=0$. In dimension eight, the same exactness criterion characterizes the real parts of octonionic regular functions and yields the corresponding equivalence with $H_{\mathrm{dR}}^7(\Omega)=0$. The proofs use global solvability of the Poisson equation and elementary identities for differential forms.

We prove the (uniform) Yau-Tian-Donaldson Conjecture for transcendental K\"ahler classes in the case of trivial automorphisms. Indeed, we show the existence of Special K\"ahler Fujita Approximations of big cohomology classes associated to big test configurations, establishing the equivalence between the uniform $K$-stability and the strengthened version for models. More generally, we show that any big cohomology class on a compact K\"ahler manifold admits such Special K\"ahler Fujita Approximations: the volume and the analogue of the Riemann-Roch coefficient of the big class are both approximated by those of K\"ahler classes on higher compactifications.

We address the problem of real-time multi-agent path planning under prescribed initial and final configurations, formulated as a high-dimensional optimal control problem subject to inter-agent collision avoidance and obstacle constraints. Classical solvers based on the Hamilton--Jacobi--Bellman equation or Pontryagin's maximum principle (PMP) become computationally intractable as the number of agents grows, due to the curse of dimensionality and the non-convexity introduced by interaction constraints. We propose OCFormer, a transformer-based neural network that parameterizes the inverse of a coordinate transformation mapping a simple, analytically tractable latent space, in which agents evolve independently under a decoupled Hamiltonian, to the physical space of the coupled, interacting system. The model is trained by minimizing a physics-informed loss comprising the Hamiltonian ODE residual from the PMP and a boundary matching term, without requiring pre-computed trajectory data. The self-attention mechanism endows OCFormer with permutation equivariance over agents, a structural symmetry that prior symplectic network approaches do not natively enforce. When trained on a distribution of boundary conditions, the amortized OCFormer produces near-optimal trajectories for unseen initial and final configurations in a single forward pass ($\sim$220ms), enabling real-time inference without retraining. We further introduce a lightweight transfer learning framework that adapts a pretrained OCFormer to out-of-distribution perturbations in seconds via L-BFGS fine-tuning. Numerical experiments on two- and three-dimensional multi-agent navigation with up to 128 agents demonstrate the scalability and efficiency of the proposed approach.

In this work, we study the Markov--Krein correspondence in both the univariate and multivariate settings. Our main result provides an explicit moment--moment formula expressing the multivariate moments of the inverse Markov--Krein transform in terms of the multivariate moments of the Markov-Krein transform via cyclic interval partitions. We also derive moment--cumulant relations involving classical, Boolean and free cumulants.

We construct a second-order, linear, and fully decoupled scheme for Rayleigh--B\'enard convection between Navier-slip walls. The nonlinear terms are treated explicitly and multiplied by a scalar with exact value $1$ and no energy contributed to the continuous problem. As a result, each BDF2 step, with a rotational pressure correction, solves only constant-coefficient linear problems. For the time discretization, we prove that, for any time-step, the energy decays when the walls have equal temperatures, under a condition on the thermal weight, and the energy is bounded on finite time intervals when heated from below. Relaxing the scalar towards $1$ preserves both estimates. Energy stability does not control the accuracy of the scalar; the velocity and temperature fields are computed with its unrelaxed value, whose deviation from $1$ scales like $(E/C)(\delta t/\tau)^3$ for kinetic energy $E$, time-scale $\tau$ of the flow, and weight $C$ of the scalar equation. In our tests, choosing $C$ comparable to $E$ keeps this deviation small and doubles the largest time-step for which steady rolls are reproduced to a relative difference of $10^{-8}$. With Taylor--Hood elements, we observe second-order convergence in time, recover the critical Rayleigh numbers, and match published Nusselt numbers to within $0.5\%$. A three-dimensional implementation passes corresponding validation tests.

The canonical double cover $BX$ of a graph $X$ is the direct product $X\times K_2$. A graph $X$ is stable if every automorphism of $BX$ is induced by automorphisms of $X$ and $K_2$, and unstable otherwise. We prove several asymptotic results on stable and unstable graphs. In particular, we show that the automorphism group of the canonical double cover of almost every graph has order $2$, and consequently that almost all graphs are stable. We also prove that the dominant sources of instability are having twins and being disconnected, in that order, and that graphs arising from a known construction are the next step in this hierarchy.

Many Bayesian inverse problems are characterized by two parameter vectors with distinct roles. A medium property, operator coefficient, or geometry block enters the forward model nonlinearly, whereas a volumetric, initial condition, or boundary condition source enters linearly. General-purpose nonlinear surrogates must relearn this known source dependence from data and may not preserve the correct source scaling outside the training regime. We introduce a structure-preserving partially linear (SPPL) framework that constructs separate reduced spaces for outputs, (linearly appearing) sources, and (nonlinearly appearing) medium variables and represents the reduced forward map as an affine function of the retained source coordinates, with coefficients that depend on the medium variables. Here, "affine" refers to the retained, mean-centered source coordinates: the physical forward map remains linear in the source, while the affine offset is induced by the source-prior mean and vanishes when that mean is zero. The analysis first identifies the optimal blockwise ridge approximation and then derives the optimal affine-in-source approximation under independent medium and source priors. A direct decomposition of the forward error into output, source, medium, and surrogate contributions yields blockwise spectral criteria and prior-predictive expected-KL control. For a Gaussian source prior, the optimal affine-in-source approximation coincides with the optimal blockwise ridge approximation. In a coseismic numerical example, a three-model controlled ablation on common block-reduced coordinates separates the effects of coefficient-matrix supervision and structure-preserving affine-in-source parameterization. Reduced-space surrogate MCMC provides orders-of-magnitude online acceleration relative to high-fidelity posterior sampling.

We prove the all-exp property for the Traveling Salesman Problem with the $2$-opt neighborhood, and for $(3,2,2)$- and $(2,3,2)$-Maximum Constraint Assignment with the flip neighborhood. The all-exp property states that infinitely many instances have an initial solution from which every sequence of improving moves to a local optimum, and hence every run of local search, has exponential length. For the Traveling Salesman Problem, the all-exp property was previously known for every $k$-opt neighborhood with $k \ge 3$, so our result for $2$-opt completes the picture for all $k$-kopt neighborhoods. In $(p,q,r)$-Maximum Constraint Assignment, each constraint involves at most $p$ variables, each variable occurs in at most $q$ constraints, and each domain has size $r$. Decreasing any of the parameters of $(3,2,2)$ or $(2,3,2)$ yields a problem in which every sequence of improving flips has polynomial length, so both Maximum Constraint Assignment results are tight. All three results settle long-standing open problems. Our proofs build on a common construction idea, which reveals a close connection between the local-search dynamics of the three problems.