数学前沿

We use Stein's method to prove general upper bounds on the second order Wasserstein distance in the CLT for general dependent partial sums $S_N=X_1+\ldots +X_N$. Then we use these estimates to provide optimal $O(N^{-1/2})$ rates for a variety of chaotic dynamical systems and $\phi$- and $\psi$-mixing sequences. Our results for $\psi$-mixing sequences are obtained under the fourth moment condition, which matches the best known results in the iid case. This is the first time that optimal CLT rates of any kind are obtained for unbounded weakly stationary variables under general mixing conditions. We then apply this result and obtain optimal CLT rates $O\left((\text{Var}(S_N))^{-1/2}\right)$ for non-stationary $\psi$-mixing sequences without any growth rates on $\text{Var}(S_N)$. We also obtain close to optimal CLT rates for nonconventional sums and some classes of random dynamical systems with non-uniform decay of correlations

The learning dynamics of Stochastic Gradient Descent (SGD) are believed to play an important role in influencing the generalization properties of trained models. While the precise dynamics of how online (one-pass) SGD optimizes the population risk are now understood in a broad range of high-dimensional problems, a comparable understanding of multi-pass SGD for empirical risk minimization is still largely lacking. In this work, we develop an asymptotically exact characterization of the dynamics of multi-pass SGD for a non-convex multi-index model, in the high-dimensional regime where sample size $n$ and data dimension $d$ increase proportionally. We consider any sub-linear batch size $\kappa \asymp n^\alpha$ with $\alpha \in [0,1)$ and a commensurate ``critical'' scaling of the learning rate, and show that the coordinate-wise dynamics of SGD admit a novel continuous-time scaling limit, given by a system of dynamical mean-field equations that are driven by a scalar Poisson jump process representing the asymptotic limit of SGD sampling noise. This scaling limit connects several previously disparate theories around SGD, namely (i) the classical interpretation of SGD as an approximation to gradient flow on the empirical risk, (ii) the characterization of online SGD via an autonomous ODE in its overlap statistics, and (iii) the coincidence between multi-pass SGD dynamics and diffusion approximations in least-squares problems. We illustrate how these perspectives are recovered by specializing our result to small learning rates, large sample sizes, and linear models, respectively. Outside of these specializations, the scaling limit of SGD is distinct both from diffusion approximations and gradient flow, and may enable SGD to converge to local optima of the non-convex empirical risk with different statistical properties.

We prove that the Hasegawa-Mima equation (HME) is the limit of a magnetized pressureless Euler-Boltzmann system in a special scaling regime. The proof relies on the derivation of a uniform estimate independent of the scaling parameter, using the particular structure of the singular system and the identification of a combination of the unknowns that compensates for fast oscillations in time. We also show that HME allow us to investigate a conditional property of inverse energy cascade and drift waves.

We study two fundamental properties of the intersection pairing between geodesic currents on infinite-type surfaces which are classical in the finite-type closed setting: injectivity of the marked length spectrum and continuity of the intersection form. We show that geodesic currents on infinite-type surfaces of the first kind are determined by their intersection numbers with closed curves, extending Otal's marked length spectrum rigidity argument from finite-type to infinite-type surfaces. In contrast, we prove that the intersection pairing is not continuous in general for the natural topologies by constructing explicit counterexamples. We then prove sufficient criteria for continuity of intersection numbers for infinite-type surfaces.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $n\geq2$. The well-known Payne-P\'olya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$\frac{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}\leq\left(\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\right)^2,\quad m=1,2,3,\cdots.$$ Here, $j_{p,k}$ denotes the $k$-th positive zero of the Bessel function $J_p$, and $\lambda_j(\Omega)$ are the Dirichlet eigenvalues of $\Omega$, counted with multiplicity and ordered as $$0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots.$$ The cases $m=1,2,3$ were established by Ashbaugh and Benguria in their seminal works. In this paper, we give an affirmative answer to the Payne-P\'olya-Weinberger conjecture for $m=4$ by combining a quadratic-form identity with a Brouwer degree argument. We also extend the corresponding spectral inequality to arbitrary bounded open sets in $\mathbb{R}^n$.

We develop algorithms to construct a basis of de Rham cohomology for modular curves and a matrix representing the Hodge filtration required in the quadratic Chabauty method. We do this without requiring an explicit plane model of the curve and instead compute these objects in terms of modular forms. Our approach works for any modular curve; to demonstrate its efficiency, we implement our method for nonsplit Cartan modular curves.

We prove that every compact orbifold with corners admits a triangulation in which the image of every codimension one face is a subcomplex. Faces need not be embedded, and the local groups may permute boundary components. The proof turns the boundary hyperplanes of linear charts into mirrors by means of the squaring map, which, unlike the folding $y\mapsto|y|$, is compatible with smooth changes of charts, and then applies the triangulation theorem for orbit spaces of proper Lie groupoids.

We introduce a new class of unitary set-theoretic Yang--Baxter solutions, called \emph{bipartite} solutions. This class, which is purely combinatorial in nature, forms a full subcategory of $\mathcal{LYB}$, the category of Lyubashenko type solutions of Agore, Chirvasitu and the author, and is particularly amenable to explicit classification results. We give an intrinsic necessary and sufficient condition for a solution of the Yang--Baxter equation to be isomorphic to a bipartite solution, we settle the isomorphism problem within this class and, as an application, we determine their automorphism groups. If $X$ is a finite set with $|X| = n$, the number of isomorphism classes of bipartite solutions on $X$ is $\lfloor n/2 \rfloor + 1 + \sum_{m=2}^{n} \bigl( o(m, n-m) - 1 \bigr)$, where $o(m, k)$ denotes the number of orbits of the symmetric group $S_m$ acting by conjugation on the $k$-element multisets of elements of $S_m$. The paper also offers a comparative study: for small $|X| = n$ we count, up to isomorphism, all set-theoretic Yang--Baxter solutions on $X$, together with five distinguished families of solutions. Throughout the range we were able to compute, the class $\mathcal{LYB}$ accounts for a large share of all bijective and of all non-degenerate solutions, strikingly close to the total.

Program optimization aims to improve performance while preserving behavior. Coding agents can already make programs faster by changing algorithms, data structures, and execution strategies. However, existing equivalence checks mostly rely on tests, which agents have been shown to exploit, passing the tests without preserving the intended behavior. We propose agentic verified optimization, in which an agent produces a faster implementation together with a machine-checked proof of equivalence to the existing one. Because the existing implementation serves as the specification, the task avoids the specification-writing bottleneck of verified code generation, and recent advances in theorem proving make the proofs attainable. We formulate the task at the function and repository levels in Lean 4, where programs can be both executed and reasoned about. Few real-world applications are written in Lean, so we develop Aqueduct, an agentic framework that migrates C, C++, Go, and Rust programs into native Lean, validates each port by differential testing against its original. On the 189 reference implementations in Verina, agents produce faster proved rewrites for 156, including algorithmic changes such as replacing exponential recursion with dynamic programming. On 17 migrated applications, their 92 proved rewrites speed up 47 of 51 application scenarios. These application-level gains are smaller than the function-level ones, making the choice of functions to optimize an important problem. Current agents can thus already carry out verified optimization on both standalone functions and applications migrated from real programs. We discuss the remaining challenges and the steps toward verified optimization in production.

We show that there exist exotic spheres with metrics of strictly positive sectional curvature in infinitely many dimensions: in every dimension $n\equiv3\pmod4$ with $n\geq11$ and $n\not\equiv255\pmod{256}$. Previously, every known closed simply connected manifold of positive sectional curvature in a dimension above $24$ was diffeomorphic to a compact rank-one symmetric space. The spheres we consider are certain homotopy spheres arising as quotients of special $S^3$-$S^3$ bundles introduced by Speran\c{c}a. These homotopy spheres are determined by an orthogonal representation $\rho$ of $S^3$ and an $S^3$-conjugation-equivariant map $\beta$, and are denoted $\Sigma_{\rho,\beta}$. By recent constructions, such spheres admit Riemannian metrics of strictly positive sectional curvature. In this paper we consider the spheres $\Sigma_{\rho_b,\beta}$ of dimension $4b+3$, $b\geq2$, where $\rho_b$ is the representation of $S^3$ on $\mathbb{R}^3\oplus\mathbb{H}^b$ by left multiplication on the second summand. For these spheres, we show that the Eells-Kuiper invariant is the product of an integer $k(\beta)$ depending only on $\beta$ and a rational number $r_b$ given by one coefficient of a power series built from the $\hat A$- and $L$-genera. We then show that, for every $b\geq2$ with $b\not\equiv63\pmod{64}$, there is a map $\beta$ for which the Eells-Kuiper invariant does not vanish, and thus the corresponding sphere is exotic. A computer calculation also gives the dimensions $n\equiv255\pmod{256}$ below $2^{16}$.

The Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory, or more specifically, with the aid of Selberg's theorem from covering theory. In this paper, we point out that the argument in that previous study has already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Ma\~n\'e equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H, which has recently been settled by Przytycki for unicritical polynomials. As an application, we sharpen the quantitative results on the value distribution of the (first) derivatives of iterated polynomials.

We determine the global minimizers of the three-dimensional Yukawa lattice energy at every fixed density. At unit covolume, BCC is the unique minimizer below a transition parameter $\kappa_*\approx1.7179218684699$, FCC is the unique minimizer above it, and exactly these two lattice classes minimize at the transition. Mathematically, denote the Yukawa potential and the energy per particle under this potential by $$v_\kappa(r)=\frac{e^{-\kappa r}}r,\qquad E_\kappa(L)=\frac12\sum_{v\in L\setminus\{0\}} \frac{e^{-\kappa|v|}}{|v|}.$$ Then, $$\arg\min_{|L|=1}E_\kappa(L)= \begin{cases} \hbox{BCC lattice},&0<\kappa<\kappa_*, \hbox{BCC or FCC lattice},&\kappa=\kappa_*, \hbox{FCC lattice},&\kappa>\kappa_*. \end{cases}$$ As a consequence, the zero-screening comparison recovers the unique BCC minimum for the renormalized Coulomb energy.

We present a numerical method for recovering rigid polygonal obstacles in two dimensions, based on the diffraction-based inverse scattering theory of Bao, Chen, Lu, and Xiong. A single pulse is emitted from a point source, and the resulting wave field is measured on a receiver curve. After isolating the diffracted signals, we recover the visible vertices from arrival times and the corner orientations and opening angles from amplitudes. The use of amplitude information permits local reconstruction in the reflection shadow, where reflected waves do not reach the receiver. The amplitudes also determine the direction of a shadowed edge adjacent to a visible vertex, enabling local non-line-of-sight imaging. Numerical experiments for a convex corner, a concave corner, a triangle, a pentagon and a star-shaped obstacle confirm that the visible portion of the boundary can be recovered stably from diffracted signals. A further convex-corner test with additive noise at level $0.2\%$ on the single arc $L_1$ confirms that the tip and opening remain recoverable.

To solve a large class of noncompact problems ubiquitous in random functional analysis and its applications, most of compactness-based fundamental theorems established for normed spaces or locally convex spaces are required to be extended for random normed modules and random locally convex modules in an appropriate way of weakening compactness. To achieve this goal, in this paper we first develop stable set theory by introducing some new set-theoretic tools, then propose a class of topological spaces with a stable structure, called stable topological spaces, and in particular propose and systematically develop the theory of stably compact spaces, which is a natural generalization of that of compact spaces. The resulting theory is of independent interest from the perspective of general topology since its investigation requires an analysis of stratification structures coordinated with partitions of unity in a complete Boolean algebra. Finally, we show the power of the theory of stable compactness by establishing random Banach-Alaoglu, random James and random Banach-Bourbaki-Kakutani-\v{S}mulian theorems.

The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, $\text{BCH}(2,m)$, were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders $k$ has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of $\rho_2(\text{BCH}(2,m))$ and $\rho_3(\text{BCH}(2,m))$, and we establish a new lower bound for $\rho_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le \rho_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$.

The optimal control of a discrete Schr\"odinger bridge is a discrete gradient, a difference of vertex potentials, but recent models such as MadSBM instead learn a free control on the edges by cross-entropy. We prove that this leaves the action uncontrolled: cross-entropy identifies the control only up to a state-dependent shift (a gauge), and the action is $\operatorname{KL}(r\|p)$ plus a gauge term the loss never constrains. We propose GDSBM, which writes the control as a difference of vertex potentials and adds the action to the loss; pinning the gauge costs no expressiveness and makes the action a property of the model, so the penalty acts on the model and not on the gauge. On peptide generation our MadSBM reimplementation has a free-gauge action of order $10^{8}$ for a gauge-fixed value of $6.5$, against $1.5$ for GDSBM, with no quality loss that three seeds can detect. The gradient form also fits guidance, which becomes the addition of a potential: guided toward a net charge of $+3$ at pH~7, GDSBM is closer than MadSBM, over three seeds, to its exact tempered target in total variation at every guidance strength (at $\beta=1$, $81\%$ of samples in the target window against $74\%$, for a target of $97\%$); we claim no statistical significance.

In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degrees $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions.

In this article, we consider the Israel-Stewart equations of relativistic viscous fluid dynamics with bulk viscosity. We investigate the evolution of the equations linearized about solutions that satisfy the physical vacuum boundary condition and establish local well-posedness of the corresponding Cauchy problem.

For over two centuries, the esteemed mathematicians Gauss and Fourier conjectured that each real critical point of the rational function $p^{'}(s)/p(s)$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The conjecture remains unresolved. The real zeros of $(p^{'}(s)/p(s))^{'}$ that are real multiple zero of $p^{'}(s)$ are defined as the second kind of zeros of $(p^{'}(s)/p(s))^{'}$. In addition to the second kind of zeros, the other real zeros of $(p^{'}(s)/p(s))^{'}$ are referred to as the first kind of zeros of $(p^{'}(s)/p(s))^{'}$. We demonstrate that the first kind of zeros of $(p^{'}(s)/p(s))^{'}$ emerge from the intersection of root loci emitted from non-real zeros of $p(s)$, thereby providing a comprehensive geometric interpretation of their relationship. Moreover, each first kind of zero of $(p^{'}(s)/p(s))^{'}$ is connected via root loci to a pair of conjugate non-real zeros of $p(s)$. This establishes the precise geometric relationship conjectured by Gauss and Fourier, fully resolving their original problem for first-kind zeros and completing a foundational chapter in polynomial theory. We have proven Gauss-Fourier conjecture about the first kind of zeros.

We construct a closed surface of genus $2$ with nonpositive Gaussian curvature and a complete nonperiodic flat geodesic asymptotic to a flat closed geodesic. The metric is of class $C^{3,1/3}$ and its curvature is of class $C^{1,1/3}$, but not $C^2$. Thus the periodic-asymptote obstruction for smooth surfaces does not extend to $C^1$ Gaussian curvature.