数学前沿

For an integer $q\ge5$, let $A_q(n,\{2,4\})$ be the largest cardinality of a subset of a $q$-ary Hamming space of length $n$ in which every two distinct words have distance $2$ or $4$. Set $$N(q)=\left\lceil\max\left\{2q+1, \frac{17+\sqrt{8q^2+96q-95}}2\right\}\right\rceil.$$ We prove that $A_q(n,\{2,4\})=1+\binom n2$ for every $n\ge N(q)$. In particular, $N(q)=2q+1$ for $q\ge26$. An explicit code of size $q(n-2)$ shows that the least eventual threshold $N_*(q)$ satisfies $N_*(q)\ge2q-1$; consequently $2q-1\le N_*(q)\le2q+1$ for every $q\ge26$. The argument applies to arbitrary integer alphabet sizes, without a linearity assumption. Its main ingredients are synchronization of symbols on intersecting supports, weighted bounds for $2$-intersecting families of four-sets, and a joint estimate for the weight-two and weight-four layers in a common-three-core configuration. No computational verification is required.

These lecture notes are based on lecture series by the authors for the Summer School 2025 on ``Physical mathematics of quantum field theory" at the University of Massachusetts Amherst organized by Owen Gwilliam, Chris Elliott, and Matteo Lotito. The lectures centred around algebraic structures arising from twists of supersymmetric quantum field theories, correlation functions of topologically twisted supersymmetric gauge theories, and their mathematical counterparts as intersection-theoretic invariants associated to moduli spaces of instantons, semistable sheaves, or Higgs sheaves on four-manifolds or complex projective surfaces.

This work proves a theorem regarding injectivity of continuous flows that are defined via discrete vector fields on triangle meshes. In similar fashion to Tutte's embedding, the theorem shows a connection between harmonicity and injectivity. Specifically, a discrete harmonic function is computed over a spherical mesh, and its $90^\circ$-rotated gradients are taken as the vector field. We show that this leads to an injective flow over the mesh, i.e., when two points are advected along the vector field, their trajectories are guaranteed to not intersect. The core insight behind this theorem is that the characterization of the Poincar\'e-Hopf indices of discrete harmonic 1-forms is paralleled in the combinatorial rules for injective vector fields. Similarly to Tutte's embedding, the injective vector field can be computed by simply solving a sparse linear system. Furthermore, it is differentiably dependent on the entries of the Laplacian, enabling unconstrained gradient-based optimization of the vector field while preserving injectivity. The advection itself can be computed by a quick and accurate algorithm. We experimentally validate the injectivity of the results, as well as show examples of differentiable optimization, such as computing an injective approximation to a given non-injective vector field, and fitting an injective vector field to satisfy constraints.

This manuscript studies a coupled system of nonlinear parabolic and hyperbolic equations that models thermoviscoelastic materials with explicit internal stress relaxation, $$\begin{cases} \tau u_{ttt} + u_{tt} = (D(\Theta)u_{xt})_x + (C(\Theta)u_x)_x, \\ \Theta_t = \Theta_{xx} + |E(\Theta)u_{xt} + F(\Theta)g_t|^2, \\ \tau g_t + g = D(\Theta)u_{xt} + C(\Theta)u_x. \end{cases}$$ The system arises from a stress-strain constitutive relation of Zener type, where the internal variable $g$ encodes mechanical memory effects. A key feature is the explicit coupling of stress relaxation to the heat equation through the dissipation term, which differs from standard approaches that eliminate $g$ via algebraic substitution. Under the assumptions that $\tau>0$ and that $D,C\in C^2([0,\infty))$ are positive and $E,F\in C^1([0,\infty))$, and for suitably regular initial data, we establish local existence and uniqueness of strong solutions. Under additionally strengthened regularity requirements on the initial data and on $D,C$, this solution is moreover shown to be classical. These results extend a corresponding local strong and classical solvability theory developed for the underlying system without the stress variable $g$ to the present, stress-augmented setting.

We consider a bounded operator on a separable complex Hilbert space, reached only through its action on vectors and that of its adjoint, and split it along a subspace of finite dimension into a compression, a far block acting on the complement, and the two couplings between them. Given upper bounds on the norms of the couplings and of the far block, we show that outside a disc a pseudospectrum of the operator lies in a pseudospectrum of the compression, with an explicit loss in the level; this reverses the classical inclusion of the Ritz values in a pseudospectrum of the operator. The same bounds, with the smallest singular value of the shifted compression, bound the resolvent of the operator, count its eigenvalues with algebraic multiplicity inside every contour outside the disc on which a computed inequality holds, and enclose its spectral projector; no normality, self-adjointness or positivity is used.

We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most $\sigma>0$. For intervals of radius $(n-\gamma_n)/\sigma$, where $0\le\gamma_n<n$ and $\gamma_n=o(n)$, one margin sequence gives this limit for every convex function of polynomial growth if and only if $\gamma_n/[n^{1/3}(\log n)^{2/3}]\to\infty$. When this condition fails, a fixed uniformly strictly convex entire function has approximation errors with unbounded limsup. We apply the limit theorem to prove existence of the convex Bernstein constant for $|x|^\lambda$ for every $\lambda\ge1$, and identify it with an attained entire-function minimum. We also give a sufficient margin for growth bounded by $\exp(a|x|^\alpha)$, $0<\alpha<1$. For arbitrary subexponential growth on the real line, each target with finite entire-function error admits a suitable margin, whereas no single margin is sufficient for the whole class. Further results concern attainment, right continuity with respect to the exponential type, weighted growth conditions, and the endpoints of fixed-fraction approximation intervals.

Over any number field with sufficiently many roots of unity, we construct homogeneous spaces of $\mathbf{SL}_n$ with finite geometric stabilizers of nilpotency class $2$, whose unramified algebraic Brauer group consists only of constant classes, and which has no rational point but has local points in every completion of the ground field. This provides the first known examples of failures of the Hasse principle for homogeneous spaces of connected linear algebraic groups that are detected only by transcendental Brauer classes. The construction builds on the failure of a global--local principle for cup-product decomposition, based on a previous idea of Borovoi and Kunyavskii.

For a closed embedded minimal hypersurface in a unit sphere, we show that the existence of a uniform vanishing angle in Lawlor's criterion suffices for its cone to minimize area. The normal-wedge condition follows from Howard's rolling theorem. We derive explicit curvature bounds and apply them to the reported Hsiang-Sterling curvature estimates for topologically trivial cones. The resulting area-minimizing cones give an affirmative answer to the existence question raised by Ilmanen and White (Camb. J. Math., 2015).

Let $u$ be the torsion function of a bounded domain $\Omega$ in $\mathbb{R}^n$ for the Ornstein--Uhlenbeck operator, that is, the solution of $-\Delta u + x\cdot\nabla u = 1$ in $\Omega$ with $u=0$ on the boundary. We prove that for every $n\ge2$ there are bounded, centrally symmetric domains whose torsion function is not concave. These domains have boundary with a flat piece and can be smoothed so that the boundary is $C^\infty$, strictly convex with principal curvatures bounded below by a positive constant, while keeping the torsion function not concave. In contrast, the torsion function for every ellipse centered at the origin is concave, and for every ball centered at the origin we prove the sharp bound $D^2u\leq-\frac{1}{n}I$. The ellipse result follows by deformation from the disk, using a boundary flux estimate and third-order boundary identities to exclude a first zero eigenvalue of the Hessian.

We introduce a novel method for verifying Lean formalizations that goes beyond LLM-judged faithfulness. Unlike most verification methods, which rely on LLMs or trained models, our method combines LLM-based semantic review with mechanical numerical and vacuity checks, providing checkable feedback and detecting errors missed by LLM judges. We apply this method, along with our Extract-Verify-Proof autoformalization pipeline, to construct the Formalized Library of Mathematical Functions (FLMF), a partial formalization of the NIST Digital Library of Mathematical Functions (DLMF). On three labeled datasets, our method achieves the lowest accepted-error rate among the evaluated methods, measured as the fraction of incorrect statements accepted by the verification method. Expert review of the results also identifies translation and source defects in existing datasets. Code available at https://github.com/annamei1/FLMF-Ch7ErrFunc-Release.git.

Let $G$ be a quasi-split $p$-adic group, defined over a non-archimedean local field of residue cardinality $q$, and $R$ be a $\mathbb{Z}[\sqrt{q}^{-1},\mu_{4p^\infty}]$-algebra. We generalize the constructions of Shahidi local coefficients and Langlands-Shahidi gamma factors to the setting of co-Whittaker $R$-representations. For classical $p$-adic groups, we prove there is a unique integral morphism of endomorphism rings of Gelfand--Graev representations, interpolating the (standard) local generic transfer to a general linear group. This yields a local transfer morphism in families, which allows us to deduce properties of the local $\ell$-adic generic transfer and congruences modulo primes $\ell\neq p$.

We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension $n\ge3$. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to $L^2$ in time. A uniform curvature bound in the evolving $L^p$ norm, for one fixed $p>n$, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial $L^p$ curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern--Ricci form gives exponential convergence on Chern--Ricci-flat Hermitian and positive K\"ahler--Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.

To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman--Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate--Shafarevich twists. We give a systematic interpretation of known results about Tate--Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda--Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to $U^2\oplus\langle-2\rangle^2$ to a smooth cubic surface with a choice of a Schl\"afli double-six. Finally, we construct a noncompact six-manifold with an abelian fibration associated to a cubic surface with a Schl\"afli double-six, and give its interpretation in terms of the K3 surfaces we introduced.

We extend Chern-Weil theory to Lie groupoids carrying a Cartan connection. For a principal $(\Sigma, \omega)$-bundle $(P, {\theta})$ over $X$ we construct a map of differential graded algebras from an $\omega$-basic subcomplex of the Weil algebra of the Lie algebroid of $\Sigma$ to $\Omega^\bullet(X)$, whose map in cohomology does not depend on $\theta$. It recovers the classical Chern-Weil map for Lie groups, the Cartan model for action groupoids, and the characteristic classes of Cartan geometries for gauge groupoids.

Consider the well-known Langevin diffusion on $\mathbb{R}^d$ $$\mathrm{d} X_t = -\nabla U(X_t)\,\mathrm{d} t + \sqrt{2}\mathrm{d} B_t,$$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-\eta \nabla U(X_k)+\sqrt{2\eta }\xi_{k+1},$$ where $\eta$ is the step size. Under mild conditions, the Langevin diffusion admits $\pi(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$W_n = \eta^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,\pi(\mathrm{d} x) \right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,\gamma)$ via the exchange pair approach, where $W$ is any random vector of interest and $\gamma$ is a $d$-dimensional standard normal random vector.

For a graph $G$, let $\text{cg}(G)$ be the maximum $k\in \mathbb N$ for which $G$ contains a $k$-vertex cograph as an induced subgraph. The Erd\H{o}s-Hajnal conjecture asserts that for every proper hereditary graph class $\mathcal{G}$, there exists $\gamma\in(0,1]$ such that every $n$-vertex graph $G\in\mathcal{G}$ satisfies $\text{cg}(G)\geq n^\gamma$. This is still open; however, for some hereditary classes, one may guarantee much stronger bounds. For instance, every chordal graph $G$ on $n\geq 2$ vertices satisfies $\text{cg}(G)\geq n/(2\log n)$. Paul Seymour asked whether this bound is optimal up to a constant factor. We answer this affirmatively, even for interval graphs: For every integer $n\geq 2$, we construct an $n$-vertex interval graph $G$ with $\text{cg}(G)<64n/\log n$.

This work constructs a non-trivial solution to the ill-defined two-dimensional Kardar-Parisi-Zhang (KPZ) equation at criticality: a distribution-valued stochastic process on the plane, which we call the Critical 2D KPZ Process. It arises as the limit in law of centred log partition functions of two-dimensional Gaussian directed polymers at criticality, viewed as discretised KPZ solutions. It is also a measurable function of the Critical 2D Stochastic Heat Flow (SHF) via a renormalised Cole-Hopf transformation, obtained as the limit of the centred logarithm of the spatially mollified SHF as the smoothing is removed. The Critical 2D KPZ process is non-Gaussian and its covariance displays a log-log blow-up near the diagonal. Gaussian fluctuations emerge when averaging on shrinking spatial scales. The results presented here were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6); see Appendix C for more details. The proof builds on key recent and older advances in our field, combined in novel ways with substantial AI input. We view this manuscript as a "community project" that represents more than a decade's collective effort. We have worked on making the presentation as accessible as possible, highlighting the main ideas and their relations to the literature.

We introduce polytopal substitutions, a recursive procedure for constructing polyhedral approximations of self-similar topological spaces. For a class of substitutions called sub-hyperbolic, we show that the limit of these approximations is naturally homeomorphic to the limit space of the associated iterated monodromy group. We establish a relationship between the $\ell^p$-contraction coefficients of sub-hyperbolic substitutions and the conformal dimension of their limit spaces. In particular, we characterize the conformal dimension in terms of asymptotic moduli and relate these moduli to $p$-energies of graphs.

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal sum of Betti numbers. We prove that every full subcomplex of them is homotopy equivalent either to a point or to a sphere. Moreover, we prove that such kind of simplicial complexes coincide with those having minimal Taylor resolutions.

PDECheck audits learned PDE simulators through callable propagators or saved predictions. It evaluates relative $L^2$ error, temporal composition, finite-difference generators, and equation-specific physical residuals, retaining horizons and missing prerequisites. Analytic and manufactured defects verify the kernels. A heat-equation sweep quantifies generator sensitivity to timestep, frequency, grid, and precision. A three-equation, five-seed FNO study compares supervised, PINO-style, and composition-regularized training. An accuracy-matched advection comparison has state errors below 0.3% and a semigroup-error ratio of 1.48 (95% interval 1.34-1.62). The composition penalty lowers semigroup error but increases composed-state and longer-rollout prediction error. A time-reparameterization check produces identical low-error endpoints with different compositions. Applications to five external equation families exercise the readers and physical contracts on coarse grids. The release includes reproducibility artifacts and measured CPU/CUDA costs.