数学前沿

We construct a smooth Riemannian metric on an arbitrary product of spheres whose geodesic flow is Bernoulli with respect to the Liouville measure.

We give an explicit partition of $\{1,\ldots,537\}$ into six sum-free sets, establishing $S(6)\ge 537$. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds $S(8)\ge 5\,372$, $S(11)\ge 204\,208$, $S(13)\ge 2\,042\,082$, and $R_6(3)\ge 539$. We also report 131 pairwise inequivalent partitions found by the search. We describe their structure and provide the complete partitions in the supplementary material. These partitions provide starting points for future searches and structural studies of Schur partitions.

We establish the central limit theorem (CLT) for Smoluchowski--Kramers approximation of Langevin equations with state-dependent, possibly non-symmetric friction and multiplicative noise. To the best of our knowledge, this is the first such result, even in the case of constant friction and additive noise. The result differs in an essential way from classical averaging CLTs and reveals a new two-scale fluctuation phenomenon. The normalized difference process retains a persistent fast component and is generally not tight in $C([0,T];\mathbb{R}^d)$, thus the usual continuous-path CLT cannot hold. We derive an exact decomposition of the normalized difference into an accumulated slow fluctuation and a rapidly relaxing velocity correction. The auxiliary slow fluctuation converges to a linear diffusion driven by both the original Brownian motion and an additional independent Brownian noise whose effective covariance is characterized explicitly through a quadratic Poisson corrector. For every fixed $t>0$, the fast correction converges conditionally to its local Gaussian equilibrium. Consequently, the fixed-time fluctuation limit is identified as a Gaussian mixture: $$\frac{\bar X_t-X_t^\varepsilon}{\sqrt{\varepsilon}} \Longrightarrow \bar Y_t+N^{1/2}(\bar X_t)\xi,$$ where $\bar X_t$ is the overdamped limit, $\bar Y_t$ is the limiting slow fluctuation, $N$ is the covariance of the terminal velocity correction and solves a Lyapunov equation, and $\xi$ is a standard Gaussian random variable independent of $(\bar X_t,\bar Y_t)$. A quantitative convergence rate is also provided. The result fills the remaining gap in the asymptotic theory of the overdamped approximation at the central limit scale, complementing the existing law of large numbers, moderate deviation and large deviation results.

We study policy evaluation (PE) and policy gradient (PG) for continuous-time stochastic control with Markov regime-switching. The main challenges are the coexistence of diffusion and regime jumps, the coupling of regime-dependent value functions, and the construction of model-free learning rules from the coupled generators. We establish a martingale characterization of PE that incorporates both Brownian and switching martingales, and extend martingale loss and orthogonality methods to this setting. We further derive a coupled PG equation and a trajectory-based gradient representation that, under an action-independent switching generator, avoids explicit knowledge of the system coefficients and transition rates. These results lead to offline and online actor-critic algorithms. We also establish convergence and study the finite-grid approximation required for implementation. Numerical experiments for a regime-switching linear-quadratic problem illustrate the proposed methods.

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the decidability of Diophantine equations and has been answered negatively by Davis, Putnam, Robinson and Matiyasevich. It is natural to ask for which subclasses of Diophantine equations H10 remains undecidable. Such subclasses can be defined in terms of universal pairs: bounds on the number of variables $\nu$ and degree $\delta$ such that all Diophantine equations can be rewritten in at most this complexity. Our work develops explicit universal pairs $(\nu, \delta)$ for integer unknowns, focused on reducing the number of unknowns to $\nu = 11$. This achieves new bounds that cannot be obtained by naive translations from known results over $\mathbb{N}$. In parallel, we have conducted a formal verification of our results using the proof assistant Isabelle. While formal proof verification has traditionally been applied a posteriori to known results, this project integrates formalization into the discovery and development process. In a final section, we describe key insights gained from this approach and its implications for mathematical practice. Our work contributes both to the study of Diophantine equations and to the broader question of how mathematics is conducted in the 21st century.

We classify locally symmetric hypersurfaces, understood in the intrinsic sense, in Riemannian globally symmetric spaces.

A hard theorem of Givant classifies the finitary algebraic theories for which every algebra is free. From it, one easily deduces that all such theories are minimal (adding any further equations makes them trivial) and have a well-defined notion of dimension (free algebras on sets of different cardinality are never isomorphic). We prove both these properties directly, without the aid of Givant's theorem.

We prove a triple quantum Pieri formula: the product of a quantum double Schubert polynomial $\widetilde{\mathfrak{S}}_u(x;y)$ by a quantum factorial elementary polynomial $\widetilde{E}_{p,k}(x;z)$ in a second set of variables $z$ is a sum over chains in the quantum Bruhat graph, indexed by a quantum analogue of Sottile's relation, whose coefficients are monomials in the quantum parameters times polynomials in the differences $y_i-z_j$ with nonnegative integer coefficients, and which is Graham-positive at $z=y$. At $y=z=0$ it is the quantum Pieri rule of Ciocan-Fontanine and Postnikov; at $z=y$ it is an equivariant quantum Pieri rule for the complete flag variety, indexed by chains in the quantum Bruhat graph rather than, as in the rule of Huang and Li, by the data of a reduction to classical cohomology, and proved directly. A doubled Pieri formula, positive in the same senses, treats multiplication by the quantization of a product of two elementary polynomials in the same variables, which is not the product of the quantizations: it is two iterated Pieri steps with the incompatible pairs of chains omitted. More generally, the triple quantum Schubert structure constants $c_{u,v}^{w,d}(y;z)$, the coefficients of $\widetilde{\mathfrak{S}}_u(x;y)\widetilde{\mathfrak{S}}_v(x;z)$ in the basis $\{\widetilde{\mathfrak{S}}_w(x;y)\}$, are polynomials in the $y_i-z_j$ with nonnegative integer coefficients; the proof of this appears as an appendix. We extend the Peterson-Woodward comparison formula and the quantum-to-classical reduction of Leung-Li and Huang-Li to the triple constants, and we describe and prove correct the algorithms of schubmult, which compute products of arbitrary quantum double Schubert polynomials well beyond the reach of previous methods.

We address a short-wave asymptotic for one class of quasi-linear second-order PDE systems involving the cross-diffusion described by the so-called Patlak-Keller-Segel law. It is common to employ these equations for modeling the predator-prey community with the prey-taxis, which means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a cue produced by the prey. Additionally, the production of the cue is assumed to be sensitive to the intensity of an external field, which is independent of the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete formal asymptotic expansion for the short-wave solutions (modulo solving a chain of the so-called slow systems) with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions. Further, we apply the short-wave asymptotic to studying the stability or instability induced by the external signal following Kapitza's theory for the upside-down pendulum. Applying the general results to some special classes of external signals, we get examples of suppressing the taxical transport, examples of the robustness of the species equilibrium to the signal up to a very strong stabilization or, oppositely, destabilization, and something like blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium.

We construct and describe the moduli stack $\mathrm{Isog}$ of isogenies between oriented elliptic curves. We show that $\mathrm{Isog}$ is a spectral Deligne-Mumford stack and a relative spectral algebraic space over $\mathcal{M}_{\mathrm{ell}}^{\mathrm{or}}$, and give a fiber product decomposition of $\mathrm{Isog}$ corresponding to a connected-\'etale factorization on oriented elliptic curves. In the process of proving these results, we construct a theory of duality and quotients for spectral abelian varieties, prove a higher-algebraic version of Nakayama's lemma, and lift the rigidity theorem for morphisms of elliptic curves to the setting of spectral algebraic geometry.

We develop a unified framework for high-order, fully discrete entropy-stable discontinuous Galerkin (DG) methods with compact explicit single-step time discretizations. At the end of each time step, the preliminary DG polynomial is scaled toward its cell average to enforce a strong cell entropy inequality while preserving conservation. This correction is feasible if and only if the updated cell average satisfies a weak cell entropy inequality, which standard third-order strong-stability-preserving Runge--Kutta DG updates can violate at arbitrarily small positive CFL numbers. For compact single-step schemes, including compact Runge--Kutta, Lax--Wendroff, and ADER-DG methods, we derive an explicit interval of feasible numerical entropy fluxes at each interface and prove that it is nonempty under a CFL condition. Projecting a compatible high-order target flux onto this interval guarantees the weak cell entropy inequality and preserves single-step high-order accuracy on smooth solutions, including near nondegenerate extrema. The limiter also enforces multiple entropy inequalities simultaneously. We also establish a Lax--Wendroff-type theorem. Numerical experiments for scalar conservation laws and the compressible Euler equations demonstrate high-order accuracy, multi-entropy control, and robustness for strong shocks.

We derive exact formulas for the diagonal distribution of half-space geometric last-passage percolation with a two-parameter family of stationary initial conditions throughout the phase diagram. Unlike previous results, our formulas remove the random shift at the origin. Under critical scaling, we obtain new formulas for the half-space stationary KPZ fixed-point distribution at $x=0$ in the high-density and maximal-current phases. In the low-density phase, we derive a new representation and show that it matches the corresponding formula from the stationary log-gamma polymer.

We study the fixed-particle limit of empirical measures for a unified self-dual family of conservative particle systems on $\mathbb{Z}^d$: independent random walkers, the symmetric exclusion process, and the symmetric inclusion process. For a fixed number of particles $k$, under diffusive space-time scaling parametrised by $n$, we prove that as $n\to\infty$ the normalised empirical measure converges in $D([0,T],\mathcal{M}_1(\mathbb{R}^d))$ to an atomic empirical measure carried by $k$ independent diffusions, with constant covariance determined by the microscopic jump kernel. Equivalently, the limit is the unique atomic solution of the Dean-Kawasaki martingale problem with mass quantisation $1/k$. In particular, exclusion and inclusion have the same fixed-$k$ atomic limit as independent walkers: the interaction contributes only through microscopic near-collisions, whose rescaled occupation time vanishes. The proof combines self-duality, consistency-based reduction to two particles, and Green-function estimates for finite perturbations of symmetric random walks.

We consider the quantization of linearized Einstein equations on 4-dimensional asymptotically flat spacetimes. We prove the existence of fully gauge invariant Hadamard states in the harmonic gauge for any Einstein metric asymptotic to the Minkowski metric in a conic neighborhood of spatial infinity.

In 1999, Arakawa and Kaneko introduced a zeta function connecting multiple zeta values with poly-Bernoulli numbers through its special values. Log-sine integrals are also known to be related to zeta values at positive integers. In this article, we study the values of a shifted log-sine integral at negative integers and establish a connection with anti-diagonal sums of poly-Bernoulli numbers. This gives a refinement of Stephan's observation on the central binomial series.

Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q >= 3. For q = p^s and n >= 1 put P_{q,n} = (theta^{q^n} - theta)^{q-2}. We show that the directed graph of the matrix M(P_{q,n},1,k) of Grishkov-Logachev has a completely explicit set of circuits: they are pairwise disjoint, none of them meets an entry involving t, and they are in bijection with the orbits of E^n under cyclic rotation, where E = {l in [0,q-2] : p does not divide l+1}. Consequently L(C_{P_{q,n}},t,T) does not depend on t, factors as a product of one term 1 - wt(nu) T^{per(nu)} per orbit, and the analytic rank equals the sum of p^{v_p(per(nu))} over the orbits of weight 1. When n is a power of p this equals (q/p)^n (p-1)^{n-1}, which is unbounded. The argument also answers questions 3.6.1-3.6.4 of the survey for this family. We then compute the L-function for arbitrary nonzero coefficients on supports of the same shape: taking all coefficients equal to -1 multiplies the rank by p-1, and the full support gives twists of degree (q-2)q^n and analytic rank (q-1)^n, the largest rank attainable on such supports. The same method applies to the tensor powers of the Carlitz module: for 1 <= e <= q-2 we obtain twists of the e-th tensor power of analytic rank (q-e)^n, and by a formula of Grishkov and Logachev the same holds for every exponent e = c q^j with 1 <= c <= q-2, with rank (q-c)^n; in particular these analytic ranks are unbounded as well.

We give a determinant proof that every nonzero real period of a nonsingular elliptic curve over $\mathbb Q$ with complex multiplication, measured with a specified nonzero rational invariant differential, has irrationality exponent two. The proof uses an integral origin chart, matched local lengths, an actual compactification and weighted packet map, and arithmetic and analytic estimates for the same selected minor on a rectangular sublattice of the full CM period lattice. For the integral-defined lemniscate period $\Omega=2\int_0^1(1-t^4)^{-1/2}\,dt$, we also give the uniform bound $\mu(\Omega^n)=O(n\log(2n))$ for integers $n\geq1$. Its new arithmetic ingredient clears integration denominators prime by prime while retaining the coefficient order; a one-root norm argument and an analytic estimate at algebraic centers complete the power proof. The two arguments are presented with their different charts and exact dependencies. Lean verifies the new CM determinant argument and rational normalization relative to explicit classical uniformization, CM-lifting and theta inputs; the lemniscate-power endpoint is also verified. The formalization record states the exact classical boundary and theorem references.

We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

We prove that every hereditary Cayley permutation class with finitely many simple members has a finite basis and an algebraic generating function. We first show that every simple Cayley permutation of length at least two contains a simple pattern obtained by deleting one or two letters, and that the only exceptional Cayley permutations are the simple parallel alternations and their symmetries. We then characterize substitution-closed classes by the simplicity of their basis elements and show that classes with finitely many simple members are partially well ordered. Given the simple members and the basis of such a class, we construct a finite system of functional equations and compute an annihilating polynomial.