数学前沿

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.

We establish the stability of the optimal non-endpoint Fourier restriction range under smooth flat perturbations of curved-root surfaces in $\mathbb R^3$. Specifically, we consider graphs of $$\phi(x_1,x_2)=(x_2-x_1^m)^n+R(x_1,x_2),$$ where $m,n\ge2$ are integers and $R$ is smooth and flat at the origin. On a sufficiently small neighborhood depending on the perturbation, we prove the local $L^p(\mathbb R^3)\to L^2(S,d\sigma)$ restriction estimate for $1\le p<p_c$, where $$p_c=\frac{q_c}{q_c-1}, \qquad q_c=\max\left\{2+\frac{2mn}{m+1},2n\right\}.$$ If the nonnegative localizing density is positive at the origin, an adapted-box example shows that the estimate fails for $p>p_c$. Thus arbitrary smooth flat perturbations preserve the optimal restriction range away from the critical endpoint. For $n\ge3$, the main local estimate combines planar curve decoupling with a one-dimensional finite-type extension estimate, whose principal scale factors cancel exactly. For $n=2$, we prove a uniform $C^4$ perturbation estimate using a weighted convolution bound and an anisotropic Whitney decomposition. Anisotropic rescaling and summation complete the proof.

For every right-c.e.\ real $p\in(1,2)$, $L^p[0,1]$ has a computable presentation. The proof combines an effective stability modulus for almost isometric embeddings with a Fra\"{\i}ss\'e-style construction in which the exponent varies. In the same range, if $\ell^p$ has a computable presentation, then $p$ is computable. Combining these with the known results gives the computable presentability classification for all separable Lebesgue spaces. We also formulate an effective Fra\"{\i}ss\'e theorem for Banach spaces and apply it to obtain a computable presentation of the Gurarij Banach space.

In this paper we study the convergence of solutions of the Stokes equations in a domain perforated by slender bodies placed on a fixed periodic grid of size $2\epsilon$ as the size of the bodies goes to zero and the number of bodies goes to infinity. We assume that the width $b_\epsilon$ of the bodies vanishes faster than the length $a_\epsilon$ of the particles. We identify the critical scaling regimes for $a_\epsilon$ and $b_\epsilon$ and prove that in this case the system is effectively described by the Brinkman equations.

For the symmetric pair $(\mathrm{GL}_n(\mathbb C),\mathrm{GL}_n(\mathbb R))$, we compute the exact Flicker--Rallis period of a distinguished vector in a cohomological minimal $U(n)$-type, normalized by the Gel'fand--Tsetlin pairing of Hara--Miyazaki--Namikawa. Our main tool is a weight-independent identity between the Whittaker period and a compact $O(n)$-average. We determine the proportionality constant using the archimedean newform formula of Humphries--Jo, and evaluate the compact functional on the normalized distinguished vector. This yields an explicit formula for the Flicker--Rallis period for every cohomological weight.