数学前沿

A Higgs bundle on a smooth complex projective variety is curve semistable if its pullback to every smooth projective curve is semistable. A 2008 conjecture asserts that this is equivalent to slope semistability for some polarization plus vanishing discriminant. One direction is known, the converse holds in rank two, and Huang recently gave a rank-four counterexample. We prove the conjecture for rank three nilpotent Higgs bundles in any dimension. We treat separately the two possible cases: if the Higgs field $\phi$ is nilpotent of order two, there is a divisorial component of a Higgs Grassmannian where nefness of the normalized tautological class forces the discriminant to vanish; otherwise, $\phi$ is regular nilpotent, and then the canonical kernel flag, commutativity and Bogomolov's lemma on line subbundles of the cotangent bundle complete the proof.

Following the celebrated breakthrough in B\'enard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on $\mathbb{R}^d$ under uniform $C^{1+\alpha}$ maps. For this, we also study almost random walks on $\mathrm{SL}(d+1,\mathbb{R})/\mathrm{SL}(d+1,\mathbb{Z})$ and prove their effective equidistribution.

The article is devoted to the parametric solution of some Diophantine equations with two variables. In essence, we find an additional parametric relationship between the solution of the original equation and the simpler linear equation ax - by = 1. The method of hidden parameters helps to obtain a parametric representation of integer solutions for a number of problems in number theory. The method consists in that, first, instead of each quantity included in the original equation, we introduce an additional number of parameters using a linear or non-linear function, which allows us to find both connections between the parameters and some parametric solution of the desired equation. Then we reset (nullify) the introduced linear or non-linear functional dependence (hidden parameters), which in the original formulation of the problem would mean just changing the letter designation of the parameters. However, as a result of these actions, a simplified parametric solution is obtained and the found relationship between the parameters remains. In this paper, the method is discussed in detail using the example of parametric solving such Diophantine equation as the Pell equation. The method of such parameterization is also applied to problems related to Pell's equation. The methods of parameterization are discussed for the Delaunay equation, the Nagell equation and similar equations of higher powers.

We prove, by a straight construction, that the automorphism group of the measure algebra and the subgroup of the measure preserving ones cannot be isomorphic to the trivial automorphisms of P(N)/fin.

The branch-and-cut paradigm has led to effective solvers for a wide range of mathematical optimization problems. Despite significant theoretical work on pure branching and pure cutting, less is known about their interaction. We study an extended abstract branch-and-cut model to investigate tradeoffs between branching and cutting, including the time to generate cuts. We prove that, under symmetric branching, some minimal-time tree has all cuts at the root, before the first branching node, and that there is an asymmetric setting where some minimal-time tree has no internal cuts, i.e., cuts both preceded and followed by branching. For other settings, although a root-cut tree may be suboptimal and differ substantially in structure from a minimal-time tree, it nevertheless provides a reliable approximation whose tree time does not deviate arbitrarily from the optimum. For constant cut strength and bounded time functions, we show that the best root-cut tree is asymptotically optimal as the target bound grows. In addition, under a single-cut-per-path restriction, we characterize when some minimal-time tree has an internal cut. Lastly, simulations indicate that the best root-cut tree is much closer to a minimal-time tree than our worst-case approximation guarantee suggests.

In this paper, we consider the magic positivity of the Ehrhart polynomial of dilated base polytope of minimal matroid of rank $k$ dimension $n$. We prove that for $\displaystyle 1\leq k<\lfloor\frac{n}{2}\rfloor$, the $n-k-1$ dilate of the Ehrhart polynomial is not magic positive. We also show that for sufficiently large $n$, the $n-k$ dilate of the Ehrhart polynomial is, in general, magic positive. This gives rise to a lower bound of the $m-$index of the Ehrhart polynomial.

We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.

We determine the Kazhdan--Lusztig polynomial $P_{\mathrm{id},w}(q)$ for permutations $w$ such that $P_{\mathrm{id},w}(1)=3$ and the Schubert variety $X_w$ has irreducible singular locus. Let $h=h(w)$ be the minimum height $w(a)-w(d)$ of a $3412$-occurrence in $w$ at positions $a<b<c<d$, with $h=1$ when no such occurrence exists. We show that $P_{\mathrm{id},w}(q)=1+q^h+q^{h+1}$ if exactly one of the patterns $3412$ and $4231$ occurs, and $P_{\mathrm{id},w}(q)=1+q+q^h$ if both patterns occur. In the latter case, $h>1$. This proves the irreducible singular locus case of a 2009 conjecture of Billey and Weed.

We study the one-dimensional Vlasov--Poisson system with a repulsive harmonic potential. For repulsive self-interactions, we prove modified scattering for initial data of arbitrary size, assuming only velocity weights and one derivative; no smallness, symmetry, compact support, or spatial moments are required. In the lens variables of \cite{HK26}, the self-consistent force carries a nonintegrable time weight. We remove it by a momentum correction that solves a forced Burgers equation; the repulsive sign makes this correction monotone, which gives bounds for large data. We also construct modified wave operators for both signs of the interaction. Together with \cite{VR24,BVR25,HK26}, this gives a scattering theory in every dimension.

We study variants of a stochastic SEIRS-SEID compartmental model incorporating presymptomatic infectiousness and disease-related mortality, with independent compartmental sojourn times drawn from arbitrary distributions, in particular Weibull distributions. We consider a large population of agents that occupy the states (compartments) S (susceptible to infection), E (exposed: infected but asymptomatic during the latency period and already infectious, including diseases with presymptomatic infections), I (infected and infectious, with symptoms), R (recovered, immune, and non-infectious), or D (dead). We consider (i) cyclic transmission pathways, where compartmental sojourn times are finite and immunity is temporary, and (ii) non-cyclic pathways, where immunity is permanent. In the absence of mortality, cyclic transmission pathways support an endemic equilibrium above the spreading threshold which is either globally stable or oscillatory (Hopf) unstable, whereas for non-cyclic pathways a stationary disease-free state is emerging. From a stochastic perspective, we elucidate the connection between compartmental sojourn-time distributions and the appearance of fractional operators in the evolution equations: fat-tailed compartment waiting-time distributions necessarily lead to fractional derivatives. For cyclic transition pathways we obtain an explicit expression for the expected number of infection cycles that an agent survives. Our model is validated by a combined Monte Carlo -- Runge -- Kutta simulation scheme and random-walk simulations on Barabasi-Albert networks. It has cross-disciplinary applications beyond epidemiology, including wildfire spreading, certain classes of chemical reactions, population dynamics, finance, economics, and biology.

We design and analyse an arbitrary-order augmented Lagrangian Discrete de Rham polytopal scheme for contact mechanics in fractured media. The fractures are represented as objects of codimension one in the matrix, a linear elastic medium, and a Tresca friction model is adopted when contact between fracture walls occurs. Both the normal Signorini-type/traction constraint and the Tresca friction law are recast as projection identities, and are combined into a single augmented Lagrangian bilinear form. The displacement is discretised by a DDR space of arbitrary polynomial degree $k\ge 0,$ enriched on the fracture faces by bubble degrees of freedom that ensure stability via the existence of a discrete lifting between the Lagrange multiplier space and the discrete jump of the displacement along the fracture. We establish an abstract error estimate yielding optimal order convergence for the displacement and the Lagrange multiplier variables. Extensive numerical experiments confirm the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.

We compare two cohomological constructions computing special values of partial zeta functions of totally real number fields at nonpositive integers: the Shintani cocycles arising from polyhedral cone decompositions, and the Eisenstein classes obtained from the topological polylogarithm. In particular we prove that the specialization of the topological polylogarithm at a nonzero torsion point coincides in cohomology with the Shintani class up to an explicit sign and error term, and obtain Siegel and Klingen's classical rationality Theorem.

Let $X^t$ be a volume-preserving Anosov flow of class $C^7$ on a closed $3$-manifold. We consider a sequence of periodic orbits which shadow a homoclinic loop of a fixed periodic orbit $\gamma$, and derive higher-order asymptotic expansions for their periods $T_n$ and for the logarithms of their unstable multipliers $\log\lambda_n$. We give geometric interpretations of the coefficients in these expansions, in particular in terms of the Anosov cocycle (\`a la Hurder-Katok) and the longitudinal Anosov cocycle (\`a la Foulon-Hasselblatt), evaluated on $\gamma$. Following the idea of De Simoi-Leguil-Vinhage-Yang, we compare these asymptotics when volume is the measure of maximal entropy. In this case, the equilibrium-state characterizations of volume and the measure of maximal entropy synchronize periods and exponents: $\log\lambda_n=hT_n$. Matching the coefficients of the two asymptotic expansions forces the cohomology classes of both cocycles to vanish. Then the work of Foulon-Hasselblatt on the longitudinal Anosov cocycle, together with Foulon's work on the contact case, yields the Foulon Conjecture for flows of class $C^7$: if volume is the measure of maximal entropy, then, up to a finite cover, $X^t$ is smoothly conjugate either to a constant-roof suspension of a hyperbolic automorphism of the $2$-torus or to a finite cover of the constant-speed geodesic flow on a closed hyperbolic surface.

We prove the Hodge conjecture for all hyper-K\"ahler varieties of generalized Kummer type of dimension $2n$ with $n+1$ a prime number, and all powers of these varieties. For any variety $X$ of generalized Kummer type of dimension divisible by 4, we show that the Lefschetz standard conjecture for $X$ implies the Hodge conjecture for $X$ and all its powers.

For $x,y>0$ let $S(x,y)$ denote the largest value of $\sum_{n\in A}\tau(n)$ over subsets $A$ of $[1,x]\cap\N$ whose elements are pairwise separated by more than $y$ (here $\tau$ is the divisor function). We determine the order of magnitude, and in a subrange the precise asymptotic, of $S(x,y)$ when the spacing parameter $y$ ranges over a polylogarithmic scale $y=(\log x)^{1+\lambda(\log\lambda-1)}$, $\lambda>1$. We show that for every fixed $\lambda>2$ the spacing constraint costs a computable power of $\log x$ in the exponent, while for every fixed $1<\lambda<2$ it costs nothing at all: $S(x,y)\sim x\log x$, the same asymptotic as the unconstrained sum $\sum_{n\le x}\tau(n)$. Near the threshold $\lambda=2$ the two regimes meet, and we show that for any fixed $\kappa>0,$ $S(x,y)\asymp_{\kappa} x\log x$ uniformly as long as $\lambda\le 2+2\kappa/\sqrt{\log\log x}$. The proof combines the Sathe-Selberg distribution of the number of prime factors, a smooth squarefree estimate obtained from a generating function together with Cauchy's formula, a sieve estimate for correlations among the squarefree parts of the integers that contribute most heavily to $\sum_{n\le x}\tau(n)$, and two combinatorial devices (the Caro-Wei bound and a concentration inequality on the symmetric group) used to extract a well-spaced independent set from a sparse auxiliary graph.

The Medvedev lattice is a tool used to classify the complexity of effective closed sets. In this article, we are interested in the complexity of using this tool. In particular, we study the complexity of deciding whether two effective closed sets have the same degree, when one is fixed and the other is given as input. This problem is naturally~$\Sigma^0_3$, and among other results, we show that for all degrees except the maximal one, the problem reaches its maximal complexity. It turns out that the argument is not uniform and depends on the self-similarity of the fixed degree, a notion that we introduce in this article. The complexity of having maximal degree is particularly difficult to establish, but we obtain a lower bound that almost matches the upper bound. We also show that deciding whether an effective closed set has a degree incomparable with a fixed degree $d$ is $\Pi^0_3$-complete whenever $\mathbf{0}_s <_s d <_s \mathbf{1}_s$.

We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load $\alpha$ and temperature $T$, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto $N$ Ising spins coupled to $M=\alpha N$ Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies $T_g=1+\sqrt\alpha$ as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below $T_g$ we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.

Inverse problems often admit multiple solutions consistent with the same noisy, indirect observations, especially with sparse data and/or strongly nonlinear forward models. Ensemble Kalman inversion (EKI) provides a robust, derivative-free framework to solve inverse problems, but its global empirical covariances couple all ensemble members across distinct solution modes, limiting its effectiveness for multimodal problems. We introduce polarized ensemble Kalman inversion (PEKI), which constructs distance-weighted local covariances for each ensemble member. This localization weakens contributions from distant, data-consistent regions and enables different clusters of ensemble members to evolve toward distinct solution modes. For the practical algorithm with finite ensemble size and polarization radius, we also establish non-asymptotic bounds and perform numerical studies, showing that cross-cluster effects decay exponentially with cluster separation relative to the polarization radius. For linear forward maps, we further establish particle-wise residual contraction and derive an exact cluster-mean decomposition that separates cross-cluster leakage, within-cluster gain variation, and observation perturbations. These results further motivate a one-step procedure for choosing the polarization radius by reusing stored forward-model evaluations. We studied the performance of PEKI on several numerical experiments, including low-dimensional nonconvex benchmarks, a problem equipped with many-mode landscape, and a high-dimensional inverse problem governed by Navier--Stokes equations. The results confirm the theoretical exponential decay of cross-cluster effects and show that PEKI can serve as a robust and derivative-free method for inverse problems with multiple solution modes.

We study the early training dynamics of nonlinear residual networks with blocks of zero hidden layers as a model of more general residual networks. We analyze networks in a sequential limit where first width and depth diverge, then the number of training data points diverges, and finally the input dimension diverges. This is a more refined setup compared to the typical mean-field theory analysis in which depth and width diverge but the number of training datapoints and input dimension are fixed. Our main technical result is a diagrammatic computation of both the network output and the hidden representations after the few steps of gradient descent in terms of the statistics of inner products of inputs from the training set. One of our principle conceptual findings is a negative result: in the limit of large input dimension the network output can learn only a linear function of its inputs early in training since higher learned monomials are more and more suppressed in the input dimension and vanish in the large dimension limit. This suggests that mean-field limits of ResNets are surprisingly rigid in the regime of high input dimension and may not be sufficient toy models for understanding feature learning. Finally, we investigate how to ensure hyperparameter transfer for the learning rate as a function of dataset size and input dimension. We test our predictions numerically and find excellent agreement with our theory.

We define and study a spin version of the open Toda integrable system, using the Iwasawa decomposition of a connected split real Lie group. We study spherical Whittaker functions as solutions of the spin Toda system. We also construct their N-point version and find the corresponding factorisation of the Casimir element, which yields asymptotic boundary KZB type equations.