数学前沿

We begin with a review of Tutte's homotopy theory, which concerns the structure of a certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path theorem asserts that this graph is connected, and his homotopy theorem asserts that every cycle in the graph is a composition of ``elementary cycles'', which come in four different flavors. We present an extended version of the homotopy theorem, in which we give a more refined classification of the different types of elementary cycles. We explain in detail how the path theorem allows one to prove that the foundation of a matroid (in the sense of Baker--Lorscheid) is generated by universal cross-ratios, and how the extended homotopy theorem allows one to classify all algebraic relations between universal cross-ratios. The resulting ``fundamental presentation'' of the foundation was previously established in [Baker--Lorscheid], but the argument here is more self-contained. We then recall a few applications of the fundamental presentation to the representation theory of matroids. Finally, in the most novel but also the most speculative part of the paper, we discuss what a ``higher Tutte homotopy theorem'' might look like, and we present some preliminary computations along these lines. The appendices include modern language translations of Tutte's proofs of the path theorem and the homotopy theorem (the latter is a contribution by Ju\v{s} Kocutar).

We study a nonlocal Fisher--KPP equation on the real line with compactly supported initial data. The reaction satisfies $f(s)\le f^{\prime}(0)s$ for $0\le s\le1$, with a power bound on $f^{\prime}(0)s-f(s)$ near zero. For continuous even probability kernels $J$ comparable to $|x|^{-\gamma}$, $\gamma>1$, we prove a global two-sided estimate and locate every fixed level set with a bounded error in $\log|x|$. The propagation scale is $\exp(f'(0)t/\gamma)$ for every $\gamma>1$. This differs sharply from the free-boundary propagation studied by Du and Ni \cite{DuNiJEMS}. For polynomial kernel tails, spreading in their model has linear, algebraic or $t\log t$ growth. We also obtain arrival-time and weighted-integral estimates and identify a spatial tail coefficient when $|x|^\gamma J(x)$ has a positive limit.

Quantum key distribution (QKD), whose security is derived from quantum mechanics rather than computational assumptions, is moving into standardization and commercial deployment. Yet its security guarantees rest on technically delicate proofs checked by hand. Formal verification closes this gap, establishing a two-factor assurance: unconditional security from physical principles, and machine-checked proofs against reasoning errors. We present FVQKD, a Rocq library for machine-checking QKD security proofs. Its first result is a complete, end-to-end machine-checked proof of composable, finite-key security for BB84. The mechanization reconstructs the complete proof infrastructure from first principles, including the smooth entropy framework, the entropic uncertainty relation, and the Quantum Leftover Hash Lemma. It uncovers necessary side conditions that the published argument leaves unstated, one of which reveals a missing proof branch, and it grounds the information-theoretic bound in a formal semantics of protocol execution, which the pen-and-paper analysis lacks. The proof is entirely admit-free, relying only on the Rocq kernel and standard foundational axioms. To our knowledge, this is the first end-to-end formal verification of a QKD security proof. Built on MathComp, MathComp-Analysis, and CoqQ, the development provides quantum-information infrastructure designed for reuse across broader QKD variants and quantum cryptography applications.

Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times \phi)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.

For a strongly convex unconstrained minimization problem, Newton's method with a line search offers rapid convergence to a minimizer by incorporating second-order information. However, for a general unconstrained problem the Hessian need not be positive definite, which can lead to the inability to find a search direction or to search directions that fail to produce descent. Modified Newton methods are designed to handle this situation. This paper considers modifications to a Newton-conjugate-gradient approach based on Lanczos tridiagonalization. Existing methods are extended using dynamic modification schemes that provide a descent direction while keeping the modification bounded. A Lanczos-based scheme and a conjugate-gradient (CG)-based scheme are formulated and analyzed. These methods are mathematically equivalent in exact arithmetic, but their practical behavior differs in finite precision. Finally, it is shown how the convergence and computational cost are affected by varying the amount of additional curvature introduced and by allowing the early termination of the inner iterations.

We study the problem of routing multiple commodities through a logistic transport network so as to minimise the joint cost of vehicle movements and of transshipment (reload) operations at intermediate storages. The problem combines two features that are common in practice but rarely treated together: edge costs that are nonlinear in the transported volume, because freight is carried by vehicles of fixed capacity, and hard upper bounds on the volume that can be reloaded at each node. We formulate the problem as a mixed integer linear program in both an arc-based and a path-based form, derive its linear relaxation, and propose a metaheuristic solver based on General Variable Neighborhood Search (GVNS). The method works in the path space and relies on three problem-specific neighborhoods that generate new candidate paths by (i) shortest paths under a reload-aware metric, (ii) edges carrying partially loaded vehicles, and (iii) detours around congested storages. We evaluate the method on our open benchmark for this problem (arXiv:2606.20653): a corpus of realistic logistic networks calibrated on real road graphs of nine European countries, five U.S. states and the European part of Russia, with near-planar topology (Zipf city sizes, a spatial-network budget model, and a doubly constrained gravity demand model). On 35 synthetic and 15 real-geometry instances with up to 150 storages, GVNS returns solutions within about one percent of the best dual bound on small networks -- a branch-and-bound lower bound substantially tighter than the LP relaxation -- is stable across random seeds, and for a comparable wall-clock budget is up to 55% better than a time-limited exact solver on the medium instances.

ADER-DG numerical method with the LST-DG predictor, when used to solve hyperbolic equations with stiff source terms, exhibits properties characteristic of IMEX computational schemes. With increasing stiffness, characterized by the parameter $z = \nu{\Delta t}^{n} < 0$, the maximum of the Courant number $\mathrm{CFL}_{\rm max}$ of the ADER-DG numerical method increases. In the region of strong stiffness $|z| \gg 1$, a linear increase in the Courant number $\mathrm{CFL}_{\rm max}(z) \sim |z|$ occurs. In the region of small source terms $|z| \ll 1$, the results $\mathrm{CFL}_{\rm max}(z)$ transform into known results for the problem $\mathrm{CFL}_{\rm max}(0)$ without source terms. In the transition region $|z| \sim 1$, a range of insignificant decrease in the Courant number $\mathrm{CFL}_{\rm max}(z)$ is discovered. In the linear case, instability is violated precisely when one of the matrix eigenvalues reaches $\lambda = -1$, regardless of the phase $\theta$ of the numerical solution. A rigorous stability theory of the ADER-DG numerical method for stiff problems for arbitrary degree $N$ of basis polynomials is constructed. The stability condition is simplified to solving a polynomial equation for the Courant number $\mathrm{CFL}$. Highly accurate dependencies of the Courant number $\mathrm{CFL}_{\rm max}(N, z)$ of the ADER-DG method on the parameter $z$ are obtained. A rigorous direct proof of the approximation of the ADER-DG method is presented. Approximation orders $p = N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection-reaction equation and the system of Euler equations for a two-component reacting medium. The results obtained in these calculations confirm the theoretical results well.

For graphs $G, T$ and a graph-family $\mathcal{H}$, let $ex(G,T,\mathcal{H})$ denote the maximum possible number of copies of $T$ in an $\mathcal{H}$-free subgraph of $G$. Alon and Shikhelman conjectured that for every family of graphs $\mathcal{H}$ with $\chi(H) > s$ for every $H \in \mathcal{H}$, it is NP-hard to compute $ex(G,K_s,\mathcal{H})$, and even to approximate it up to additive error $n^{s-\delta}$ on $n$-vertex graphs, for any fixed $\delta > 0$. The case $s=2$ was established in a breakthrough result of Alon, Shapira and Sudakov. We prove the Alon-Shikhelman conjecture for finite families $\mathcal{H}$. In particular, we obtain an elementary proof of the Alon-Shapira-Sudakov result when $\mathcal{H}$ is finite.

We study symmetric divergence form elliptic equations with measurable coefficients that may be degenerate and unbounded, under a uniform bound on the coarse ellipticity ratio over all ambient cubes and scales. We prove that uniform coarse ellipticity and classical $A_2$ weighted ellipticity define incomparable coefficient classes. For every admissible choice of coarse parameters, we construct a scalar weight with a logarithmically corrected, locally integrable singularity along a product Cantor set. Its codimension is chosen according to the upper coarse discount exponent, and the resulting coefficient satisfies the uniform coarse condition but is comparable to no $A_2$ weight. Conversely, an explicit $A_2$ weight fails the coarse condition for every admissible parameter choice. Under uniform coarse ellipticity we give a direct PDE treatment of boundary regularity in the approximable energy realization. Using coarse interior estimates, we prove boundary decay and solvability of the continuous Dirichlet problem in bounded NTA domains. An almost-positivity argument, with energy approximation and boundary testing justified, yields boundary Harnack and H\"older continuity of quotients of positive solutions with common zero boundary values. The argument requires no additional energy Sobolev inequality and avoids the construction of Green functions. Further examples distinguish uniform coarse control from finite coefficient moments and from coercivity on $\mathrm L^2(\mathrm{d} x)$. We also record interior H\"older estimates and continuity criteria under nonuniform control along nested scales.

For a given matrix $A$ in the real Schur form, the eigenvalue swapping algorithm is used to swap two adjacent diagonal blocks. In LAPACK, this operation is performed by the subroutine~\texttt{SLAEXC}/\texttt{DLAEXC}, which is based on a direct swapping algorithm proposed by Bai and Demmel in 1993. In this work, we present a thorough error analysis of a modified direct swapping algorithm. By a carefully designed pivoted Givens QR factorization, this algorithm can achieve an error bound of the form $O( \boldsymbol{u} \lVert A \rVert )$ under mild assumptions, where $\boldsymbol{u}$ is the unit roundoff. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.

Let $\mathcal{A}$ be an arrangement of $d$ lines in $\mathbb{P}^2_\mathbb{C}$ and let $\mathcal{T}\langle\mathcal{A}\rangle$ be its bundle of logarithmic vector fields. On a general line, $\mathcal{T}\langle\mathcal{A}\rangle$ splits as $\mathcal{O}(-k)\oplus\mathcal{O}(k+1-d)$ with $k=\min\{\mathrm{mdr}(\mathcal{A}),\lfloor\frac{d-1}{2}\rfloor\}$, where $\mathrm{mdr}$ is the minimal degree of a Jacobian syzygy. We prove that $k$ is determined by the intersection lattice whenever $d\le16$, answering in this range a question of Cook, Harbourne, Migliore and Nagel. Since freeness can be read off from $k$ and $c_2(\mathcal{T}\langle\mathcal{A}\rangle)$, this strengthens Terao's freeness conjecture in this range; in particular we obtain a proof of Terao's conjecture for $14$ lines that uses it only for $13$ lines. For any number $d$ of lines, it follows that $\mathrm{mdr}$ is determined by the lattice whenever $\mathrm{mdr}\le6$ and $2\mathrm{mdr}\le d-3$, and that Terao's conjecture holds for every free arrangement whose smaller exponent is at most $7$. The proof reduces the question to critical arrangements: $2s+3$ lines, each containing at most $s+1$ intersection points, with $\mathrm{mdr}=s$. For these, the inflection polynomial of the minimal derivation factors as $Q_\mathcal{A} R$ with $R\in H^0(\mathcal{I}_Z^2(s-3))$, where $Z$ is the zero scheme of the derivation, of length $\nu=c_2(\mathcal{T}\langle\mathcal{A}\rangle)-s(s+2)$. For $s\le5$ this forces $\nu\le1$, and Chern class arguments then reduce everything to Terao's conjecture for $13$ lines. For $s=6$ we show $\nu\le3$, using a Riccati equation along multiple components of $R$, Ziegler's multirestriction, and local computations verified by computer algebra; an exhaustive enumeration of the 26,265,318 candidate lattices with $\nu\in\{2,3\}$ that satisfy Bojanowski's inequality completes the proof.

The Chowla--Milnor conjecture predicts the linear independence of certain Hurwitz zeta values. In this paper, we prove that for any fixed integer $k \geqslant 2$, the dimension of the $\mathbb{Q}$-linear span of $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) is at least $(c -o(1)) \cdot \log q$ as the positive integer $q \to +\infty$ for some absolute constant $c>0$. It is well known that $\zeta(k,a/q)+(-1)^{k}\zeta(k,1-a/q) \in \overline{\mathbb{Q}}\pi^k$, but much less is known previously for $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$. Our proof is similar to those of Ball--Rivoal (2001) and Zudilin (2002) concerning the linear independence of Riemann zeta values. However, we use a new type of rational functions to construct linear forms.

We study two notions of pluripotency for dynamical systems. Strong pluripotency refers to the possibility of reprogramming, by arbitrarily small perturbations, the forward orbits of points in a set of positive Riemannian volume so that they track a prescribed orbit in a uniform time-averaged sense. In contrast, pluripotency, also called ``statistical pluripotency" in this paper, requires only asymptotic agreement of empirical measures. Strong pluripotency implies statistical pluripotency, but the converse need not hold. For $C^r$ transitive Anosov diffeomorphisms, $r\ge2$, we prove that strong pluripotency fails for every nonempty reference set. On the other hand, statistical pluripotency holds for a nonempty reference set if and only if it is contained in the basin of the unique SRB measure. This basin is invariant and has full Riemannian volume, but contains no periodic points. These results reveal a sharp distinction between the two notions even within uniformly hyperbolic dynamics.

Let $X$ be a finite set and $\mathcal{F} \subseteq 2^X$ a non-trivial increasing property. We show that the bias-threshold $b(\mathcal{F})$ for the Picker-Chooser $\mathcal{F}$-game satisfies $b(\mathcal{F}) \ge \alpha / p_c(\mathcal{F})$, where $p_c(\mathcal{F})$ is the threshold for the property $\mathcal{F}$ and $\alpha > 0$ is an absolute constant. This proves that Picker-Chooser games satisfy one-sided ``probabilistic intuition''. The proof is based on the notion of influence from the theory of boolean functions.

We characterize compactly supported functions $f$ in Dunkl--BMO spaces as those that can be written as $$\begin{equation*} f=g+\widetilde{S}_{\mu,\mathcal{P}}, \end{equation*}$$ the sum of some bounded function $g$ and the generalized balayage $\widetilde{S}_{\mu,\mathcal{P}}$ of a Carleson measure on $\mathbb{R}^{N}\times(0,\infty)$ associated with the Dunkl--Poisson semigroup. As a direct application of this representation of Dunkl--BMO functions, we establish a new characterization of the Hardy space in the Dunkl setting via a principal value integral related to $\widetilde{S}_{\mu,\mathcal{P}}$. Moreover, analogous conclusions hold for the Dunkl--heat semigroup as well.

Let $K$ be a non-Archimedean local field of characteristic different from $2$ and let $V$ be a four-dimensional symplectic $K$-vector space. We study the Penrose incidence correspondence $\mathbb{P}(V) \leftarrow \mathrm{Flag}(1,2;V) \rightarrow \mathrm{Gr}(2,V)$ after Berkovich analytification. The Klein quadric $\mathrm{Gr}(2,V)$ decomposes into the open orbit $\mathrm{Spin}(5)/\mathrm{Spin}(4)$ and the Lagrangian quadric, which parametrize respectively the projective lines transverse and tangent to the standard contact distribution on $\mathbb{P}(V)$. We construct explicit free discrete subgroups $\Gamma\subset\mathrm{PSp}(V)$ and analytic domains, cut out by Drinfeld upper half-spaces, in all three members of the incidence diagram. Both incidence projections are surjective on these domains, the three $\Gamma$-actions are free and properly discontinuous, and the Penrose double correspondence descends to the analytic quotients. The quotient of the twistor domain is a contact analytic threefold, and the Grassmannian domain is contained in the analytification of the $\mathrm{Spin}(5)/\mathrm{Spin}(4)$ open orbit.

We present a definition of a local height of nef toric divisors in the sense of Yuan and Zhang by approximating a height for canonical divisors and then using the local energy to extend to other metrics on the same geometric divisor.

How difficult is it to find an efficient communication protocol when the entire truth table of the function is given? We give a unified interlacing proof of NP-completeness for exact optimisation and of hardness of approximation. Assuming the Exponential Time Hypothesis, there is a computable, nondecreasing additive error allowance tending to infinity for which approximation is impossible in polynomial time. The proof uses interlacing, which asks the players to solve one problem selected by Alice from a collection of inputs held by Bob. For suitable matrices, a short protocol must reveal enough of Alice's selection to encode a solution to a combinatorial problem. This produces a one-bit communication gap in a polynomial-size matrix and proves exact hardness. Repeating the construction is harder: earlier messages can remove the inputs on which a later lower bound depends. Restriction bounds that survive the construction make the one-bit gain repeatable. A smaller, balanced family of interlaced columns controls the truth-table size. For every fixed number of applications, the construction takes subexponential time in the source length. This rules out every fixed additive error under ETH. A general computability argument then yields the stated unbounded allowance.

Many models of spatial populations use directed motion towards environmental attractants as a mechanism for spatial pattern formation. While prior approaches typically focus on partial differential equation models for directed motion in continuous space, it is natural to consider the dynamics of biological and social systems on a metapopulation featuring well-mixed local patches and spatial motion between patches. In this paper, we formulate a version of the Keller-Segel model for chemotaxis on a graph metapopulation, modeling a chemoattractant that can diffuse across patches and a population of individuals who move through a biased random walk through comparing chemoattractant levels along edges of the metapopulation network. We also derive a continuum limit for our model describing chemotaxis in a population on patches connected by a graphon, showing that our resulting integro-differential equation model arises as the limit of our discrete patch model for families of large deterministic or random graphs. Through linear stability analysis, we derive conditions on the chemotactic sensitivity required to produce instability of a spatially uniform equilibrium for our Keller-Segel models in terms of the reaction parameters and the eigenvalues of the Laplacian operator for the metapopulation graph or graphon. Through a mix of weakly nonlinear stability analysis, numerical simulations, and analytical bounds on solutions, we can also characterize the form of pattern-forming instability in our graphon model and show how our choice of demographic dynamics for the population and the graphon adjacency function can impact the long-time total population size and chemoattractant concentration achieved in emergent spatial patterns.

We give a constructive representation of the solution of the KdV equation $u_t=-u_{xxx}+6uu_x$ on $\mathbb R$ for real initial data in $H^3(\mathbb R)$. The approximations couple an exact, unitary centered-difference Chernoff product for the Airy group with a skew-symmetric discretization of the quadratic term, and are computed by Picard iteration; they preserve the $L^2$ norm exactly, as does the limit. A two-mesh Cauchy argument shows that the whole sequence converges in $C([0,T];H^2)$, at rate $O(n^{-\frac{2(s-2)}5})$ for $H^s$ data ($3\leq s\leq 7$), and in $L^2$ at rate $O(n^{-6/5})$ for $H^3$ data. This scheme, to the best of our knowledge, is novel and is of independent interest. As a by-product we obtain existence and uniqueness of strong solutions, Lipschitz dependence on the data in the $H^2$ norm, uniformly on bounded subsets of $H^3$ and compact time intervals, and joint locally Lipschitz dependence of $(t,f)\mapsto S(t)f$ into $L^2$; the classical conservation laws extend the flow globally in time.