数学前沿

The canonical double cover $BX$ of a graph $X$ is the direct product $X\times K_2$. A graph $X$ is stable if every automorphism of $BX$ is induced by automorphisms of $X$ and $K_2$, and unstable otherwise. We prove several asymptotic results on stable and unstable graphs. In particular, we show that the automorphism group of the canonical double cover of almost every graph has order $2$, and consequently that almost all graphs are stable. We also prove that the dominant sources of instability are having twins and being disconnected, in that order, and that graphs arising from a known construction are the next step in this hierarchy.

Many Bayesian inverse problems are characterized by two parameter vectors with distinct roles. A medium property, operator coefficient, or geometry block enters the forward model nonlinearly, whereas a volumetric, initial condition, or boundary condition source enters linearly. General-purpose nonlinear surrogates must relearn this known source dependence from data and may not preserve the correct source scaling outside the training regime. We introduce a structure-preserving partially linear (SPPL) framework that constructs separate reduced spaces for outputs, (linearly appearing) sources, and (nonlinearly appearing) medium variables and represents the reduced forward map as an affine function of the retained source coordinates, with coefficients that depend on the medium variables. Here, "affine" refers to the retained, mean-centered source coordinates: the physical forward map remains linear in the source, while the affine offset is induced by the source-prior mean and vanishes when that mean is zero. The analysis first identifies the optimal blockwise ridge approximation and then derives the optimal affine-in-source approximation under independent medium and source priors. A direct decomposition of the forward error into output, source, medium, and surrogate contributions yields blockwise spectral criteria and prior-predictive expected-KL control. For a Gaussian source prior, the optimal affine-in-source approximation coincides with the optimal blockwise ridge approximation. In a coseismic numerical example, a three-model controlled ablation on common block-reduced coordinates separates the effects of coefficient-matrix supervision and structure-preserving affine-in-source parameterization. Reduced-space surrogate MCMC provides orders-of-magnitude online acceleration relative to high-fidelity posterior sampling.

We prove the all-exp property for the Traveling Salesman Problem with the $2$-opt neighborhood, and for $(3,2,2)$- and $(2,3,2)$-Maximum Constraint Assignment with the flip neighborhood. The all-exp property states that infinitely many instances have an initial solution from which every sequence of improving moves to a local optimum, and hence every run of local search, has exponential length. For the Traveling Salesman Problem, the all-exp property was previously known for every $k$-opt neighborhood with $k \ge 3$, so our result for $2$-opt completes the picture for all $k$-kopt neighborhoods. In $(p,q,r)$-Maximum Constraint Assignment, each constraint involves at most $p$ variables, each variable occurs in at most $q$ constraints, and each domain has size $r$. Decreasing any of the parameters of $(3,2,2)$ or $(2,3,2)$ yields a problem in which every sequence of improving flips has polynomial length, so both Maximum Constraint Assignment results are tight. All three results settle long-standing open problems. Our proofs build on a common construction idea, which reveals a close connection between the local-search dynamics of the three problems.

We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients whose regularity is formulated in terms of horizontal and vertical Dupire derivatives. Under the same regularity assumptions as in Ananova~\cite{An23}, we develop a new causal Taylor argument and establish a second-order remainder estimate. This removes the regularity loss responsible for the restriction $2\leq p<1+\sqrt{2}$ in~\cite{An23} and extends the existence result to the full range $2\le p<3$.

This article studies a finite-horizon American put option with up to $m$ exercise rights under the risk-neutral Black--Scholes model, where consecutive exercises must be separated by a prescribed refractory period $\delta>0$ and unused rights may expire. For each number of remaining rights, we show that optimal exercise is characterized by a single exercise boundary and that these boundaries are ordered by the number of remaining rights. We prove optimality of the first boundary-hitting time and establish local Lipschitz continuity of the exercise boundaries within each open deadline strip. Away from the deadlines, we derive smooth fit, the associated quasi-variational inequality and free-boundary formulation, an early-exercise-premium representation, recursive Volterra-type boundary equations, and a verification theorem. We also extend the recursive formulation to an independent random termination time, thereby allowing for exogenous termination of the option contract.

We compare relative Abelian BF torsion with the Alexander-graded Euler characteristic of knot Floer homology. A meridional annulus boundary condition identifies the normalized relative torsion with the Alexander polynomial. One exact prime-cyclotomic torsion value recovers all its coefficients under a sharp genus bound, compatibly with connected-sum gluing and branched-cover formulas. We compute interface contractions and obstructions to scalar comparisons, including the pairing defect of two-sector reduction for the degree-zero torus algebra. A six-generator dg resolution yields an equivalence of perfect-module categories preserving derived pairings and admits a classical one-dimensional BV-BFV realization. In the fixed nonresonant polynomial quantum BFV sector, over characteristic zero with $\hbar$ inverted, both states and Weyl observables reduce to a vacuum field; a Hochschild invariant obstructs a Morita equivalence with the degree-zero torus algebra. These results do not establish a full geometric BV-BFV/Floer equivalence.

This paper addresses the longstanding problem of determining the structure of the $\leq_{\mathrm{LT}}$-order in the Effective Topos, known to effectively embed the Turing degrees. In a surprising discovery, we show that the $\leq_{\mathrm{LT}}$-order is in fact tightly controlled by the combinatorics of filters on $\omega$, raising deep questions about how combinatorial and computable complexity interact, both within this order and beyond it. To make the connection precise, we introduce a game-theoretic (''gamified'') variant of the Kat\v{e}tov order on filters over $\omega$, which turns out to exhibit a striking mix of coarseness and subtlety. For one, it is strictly coarser than the classical Rudin-Keisler order and, when viewed dually on ideals, collapses all MAD families to a single equivalence class. On the other hand, the order also supports a rich internal structure, including an infinite strictly ascending chain of ideal classes, which we identify by way of a new separation technique. From the computability-theoretic perspective, we show that a computable (and extended) variant of the gamified Kat\v{e}tov order is isomorphic to the original $\leq_{\mathrm{LT}}$-order. Moreover, our work brings into focus a new degree-spectrum invariant for filters $\mathcal{F}$, $$\mathcal{D}_{\mathrm{T}}(\mathcal{F}):=\{\,[f\colon\omega\to\omega] \mid f\leq_{\mathrm{LT}} \mathcal{F} \},$$ which is shown to always determine a proper initial segment of the Turing degrees. Extending this, given any non-principal $\Delta^1_1$ filter $\mathcal{F}$, we show that $\mathcal{D}_{\mathrm{T}}(\mathcal{F})$ is precisely the class of hyperarithmetic degrees. This significantly generalises previous results obtained by van Oosten \cite{vO14} and Kihara \cite{Kih23}. The proofs draw on ideas from general topology, descriptive set theory, and computability theory.

Let $E/\mathbf{Q}$ be an elliptic curve, let $K=\mathbf{Q}(\sqrt{D})$ satisfy the Heegner hypothesis with $D<-4$, and fix a modular parametrization $\varphi$ with $\varphi(\infty)=O$. Under surjectivity modulo $2^L$, with $L\ge 2$, and a cohomological exclusion condition at level $4$, we construct a finite set of primes $T$ such that, writing $d_T=\prod_{q\in T}(-q)$ and $n_T=\prod_{q\in T}q$, the $2$-Selmer dimensions of $E^{d_T}$ and $E^{d_TD}$ are $0$ and $1$. The support size is bounded in terms of the initial Selmer dimensions, independently of $L$. Inspired by the work of Tian and Tian--Yuan--Zhang on congruent numbers, we construct a point $Q_T$ on the terminal twist and identify its Kummer class with the standard mod-$2$ Kolyvagin class. Using the low-corank $2$-converse and the $2$-primary Birch--Swinnerton-Dyer formula, we prove that its index $m_T$ in the full Mordell--Weil lattice satisfies $v_2(m_T)=v_2(c_\varphi)+\mathfrak{i}(E,K)$. The optimal Manin constant theorem supplies a $2$-indivisible parametrization $\varphi_0$, for which $\kappa^{\mathrm{std}}_{n_T,1}(\varphi_0)\ne 0$ if and only if $\mathfrak{i}(E,K)=0$. When the intrinsic index vanishes, the minimum terminal support size is $R=\min\{d_2(E),d_2(E^D)\}$, and the normalized standard classes have order $2^M$ for $1\le M\le L$. Full $2$-adic Galois image then gives strong nonvanishing.

The edge-girth of an edge $e$ in a simple connected graph is the length of a shortest cycle containing $e$, with $g_e = \infty$ if no such cycle exists, and the edge-girth sequence of a graph is the nondecreasing sequence of the edge-girths of its edges. We characterize the sequences that are realizable as the edge-girth sequence of a simple connected graph. A constant sequence $(g^{(m)})$ with $g$ finite and $m \geq 1$ is realizable if and only if $m = g$ or $m \geq \lceil 3g/2 \rceil$. Otherwise, writing $S = S_0 \uplus (g^{(m)})$, where $g$ is the maximum value of $S$ and $m$ its multiplicity, we prove that $S$ is realizable if and only if $S_0$ is realizable and $m$ lies in an explicit set determined by $g$ and by the maximum diameters of the graphs realizing $S_0$, alone or extended by fewer than $g$ edges of edge-girth $g$. The proof rests on a chaining lemma, which turns a shortest cycle of a realization into a lower bound on the maximum diameter of the realizations of a shorter sequence.

We examine the proof of Dongyi Wei's logarithmic circulation criterion. A trace estimate specific to $\nabla(u_r/r)$ replaces a generic squared-swirl form bound by one with one logarithm rather than two. Repeating the coupled $J$--$\Omega$ energy and comparison argument then gives a proposed threshold of order $(\log D)^{-5/4}$. Under the hypotheses stated below, the proof is intended to establish regularity when $|r u_\theta|\le C|\log r|^{-\alpha}$ uniformly near the axis with $\alpha>5/4$, and at $\alpha=5/4$ with an absolute sufficiently small coefficient. Every analytic step, including the cutoff core and exterior terms, is included for independent checking.

We present a homological construction of Dax invariants in 4-manifolds that is intrinsic to the topological category and requires neither smoothing nor generic representatives. Applications include an extension of the invariant of Cha-Kim to neat disks without assuming local flatness, and a topological definition of the Dax map for arc spaces that yields a short alternative proof of the topological Dax isomorphism theorem.

We study the exceptional theta correspondence for dual pairs of the form $\SL_3\times G$ inside $p$-adic groups of type $E_n$, $n=6,7,8$. We establish Howe duality for lifts of all Whittaker-generic unitarizable representations of $\SL_3$. Our dual pair is in a see-saw relationship with a dual pair of the form $G_2 \times H$ such that $G/H$ is a rank-two symmetric space. Using the theta correspondence for $\SL_3 \times G$, we prove a multiplicity-one result for $H$-invariant functionals on representations of $G$.

We present a syntactic cut-elimination procedure for the alternation-free fragment of the modal mu-calculus. Cut reduction is carried out within a cyclic proof system, where proofs are finitely branching but may be non-wellfounded. The structure of such proofs is exploited to directly transform a cyclic proof with cuts into a cut-free one, without detouring through other logics or relying on intermediate machinery for regularisation. Novel ingredients include the use of multicuts and results from the theory of well-quasi-orders, the later used in the termination argument.

We give a new proof of the classification of unipotent representations for split finite groups of Lie type. To do this, we first construct an (infinity, 2)-categorification of Lusztig's map from the Hecke algebra to the asymptotic algebra. Then we use the theory of weights to show that it induces an isomorphism after applying trace of Frobenius.

We present a simplified proof of the Erd\H{o}s-S\'os conjecture on trees in graphs, which is based on its recent resolution by GPT-6 Astra. At the same time, we also give another proof of the corresponding conjecture for antidirected trees in digraphs, posed by Addario-Berry, Havet, Linhares Sales, Reed, and Thomass\'e, and we simplify recent proofs of Kalai's conjecture for tight trees in hypergraphs. Our proof uses the language of random cyclic orderings in order to take advantage of symmetries that were obscured in the original argument, and we hope that the reader will find it intuitive. We also characterize the extremal examples for Kalai's conjecture, and we generalize it to the setting of so-called fully directed $r$-uniform hypergraphs, whose edges are ordered $r$-tuples.

The stationary increment measure of the open Kardar-Parisi-Zhang equation on a bounded interval can be represented as the independent sum of a Brownian motion and another process defined via an explicit Radon-Nikodym derivative with respect to the measure induced by a Brownian motion. In the fan region, its finite-dimensional Laplace transform was previously expressed in terms of the continuous dual Hahn process. The main contribution here consists of computing the finite-dimensional Laplace transform for a range of parameters in the shock region by an analytic continuation argument. As an application, we provide a direct proof of the convergence of the finite-dimensional distributions of the increment height function of open WASEP under the stationary measure in the shock region, by computing the convergence of finite-dimensional Laplace transform without relating to the open KPZ equation.

We study eventual strict log-concavity and level-set geometry for the multidimensional Fisher-KPP equation with nontrivial, nonnegative, smooth compactly supported initial data, without assuming radial symmetry or convexity of the initial superlevel sets. Our main result is that there exist $T>1$ and $c>0$ such that $D_x^2\log u(t,x)\le -(c/t)e^{-\beta_*t}\mathrm{Id}$ for all $t\ge T$ and $x\in\mathbb{R}^N$, where $\beta_*=2(\sqrt2-1)$. Consequently, after a single waiting time, $u(t,\cdot)$ has exactly one critical point, a nondegenerate global maximizer, and every level set $\{u(t,\cdot)=\lambda\}$ with $0<\lambda<\max_x u(t,x)$ is a smooth closed strictly convex hypersurface with positive principal curvatures. For levels in any fixed closed interval $K\subset(0,1)$, the level sets are radial graphs $\rho=R_\lambda(t,\omega)$, with every principal curvature bounded below by $(1-C\log t/t)/(2t)$ for all sufficiently large $t$. Unlike log-concavity, ordinary concavity exhibits a threshold determined by the inflection level of the critical travelling wave: for each fixed level above it, $u$ is eventually strictly concave on its superlevel set, whereas below it concavity eventually fails. Our proof proceeds by establishing estimates in four overlapping spatial regions. In the fixed-front region, we combine a one-sided estimate for second angular derivatives with radial and mixed derivative bounds to obtain curvature control without differentiating the Lipschitz directional phase. In the remaining regions, absorptive comparison in the invaded interior, leading-edge barriers and blow-up arguments through the diffusive scale, and Gaussian amplitude estimates combined with parabolic regularity in the exterior yield the required logarithmic-Hessian bounds.

This paper and its companion Part II prove that the incompressible Euler equations in the plane, without external force, admit two distinct renormalized weak solutions with the same initial vorticity $f$, where $f$ is radial and nonnegative and lies in $(L^1\cap L^q)(\mathbb{R}^2)$ for every $q<\infty$ but is not bounded. One solution is radially symmetric and stationary. The other is $3$-fold rotationally symmetric, is equimeasurable with $f$ at every time, and departs from $f$ in finite time; the two solutions are distinct in $L^\infty_t(L^1\cap L^q)_x$ on $[0,T]\times\mathbb{R}^2$ for every $q\in(4/3,\infty)$ and every sufficiently large $T$. In this first part we prove two ingredients. One is a compactness criterion: for a radial datum, a sequence of $3$-fold rotationally symmetric solutions with converging data, whose non-radial moment is bounded below at some bounded time, has a subsequence converging to a second distinct renormalized weak solution. The other is the linear analysis for the building blocks of our multiscale datum. We show that the linearization of the Euler equations around the basic building block plus a small point vortex has an unstable eigenvalue in a space of $3$-fold rotationally symmetric perturbations. This latter argument depends on rigorous interval arithmetic that we discuss more fully in Part II.

For the method of successive approximations (MSA), a fundamental challenge is that the stochastic maximum principle (SMP) provides an optimality condition rather than a descent rule. To overcome this difficulty for nonconvex stochastic control problems, we develop a soft-update MSA (SU-MSA) with a second-order-adjoint correction of the Hamiltonian and a neural realization called DeepSMP. Both an explicit stepsize rule guaranteeing cost descent and the convergence of the algorithm are established. Moreover, the global SMP condition is asymptotically satisfied when the corrected Hamiltonian is convex, while in a special case the cost is further proved to converge to the global optimum. To extend SU-MSA to high dimensions, we develop DeepSMP using a differentiable smooth approximation of the corrected Hamiltonian and analyze the convergence of the resulting objective and its gradient. Numerical experiments demonstrate the effectiveness of the proposed framework.

We develop a method for studying centralizers of generic elements in some large Polish groups. We identify a condition which enables us to show that the range of each power map is comeager; a strengthening of this condition allows us to prove that a generic element embeds in a flow (a continuous, one-parameter subgroup). This approach enables us to answer the question that motivated this work : we prove that a generic isometry of the Urysohn space (or the Urysohn sphere) embeds in a flow. We also obtain information on the centralizer of a generic isometry; we show that it contains an infinite-dimensional torus and that torsion elements are dense in it. To illustrate the flexibility of our method, we apply it to two other Polish groups. The first is the group of measure-preserving automorphisms of a standard probability space, and we recover results of King and of de la Rue and de Sam Lazaro. Finally we apply our technique to the group of nonsingular automorphisms of a standard probability space, and we deduce that a generic nonsingular automorphism embeds in a flow, answering a question of Danilenko and Silva.