数学前沿

Given rearrangement-invariant (r.i.) spaces $X$ and $Y$, we show that the segment multiplier, the truncated Hilbert transform, and the discrete Hilbert transform (on the associated discretized space) are bounded from $X$ to $Y$ simultaneously. Moreover, when $X = Y$ this boundedness is characterized by a condition on a pair of Boyd-type indices. We also exhibit an r.i. space on which the segment multiplier is bounded while the (non-truncated) Hilbert transform fails to be bounded.

We present a detailed study of the Hodge theory, geometry and derived categories of Verra fourfolds, and their relationship with twisted K3 surfaces and cubic fourfolds. We classify Brauer classes of Verra type on K3 surfaces of degree 2, and use this to derive an explicit counting formula for Fourier--Mukai partners of a large class of Verra fourfolds. We develop the basic theory of cohomology of Verra fourfolds and apply it to prove a derived Torelli theorem for a large class of Verra fourfolds. As an application, we prove that L-equivalent Verra fourfolds in this class are Fourier--Mukai partners, and conjecture that this implication holds for all Verra fourfolds. We then give a construction of cubic fourfolds birational to Verra fourfolds. The construction yields examples of cubic and Verra fourfolds whose associated twisted K3 surfaces have non-trivial Brauer classes. Therefore, these fourfolds are expected to be irrational.

The $S$-function method for rational second order ODEs $y''=\varphi(x,y,y')$ uses $S=I_y/I_z$, attached to a first integral $I$, which satisfies a Riccati equation along the flow; it was designed for Liouvillian $I$. We study the case of rational $S$ with non-Liouvillian $I$. From any rational $g(x,y,z)$ with $g_z\neq0$ and any linear equation $A_2w''+A_1w'+A_0w=0$ with polynomial coefficients, the substitution $w'/w=g(x,y,y')$ yields a rational 2ODE whose first integral is built from two independent solutions, and $S=g_y/g_z$: the Wronskian cancels, so $S$ does not see the linear equation, and one rational $S$ is shared by equations with and without Liouvillian first integrals. Conversely, given $(\varphi,S)$, membership in this class reduces to a rational first integral of an explicit planar vector field plus a M\"obius-invariant Riccati test in $x$, which recovers the linear equation. A transfer theorem shows that $I$ is Liouvillian if and only if this equation has a Liouvillian solution, so Liouvillianity is decidable on the class by Kovacic's algorithm: a three-variable problem with no a priori degree bound is reduced to two classical ones. On nine pull-back equations of degree up to $15$, given without information on their construction, the procedure recovers the Airy equation and the Bessel equation of order zero.

An identifying code of a graph is a dominating set $C$ such that the sets $N[v]\cap C$ are pairwise distinct. The minimum density of an identifying code has been studied extensively for several infinite grids. It is known to be $1/4$ for the triangular grid and $7/20$ for the square grid, whereas for the hexagonal grid the problem is still open, between $5/12$ and $53/126$. These three grids are the regular members of the family of the eleven Archimedean grids. In this paper, we start a systematic study of identifying codes in this family. We determine the minimum density for five of the nine Archimedean grids for which it was not known: it is $2/3$ for the truncated hexagonal grid $3.12.12$, $3/7$ for the truncated square grid $4.8.8$, and $1/3$ for the trihexagonal (kagome) grid $3.6.3.6$, the elongated triangular grid $3^3.4^2$ and the snub hexagonal grid $3^4.6$. All lower bounds are proved by the discharging method. For the three remaining non-regular Archimedean grids, we also obtain some bounds: the minimum density lies between $3/7$ and $13/30$ for the truncated trihexagonal grid $4.6.12$, between $1/3$ and $11/30$ for the rhombitrihexagonal grid $3.4.6.4$, and between $2/7$ and $9/28$ for the snub square grid $3^2.4.3.4$.

Estimating a camera pose from 3D-2D correspondences - Perspective-n-Point - is one of the most fundamental problems in computer vision. Existing methods either solve a minimal problem or replace the objective with an easier one. Surprisingly, no scalable certifiable solver for the actual Maximum Likelihood Estimation (MLE) exists. We propose PnP*, the first practical solver that guarantees efficient computation and returns a certifiable globally optimal solution. We prove that the underlying convex relaxation is exact under moderate noise, which to our knowledge is the first stability guarantee for any relaxation for the MLE of PnP. Furthermore, we reduce the relaxation by exploiting chordal sparsity pattern and quotient ring reduction such that the relaxation is tractable and scalable. The stability and scalability results are verified numerically, as PnP* returns a provably correct camera pose at over 10 Hz.

We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism $\Delta^{d-2}\times\Delta^1$. In dimension three these are exactly the cubical polytopes, so the class sits naturally between simplicial and cubical polytopes. In dimension four, $\Delta^2\times\Delta^2$ is the simplest prismatical polytope and we can construct infinitely many by using connected sums. By fibreing a triangle over a triangulated polygon we construct infinitely many connected-sum-prime prismatical $4$-polytopes. Finally, we show that there are no prismatical $d$-polytopes for $d\ge5$, answering a question of Perles and Shephard.

We construct closed smooth $4$-manifolds with finite fundamental group which are homotopy equivalent but not simple homotopy equivalent, even after taking a connected sum with copies of $S^2\times S^2$. Our explicit examples have remarkably large fundamental groups; in particular, the smallest examples we obtain have fundamental groups of order roughly $10^{10^{42}}$.

A minimal finite model of a topological space is a finite $T_0$-space of least cardinality that is weak homotopy equivalent to it. We determine, up to homeomorphism, the minimal finite models of eight wedges of one- and two-dimensional spheres. The spaces $S^{2}\vee S^{1}$ and $S^{2}\vee S^{2}$ have exactly two and three minimal finite models, respectively, all with seven points, whereas $S^{1}\vee S^{1}\vee S^{2}$ has exactly seven minimal finite models, $S^{1}\vee S^{1}\vee S^{1}\vee S^{2}$ has one, $S^{1}\vee S^{2}\vee S^{2}$ has six, $S^{2}\vee S^{2}\vee S^{2}$ has five, $S^{2}\vee S^{2}\vee S^{2}\vee S^{2}$ has three, and $S^{1}\vee S^{1}\vee S^{2}\vee S^{2}$ has two, all with eight points. To prove this we classify the connected finite $T_0$-spaces of height $2$ without beat points that have seven or eight points, and we identify their weak homotopy types by means of McCord's theorem and explicit collapse and suspension arguments for their two-dimensional order complexes. Together with the known results for spheres and wedges of circles, this determines, up to weak homotopy equivalence, all path-connected spaces whose minimal finite models have at most eight points.

We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-\mu}$, we assume $|F(z)|\leq C\Phi(|z|)$, where $\Phi$ is positive, continuous and regularly varying at zero and infinity. For strongly $A(\alpha)$-stable multistep methods, we derive error estimates and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_\beta(t)=t^{\beta1}/\Gamma(\beta)$ with $\beta>0$. These estimates explicitly retain the dependence on the memory law through the factor $\Phi(t_n^{-1})$. The framework accommodates symbols whose behavior cannot be described by a global single-power. It also preserves logarithmic and multiscale factors that would be lost under a coarser power bounds. Then we apply the to fractional viscoelastic, distributed-order, and logarithmic-power memory models. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. A distributed-order PDE further illustrates temporal convergence for smooth and nonsmooth initial data.

Decoupling inequalities for the parabola are obtained for weights in physical space. The decoupling iteration relies on a fractal bilinear Kakeya estimate. We record applications to exponential sums.

Let $\mathcal{P}$ be a finite collection of cells. Two non-attacking rook configurations on $\mathcal{P}$ are switching-equivalent if one can be obtained from the other by a sequence of switches inside rectangles contained in $\mathcal{P}$. The switching rook polynomial of $\mathcal{P}$ enumerates the resulting equivalence classes according to the number of rooks. We characterize the collections of cells whose switching rook polynomial is palindromic. More precisely, we introduce a new combinatorial property, called domino-stability, and we prove that the switching rook polynomial of $\mathcal{P}$ is palindromic if and only if $\mathcal{P}$ is domino-stable.

In this paper, we study the global injectivity of local diffeomorphisms in $\mathbb{R}^n$, extending the criterions of Braun and Llibre in [{\it Pacific J. Math.} {\bf 333} (2024) 1--15] from the planar radial case to arbitrary dimension. Then we extend the results of Llibre and Mello in [{\it Bull. Sci. Math.} {\bf 213} (2026) 103889] to arbitrary dimension, obtaining two necessary and sufficient conditions for the real Jacobian conjecture in $\mathbb{R}^n$ by qualitative theory of differential equations. Finally, using weighted Poincar\'e compactification in $\mathbb{R}^n$, we provide a sufficient condition for the real Jacobian conjecture.

Takemura conjectured that the Conway polynomials $\sum a_\ell z^\ell$ of the Turk's head link $Th(p,q)$ and $\sum b_\ell z^\ell$ of $Th(q,p)$ satisfy $a_\ell=b_\ell$ for $\ell\equiv0,1\pmod 4$ and $a_\ell=-b_\ell$ for $\ell\equiv2,3\pmod4$, and proved it for $q=2,3$. We prove it for all $p,q\ge1$, knots and links. Equivalently, $\nabla_{Th(q,p)}(z)=c\,\nabla_{Th(p,q)}(iz)$ with $c=1$ if $\gcd(p,q)$ is odd and $c=-i$ if it is even. The proof combines a product formula $\nabla_{Th(p,q)}(z)^2=\prod_{j=1}^{p-1}\prod_{k=1}^{q-1}(z^2+2\cos\frac{2\pi k}{q}-2\cos\frac{2\pi j}{p})$, which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.

We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of $$\begin{equation*} \Delta u+V(x)u=0\qquad\text{in }\Omega, \qquad u=0\qquad\text{on }\partial\Omega, \end{equation*}$$ where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(\Omega)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,\Omega)(1+\|V\|_{L^p(\Omega)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.

We consider a near-critical version of loop-erased random walk (LERW) by boosting the weight of the underlying random walk according to its length before loop-erasure. We construct the scaling limit in natural parametrization in the two-dimensional unit disk and the three-dimensional unit ball. The boosted scaling limit undergoes a phase transition in mutual absolute continuity with the classical LERW scaling limit; in the disk the transition occurs at boost rate $m^2=\pi^2/4$. As the boost increases, the curve becomes macroscopically space-filling, in the sense that unvisited cubes have side length $O(m^{-1})$, while its Hausdorff dimension at each finite boost is unchanged. These results are motivated by Makarov--Smirnov's questions on near-critical lattice models. In the planar chordal setting, we also construct a continuum loop tilt below the spectral threshold of the domain and calculate its Loewner driving function.

Let $1\le h\le n-2$ and $g=\gcd(n,h+1)$. We prove that, for every prime power $q$, a maximum $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ exists whenever $(h+1)\mid rn$ and $$r\ge\frac{h+1}{g}\bigl(h(n-h-2)+1\bigr).$$ In particular, for fixed $n$ and minimum distance $\delta\in\{2,\ldots,n-1\}$, $\mathbb{F}_{q^n}$-linear MRD codes of length $N$ exist for every $q$ and every sufficiently large $N$ such that $n$ divides $(n-\delta+1)N$, with a bound on $N$ depending only on $n$ and $\delta$. We take the $\mathbb{F}_{q^s}$-span of the columns of a random matrix $A$ over the algebra $R=\mathbb{F}_{q^n}\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$, for a suitable $s$. Such a subspace is maximum $h$-scattered if and only if the kernel of $A$ contains no nonzero vector whose expansion with respect to an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^s}$ has rank weight at most $h+1$. When $\gcd(n,s)=1$, $R$ is a field, and a first-moment argument, counting these vectors up to multiplication by $\mathbb{F}_{q^n}^*\mathbb{F}_{q^s}^*$, shows that with positive probability the kernel contains none of them if $s>h(n-h-2)$. When $d=\gcd(n,s)>1$, $R$ is a product of $d$ fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an $(h+1)$-dimensional $\mathbb{F}_q$-subspace $D$ of $\mathbb{F}_{q^n}$, the elements of $D\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$ vanishing outside $k$ given factors, $0<k<d$, form an $\mathbb{F}_{q^s}$-space of dimension at most $(h+1)k/d$, with equality only if $D$ is linear over a subfield of $\mathbb{F}_{q^n}$ larger than $\mathbb{F}_q$. Taking $s=gr/(h+1)$ gives the bound on $r$. Finally, if $(n-h-1)\mid(h+1)$, then for every $r$ satisfying the above inequality there is an $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ of dimension $\lfloor rn/(h+1)\rfloor$, without a divisibility condition on $rn$.

For a graph $G = (V, E)$ with a vertex set $V$ and an edge set $E$, a function $f : V \rightarrow \{0, 1, 2, . . . , diam(G)\}$ is called a \emph{broadcast} on $G$. For each vertex $u \in V$, if there exists a vertex $v$ in $G$ (possibly, $u = v$) such that $f (v) > 0$ and $d(u, v) \leq f (v)$, then $f$ is called a dominating broadcast on $G$. The cost of the dominating broadcast $f$ is the quantity $\sum_{v\in V}f(v)$. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $\gamma_{b}(G)$. A multipacking is a set $M \subseteq V$ in a graph $G = (V, E)$ such that for every vertex $v \in V$ and for every integer $r \geq 1$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The multipacking number of $G$ is the maximum cardinality of a multipacking of $G$ and is denoted by $mp(G)$. It is known that $mp(G)\leq\gamma_b(G)$. In 2014, Hartnell and Mynhardt proved that $\gamma_b(G)\leq 3mp(G)-2$ whenever $mp(G)\geq2$. In 2019, Beaudou, Brewster, and Foucaud improved this bound to $\gamma_b(G)\leq 2 mp(G)+3$ and conjectured that $\gamma_b(G)\leq 2 mp(G)$. We solve their conjecture by proving that $\gamma_b(G)\leq 2 mp(G)$ for every graph $G$. Our proof is constructive and yields a linear-time $2$-approximation algorithm for Maximum Multipacking problem which improves the earlier approximation factor $2+o(1)$.

We show that multifractional Brownian motions (mBms) can be lifted to geometric rough paths. One of the main ingredients in the proof is a decomposition of mBms via the Lemari\'e-Meyer orthogonal wavelet basis, which is introduced in the context of multiresolution analysis. As a consequence, we are able to construct solutions to rough differential equations driven by mBms.

We prove relative Langlands duality for $X = GL_{p+q}/(GL_p \times GL_q)$. These symmetric spaces arise in the setting of the linear period of Friedberg-Jacquet when $p=q$, and exhibit non-tempered behavior when $q>p+1$. Via the real-symmetric correspondence, we establish a derived Satake equivalence for the real group $G_{\mathbb R}=U(p,q)$, and use it to obtain a constructible realization of coherent sheaves on Kato's exotic nilpotent cones. Along the way, we describe the dual variety of $X$ in terms of the Whittaker cotangent bundle and regular centralizers, viewed as Coulomb branches.

We characterize the generating function of bipartite planar maps counted according to the degree distribution of their black and white vertices. This result is applied to the solution of the hard particle and Ising models on random planar lattices. We thus recover and extend some results previously obtained by means of matrix integrals. Proofs are purely combinatorial and rely on the idea that planar maps are conjugacy classes of trees. In particular, these trees explain why the solutions of the Ising and hard particle models on maps of bounded degree are always algebraic.