数学前沿

Let $p$ be a prime and $n\geq 1$. We prove that the Tate-Frobenius of the $\operatorname{K}(n)$-local sphere induces an equivalence between $L_{n-1}L_{\operatorname{K}(n)}Y$ and $(L_{\operatorname{K}(n)}Y)^{t\Sigma_p} \cong (L_nY)^{t\Sigma_p}$ for any finite $p$-local type $n-1$ spectrum $Y$ with trivial $\Sigma_p$-action.

We give a direct proof of the equivalence between the Muckenhoupt $\mathcal{A}_2$ and the Helson--Szeg\H{o} conditions on the unit circle $T$. A positive weight $w$ belongs to $\mathcal{A}_2(T)$ if and only if $\log w=u+Hv$, where $u,v$ are real bounded functions, $\|v\|_\infty<\frac{\pi}{2}$, and $H$ denotes the periodic Hilbert transform. The classical proof combines the Helson--Szeg\H{o} theorem with the Hunt--Muckenhoupt--Wheeden characterization of weighted boundedness of the Hilbert transform. Our proof derives the Helson--Szeg\H{o} representation directly from the Muckenhoupt $\mathcal{A}_2$ condition, without using boundedness of $H$ on $L^2(w)$.

We give two alternate formulations for the second moment formula of the Siegel transform over $\mathrm{Sp}(2n, \mathbb Z) \backslash \mathrm{Sp}(2n,\mathbb R)$, previously established by Athreya-Konstantoulas and Kelmer-Yu. We also provide several applications that demonstrate their usefulness.

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this evolution -- the percolation threshold, where the order of the largest component seemingly jumps from logarithmic to linear in size, and the connectivity threshold, where the subgraph becomes connected. Similar phenomena have been observed in many other random graph models, and in particular, works of Ajtai, Koml\'{o}s and Szemer\'{e}di and of Erd\H{o}s and Spencer determine corresponding thresholds for random subgraphs of the hypercube. The permutahedron, like the hypercube, has many equivalent representations, and arises as a natural object of study in many areas of combinatorics. In particular, as a highly-symmetric simple polytope, like the $n$-simplex and $n$-cube, percolation on the permutahedron arises naturally as an analogue to the Erd\H{o}s-R\'{e}nyi random graph and the percolated hypercube. We determine the percolation threshold and the connectivity threshold for random subgraphs of the permutahedron. Along the way we develop a novel graph exploration technique which can be used to find exponentially large clusters after percolation in high-dimensional geometric graphs, and we initiate the study of the isoperimetric properties of the permutahedron.

We prove the Hardy--Littlewood asymptotic for prime pairs of difference four from a restricted uniform two-over-two zeta-ratios formula. The formula has error $O_\varepsilon(U^{1/2+\varepsilon})$, real parts of all four shifts in $[3/64,5/64]$, and imaginary parts of absolute value at most $U^{1/4}$. For a fixed real smooth weight supported in $(1,101/100)$, we obtain an error $O_\varepsilon(x^{-1/16+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$. Adding the Riemann hypothesis improves this error to $O_\varepsilon(x^{-3/8+\varepsilon})$. The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral. Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference four. We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_4$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function. The latter is connected to a logarithmically weighted M\"obius approximant in the Nyman--Beurling criterion by finite identities.

We propose an approximate interpolation construction and a practical coarsening algorithm for algebraic multigrid (AMG) method when applied to H(curl) problems. Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows a geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. Under a uniform fine-space smoothing assumption, we prove a jump-independent two-grid convergence bound for canonical finite-element spaces. We also establish an exact commuting relation for the unshifted construction. Two-dimensional numerical experiments demonstrate sparse multilevel hierarchies and robust convergence under strong coefficient jumps in the curl--curl term. Three-dimensional experiments provide a two-level proof of concept, although the interpolation operators remain dense. Developing a practical sparse three-dimensional multilevel method is left to future work.

This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have parameters not only in the function but also in the related set. We show that the solution function of the PVI is Lipschitz continuous with respect to the upper-level decision variables and the solution set of the optimization problem is nonempty and bounded. Moreover, we prove that the metric regularity of the constraints holds automatically, which allows us to characterize stationary points without any additional assumptions. A Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI with a novel scheme for updating the smoothing parameter. We prove the convergence of SIGA to a stationary point and validate the efficiency of SIGA by numerical examples from synthetic problems and portfolio management with real data.

We show that each hyperbolic once-punctured torus bundle has exactly one veering triangulation. We also find manifolds for which the exact number of veering structures is known and is greater than one. Finally, we give new examples of manifolds without veering triangulations.

We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded.

We study the preservation of spatial concavity and log-concavity for stochastic parabolic equations posed on bounded convex domains. For the linear noise, by using Minkowski convolutions, we establish the convexity preservation of solution almost surely. For the nonlinear noise, by using It\^{o} formula, the impact of noise is given. Moreover, we establish the relationship between the weighted (stochastic) Pr\'{e}kopa-Leindler inequality and properties of (stochastic) parabolic equation on bounded domain, which is new even for deterministic case.

We propose a globally convergent Newton method for convex composite optimization problems with polyhedral regularizers. The method rests on the observation that the proximal gradient step yields not only a new point but also a subgradient, which determines the face on which the linearization of the regularizer is exact. Minimizing a second-order model of the objective over this face produces a Newton direction that involves no generalized second-order information about the regularizer and is a descent direction for the objective itself. The method is therefore globalized by an Armijo line search on the original objective which, unlike forward-backward envelope techniques, requires only evaluations of the smooth component. We establish global convergence and local quadratic convergence to tilt-stable minimizers without any nondegeneracy condition, and show that the unit step is always accepted when the smooth component is quadratic, which yields a line-search-free method. The subproblems require only projections onto faces, which are explicit for the Lasso and nonnegative least squares and computable in linear time for the fused Lasso. Numerical experiments show that the method compares favorably with state-of-the-art solvers.

K\"uhn, Osthus and Treglown [\emph{J. Combin. Theory Ser. B} \textbf{100} (2010)] proved that every sufficiently large robust outexpander with linear minimum semi-degree contains a Hamilton cycle. In this paper, we extend their result to spanning subdivisions of a digraph $F$ in a robust outexpander $D$, with prescribed branch vertices and arbitrarily specified path lengths. In particular, the digraph $F$ need not be fixed. For an $n$-vertex digraph $D$, $F$ may have as many as a constant multiple of $n/\log n$ arcs. This holds provided that $F$ has sufficiently few vertices and the prescribed lengths satisfy suitable conditions. We further establish a more general result for mixed prescribed path lengths, which allows $F$ to have linearly many arcs when the shorter replacement paths together use only a small proportion of the vertices of $D$. As applications, we obtain spanning subdivision extensions of the asymptotic Nash--Williams theorem for digraphs and an asymptotically sharp Ore-type theorem for oriented graphs.

Authors, in an earlier work, proved that prime power subsequences of the polygonal numbers corresponding to $4$, $6$, $8$, and $10$ arise naturally as the numbers of $\mathbb{F}_q$-rational points of certain double Danielewski surfaces. In this work, authors generalize the previous construction and give a new family of affine varieties over finite fields such that the number of $\mathbb{F}_q$-rational points of these varieties give the prime power subsequence of every polygonal number sequence.

We prove that the polynomials counting locally free, absolutely indecomposable, rank 1 representations of quivers over rings of truncated power series have non-negative coefficients. This generalizes the positivity of toric Kac polynomials to higher depth. The proof goes by inductively contracting/deleting arrows of the quiver and is inspired by a previous work of Abdelgadir, Mellit and Rodriguez-Villegas on toric Kac polynomials. We also relate counts of absolutely indecomposable quiver representations in higher depth and counts of jets over fibres of quiver moment maps. This is expressed as a plethystic identity involving generating series of these counts. In rank 1, we prove a cohomological upgrade of this identity, by computing the compactly supported cohomology of jet spaces over preprojective stacks. This is reminiscent of PBW isomorphisms for preprojective cohomological Hall algebras. Finally, our plethystic identity allows us to prove two conjectures by Wyss on the asymptotic behavior of both counts, when depth goes to infinity.

We study a mathematical model of in situ leaching of rare metals, in which the joint filtration of two liquids is governed by the microscopic model $\mathbb{A}^{\varepsilon}$. A key difficulty is the unknown (free) boundary $\Gamma(r)$ between solid and liquid components, determined by an additional condition on $\Gamma(r)$; no standard methods exist for this nonlinear problem. To resolve it, we apply the fixed point theorem. For a given function $r(\boldsymbol{x},t)$ from a set $\mathfrak{M}_{(0,T)}$ of sufficiently smooth functions describing the skeleton structure, we consider the auxiliary problem $\mathbb{B}^{\varepsilon}(r)$: an elliptic system for displacements of the liquid and solid components coupled with parabolic equations for the acid concentration. Selecting the weak solution of minimal smoothness, we apply the homogenization method to pass from the microscopic to the macroscopic description. The resulting macroscopic model $\mathbb{H}(r)$ contains a homogenized boundary condition that expresses the normal boundary velocity $V_{N}=\partial r/\partial t$ as a linear function of the acid concentration $c$. Since $c$ depends on $r$ via an operator $\mathbb{F}\colon\mathfrak{M}_{(0,T)}\to\mathfrak{M}_{(0,T)}$, we prove that $\mathbb{F}$ is Lipschitz continuous and, by Banach's theorem, possesses a unique fixed point $r^{*}$, which yields the unique solution $\mathbb{H}=\mathbb{H}(r^{*})$.

We study divisorial ideals and the $v$-class semigroup of a Noetherian domain satisfying Serre's condition $(S_2)$. We prove that strict closedness is equivalent to Clifford $v$-regularity and describe the structure of the associated $v$-class semigroup. We use these results to characterize finite generation of the class semigroup and Boole $v$-regularity. We also construct prime ideal decompositions of integrally closed divisorial ideals and clarify the relationship between these decompositions and local Boole $v$-regularity. Finally, we study polynomial extensions and use them to construct Boole $v$-regular and locally Boole $v$-regular domains in arbitrary dimension.

It is an open question whether the Lie algebra $\mathrm{HH}^1(B)$ of a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ is simple only if the block is nilpotent with an elementary abelian defect group of order at least $3$. We show this to be the case if $B$ has a normal abelian defect group. Another open question is whether $\mathrm{HH}^1(kS_n)$ is a solvable Lie algebra, where $S_n$ denotes a symmetric group on $n$ letters. We give some general solvability criteria for $\mathrm{HH}^1(B)$ of blocks $B$ with a normal defect group, and, making use of results of Chuang-Kessar and Chuang-Rouquier, we deduce that if $B$ is a block of a symmetric group algebra with an abelian defect group, then $\mathrm{HH}^1(B)$ is solvable.

A path or cycle in a $2$-edge-coloured graph is alternating if consecutive edges have different colours. These objects are natural analogues of paths and cycles in graphs, and even more so in digraphs. A well-studied class is that of acyclic $2$-edge-coloured graphs, that is, those forbidding alternating cycles. In particular, (acyclic) $2$-edge-coloured graphs generalize directed (acyclic) graphs (DAGs), but the complexity of problems on DAGs is much better understood than that of the corresponding problems on acyclic $2$-edge-coloured graphs. We investigate the complexity of two such problems, whose directed counterparts are well-known. Our first result is an $n^{O(k)}$-time algorithm for the alternating $k$-linkage problem on acyclic $2$-edge-coloured graphs. This problem asks for $k$ vertex-disjoint alternating paths joining $k$ prescribed pairs of vertices. This generalizes the directed $k$-linkage algorithm for DAGs. For every fixed $k$, this also yields a polynomial-time algorithm for finding $k$ internally vertex-disjoint alternating $(s,t)$-paths, solving an open problem by Abouelaoualim et al. from 2008. We then consider the alternating chromatic number of $2$-edge-coloured graphs, defined as the minimum number of parts in a vertex partition such that each part induces an acyclic subgraph. This notion is analogous to the dichromatic number of digraphs, where it is known that deciding dichromatic number $2$ is NP-complete even on tournaments. Our second result is that deciding alternating chromatic number $2$ is NP-complete on complete $2$-edge-coloured graphs. Equivalently, deciding whether a graph can be partitioned into two induced threshold graphs is NP-complete.

We establish a new dynamical convergence-divergence criterion for the existence of Lipschitz functions with continuous sub-actions, over general dynamical systems. This unifies and generalises various versions of the Ma\~n\'e cohomology lemma, in the setting of uniformly expanding maps, and systems with indifferent fixed points. One consequence is that for any compact metric space $(X,\rho)$, and Lipschitz Ruelle-expanding map $T:X\to X$, a metric transform $\omega$ satisfies Dini's criterion $\int_0 \omega(s)s^{-1}\, ds <\infty$ if and only if every function that is Lipschitz for $\omega\circ\rho$ has a continuous sub-action. Another consequence is that if $T$ is an expanding circle map of Manneville-Pomeau type $\alpha\in(0,1)$, then $\omega$ satisfies the convergence criterion $\int_0 \omega(s)s^{-1-\alpha}\, ds <\infty$ if and only if every function that is Lipschitz for $\omega\circ\rho$ has a continuous sub-action. For the latter class of maps $T$, with $\omega$ satisfying the convergence criterion, we prove a Liv\v{s}ic theorem for functions that are Lipschitz for $\omega\circ\rho$: if $\int f\, d\mu=0$ for all invariant measures $\mu$, then there exists a continuous function $\varphi$ such that $f=\varphi\circ T-\varphi$, where $\varphi$ is Lipschitz with respect to $D_\alpha(\omega)\circ\rho$, and $D_\alpha(\omega)(t)= \int_0^t \omega(s)s^{-1-\alpha} ds$.

We construct complete smooth K\"ahler metrics on $\mathbb{C}^n$ , $n\ge2$, with nonnegative Ricci curvature, Euclidean volume growth, and nonunique tangent cones at infinity. More generally, we prescribe the entire tangent cone set within a class of smooth Hopf cones, using a compact connected family of positive transverse Ricci forms on $\mathbb{C}\mathbb{P}^{n-1}$. We identify all metric blow-downs and their Ricci currents, compute the exact asymptotic volume ratio, and classify isometries between the model cones. In particular, the K\"ahler condition does not ensure uniqueness at infinity under nonnegative Ricci curvature and Euclidean volume growth, even in complex dimension two.