数学前沿

In this paper, we study perfectoid ideals using the reduction modulo $p$ of perfectoid rings. We show that perfectoid ideals correspond to ideals in the reduction modulo $p$ on which Frobenius is surjective. We establish the existence of universal $p$-adic power series whose evaluations, together with their compatible $p$-power roots, generate perfectoidizations of ideals. In the special case of the perfectoidization of $\mathbb{Z}_p^{\mathrm{cycl}}\langle t^{1/p^{\infty}}\rangle/(t-1)$, we describe a linear algebra algorithm for computing finite approximations of such generators. We illustrate this by an explicit approximation modulo $8$ of an element $\sigma\in (t-1)_{\mathrm{perfd}}\subseteq \mathbb{Z}_2^{\mathrm{cycl}}\langle t^{1/2^{\infty}}\rangle$ which admits compatible sequences of $2$-power roots and satisfies $\sigma\equiv t-1\pmod{2}$.

We study sober rings, a class of commutative rings in which every prime non-maximal ideal is not the intersection of the prime ideals strictly containing it. We establish basic structural properties of sober rings and investigate connections to other known classes of rings. We show that sobriety is preserved under quotients and localizations and we examine the behavior of sobriety for polynomial extensions.

A theory can accommodate a counterexample by adding an exception. The harder problem is deciding how far the exception should extend and which predictions should survive. We present DeFAb, a dataset and symbolic verification framework built from curated biological knowledge in the Gene Ontology. Models fill gaps in a theory or propose exceptions to its rules. A verifier checks the consequences of each edit, allowing different solutions to receive credit. Simple symbolic repairs solve all 336 tasks in the released evaluation panel. This result exposes a limitation of the revision test. An exception tied to one entity can succeed without identifying a pattern that applies to other cases. We develop a contextual task that requires an exception to reach the relevant cases and preserve predictions elsewhere. Two planned studies compare training with verified and rejected examples, and revision with and without feedback about consequences. DeFAb makes the scope of a proposed explanation testable, providing a controlled step toward studying theory revision in scientific reasoning.

This paper studies foliations on abelian surfaces in positive characteristic. We provide an explicit description of all smooth foliations, linear and turbulent, and establish criteria for being $p$-closed. We explicitly determine the moduli spaces of such foliations. We give an application to the construction of $p$-closed foliations, with mild singularities, on Kummer $K3$-surfaces in positive characteristics.

The cyclicity index is used to study the periodic behavior of powers of tropical matrices. It is determined by the structure of the critical subgraph of a matrix, which consists of the cycles with the largest average weight. In this paper bijective linear transformations of tropical matrices that preserve an individual value $k\neq 2$ of this invariant are studied.

We study the conformal Einstein-Dirac system on the round sphere $S^m$, $m\geq3$, with coupling coefficient $1+\eps k$, where $k\in C^2(S^m,\R)$ and $\eps$ is small. Morse and degree conditions yield solutions with positive scalar component, and a generic nondegeneracy condition gives multiplicity modulo spinorial symmetries. For a suitable fixed coefficient, we obtain a noncompact solution sequence with infinitely many distinct energy values. A Kazdan-Warner identity gives nonexistence for a smooth Morse potential and its small $C^2$-perturbations, even with positive coupling. The existence proofs use a local reduction in the interior of the positive cone, accommodating the negative powers occurring when $m\geq5$.

We study the stochastic harmonic map heat flow into the sphere driven by a Stratonovich noise term that formally preserves the spherical constraint by construction. Starting from a fully discrete, implementable numerical scheme, we prove its convergence to the novel solution concept of probabilistically strong, analytically energy-variational solutions. This yields the first existence result for probabilistically strong solutions in the three-dimensional setting on the prescribed stochastic basis. In addition, we establish weak-strong uniqueness and illustrate our theoretical findings with numerical experiments.

Every nonempty connected finite-dimensional CW topological semigroup admits a continuous associative multiplication on its universal cover that lifts the multiplication of the base. Such a multiplication is unique up to a deck transformation: in fact, any two associative lifts are conjugate whenever they exist. We show that the finite-dimensional hypothesis cannot be dropped in general: a connected, countable, infinite-dimensional CW topological semigroup is constructed for which continuous lifts of multiplication exist, but none is associative. The counterexample has nonabelian fundamental group isomorphic to Thompson's group $F$. Thus finite dimensionality is sufficient for associative lifting in the CW setting, whereas the unrestricted infinite-dimensional statement fails.

We prove that every finite simple maximal planar graph of maximum degree at most six admits a total eight-coloring. We first show that every proper vertex four-coloring of a subcubic graph extends to a total five-coloring. A curvature inequality for boundary walks then gives a matching decomposition of four-connected triangulations with degrees five and six, extending their prescribed vertex four-colorings to total eight-colorings. In a minimum counterexample, two local boundary colorings force the degree-four vertices to have disjoint neighborhoods. A characterization of the local extension obstructions permits these vertices to be restored simultaneously. The proof uses Hamiltonian-connectedness of four-connected planar graphs and the computer-assisted Four-Color Theorem; it requires no further computation.

Let $L$ be a thin distributive lattice of rank $m+1$, and let $I_L$ be its join-meet ideal in the polynomial ring $S_L$. For $A\subseteq\{1,\dots,m\}$, let $J_A$ be the ideal obtained from $I_L$ by identifying, for every $i\in A$, the variables of the two elements of $L$ of rank $i$. We prove that $J_A$ is prime if and only if $A$ contains no turning rank and does not contain both ranks adjacent to a turning rank. In this case, $S_L/J_A$ is the toric ring obtained from Hibi's parametrization of the Hibi ring of $L$ by identifying suitable pairs of variables, and for an arbitrary $A$ we describe the saturation of $J_A$ as a lattice ideal.

Let $0<a_1<\cdots<a_n$ be integers with $n\ge 2$ and $\gcd(a_1,\ldots,a_n)=1$. Their Frobenius number $F(a_1,\ldots,a_n)$ is the largest integer that cannot be expressed as a nonnegative integer combination of $a_1,\ldots,a_n$. Erd\H{o}s and Graham proved the classical bound $F(a_1,\ldots,a_n)\le 2a_{n-1}\lfloor a_n/n\rfloor-a_n$, which depends on the two largest generators. Dixmier subsequently proved their conjecture $F(a_1,\ldots,a_n)\le a_n^2/(n-1)$, a bound depending only on $a_n$ and $n$ that removes the factor $2$ from the corresponding quadratic estimate. Following the Erd\H{o}s--Graham approach, we remove the factor $2$ while retaining the dependence on $a_{n-1}$, and prove $F(a_1,\ldots,a_n)\le a_{n-1}\lceil a_n/(n-1)\rceil-a_n-1$. This bound can be substantially smaller than Dixmier's when $a_{n-1}$ is small relative to $a_n$. For every $n\ge3$, we also construct infinitely many instances satisfying the reverse inequality, which shows that the bound is sharp.

We generalize results of [1] concerning two partial orders on Standard Young Tableaux: the dominance order and the box order. Extending their definitions from tableaux of a fixed shape to the set Tr of all Standard Young Tableaux of a fixed size r, we prove that these orders coincide in this more general setting. We also investigate basic properties of these orders. We further show that this result extends to Littlewood-Richardson tableaux whose skew shapes are rook strips. We briefly mention applications to invariant subspaces of nilpotent linear operators.

Dirac equations describe the relativistic evolution of quantum particles. Using the point-process stochastic calculus of piecewise-deterministic Markov processes (PDMPs) on general state spaces, which naturally allows additional Feynman-Kac potentials, we develop a probabilistic representation for the nonlinear Dirac equations in $1+1$ spacetime dimension with cubic nonlinearity and $L_\infty$-type problem data using a multi-type branching telegraph process together with both continuous and jump-type multiplicative functionals. Our main theorem leads to a classification result which, when applied to the specific Soler model, gives a sharp finite-time admissibility of the probabilistic representation. By extending the one-step representation to a multi-step, time-marching representation using an analytically-derived a priori $L_\infty$ estimate, we observe a ``local probabilistic representation, but global PDE well-posedness" phenomenon. We proceed to propose a time-marching Monte Carlo algorithm as a global numerical solver for the Soler system with a rigorous convergence proof and establish an additional stability result on the PDE side. Finally, while the underlying process parameters do not affect the validity of the probabilistic representation, we observe their potential for variance reduction in the Monte Carlo methods.

We introduce the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\leq p\leq \infty$. We then establish various properties of these spaces, including their behavior under complex interpolation and duality, and their invariance under Fourier integral operators. We also obtain Sobolev embeddings, equivalent characterizations, and a molecular decomposition. These spaces are used in the companion article arXiv:2502.02511 to determine the sharp $\mathcal{H}^{1}(\mathbb{R}^{n})$ and $\mathrm{bmo}(\mathbb{R}^{n})$ regularity of wave equations with rough coefficients.

Finding the roots of a polynomial over a finite field $\mathbb F_q$ in deterministic polynomial time is a long-standing open problem. Classical deterministic methods work in the multiplicative group $\mathbb F_q^\times$ and are efficient only when $q-1$ is smooth, which is rarely the case. We propose a method that replaces $\mathbb F_q^\times$ by the groups of rational points of an elliptic curve $E$ and of its quadratic twist $\widetilde E$. The computation splits into an offline phase, which depends only on $\mathbb F_q$ and outputs a short \emph{elliptic preprocessing datum} $\mathcal D$, and an online phase, which processes the input polynomial. Writing $\Xi(\mathcal D)$ for the largest prime factor of $\lvert E(\mathbb F_q)\rvert\cdot\lvert\widetilde E(\mathbb F_q)\rvert$, the online phase is deterministic and finds the roots of any $f\in\mathbb F_q[X]$ of degree $n$ within $\widetilde O(\Xi(\mathcal D)^2\,n\log^2 q+n^\omega\log q+n\log^3 q)$ operations in $\mathbb F_q$. A precursor of our method is an algorithm by R\'onyai, which recovers the roots through Galois computations over number fields, relies on GRH, and comes with a polynomial bound of unspecified degree; in contrast, our online phase works entirely over $\mathbb F_q$ and is unconditional. Moreover, $\mathcal D$ can be certified deterministically and unconditionally in time polynomial in $\log q$ and $\Xi(\mathcal D)$. We conjecture that every $\mathbb F_q$ of large enough characteristic admits a datum with $\Xi(\mathcal D)\leq(\log q)^{O(1)}$, and support this with a heuristic and with experiments. Constructing such a datum in polynomial time remains open.

Efficient matrix-free preconditioning remains a challenge for discontinuous Galerkin spectral element discretisations of elliptic problems, particularly for general element shapes with collapsed-coordinate bases, such as prisms, tetrahedra and pyramids. Multigrid provides a promising framework, but its efficiency depends strongly on the effectiveness and computational cost of the smoother. For modal bases, point Jacobi can provide insufficient smoothing, whereas applying block Jacobi through stored dense inverses incurs substantial storage and computational costs. We propose a matrix-free approximate elemental inverse, an approximation to the inverse of each elemental diagonal block, constructed from inverse modal-to-nodal transforms and an element-dependent sparse matrix. Its application exploits generalised sum-factorisation and partial sum-factorisation in collapsed coordinates, avoiding the explicit storage and multiplication of large dense matrices. The method is evaluated as both a standalone preconditioner and a multigrid smoother on hexahedral, prismatic, tetrahedral and pyramidal elements. Although the proposed approximation generally increases iteration counts compared to exact block Jacobi, the cost per iteration is far lower. This trade-off yields substantial reductions in total solution time for hexahedral and prismatic elements, while achieving comparable performance for tetrahedral and pyramidal elements. This study can also serve as a practical guide for tuning the convergence of the discontinuous Galerkin interior-penalty Poisson solver.

We study the magnetization vector of mixed spherical spin glasses in a weak external field, in the replica symmetric phase. We derive a coordinatewise expansion of the magnetization which includes the leading cubic correction to the quadratic approximation, with an error that is $o(N^{-1/2})$ per coordinate. We also prove a corresponding high-moment bound for the full vector. Finally, our companion paper applies these estimates to the effective Hamiltonian obtained by restricting a tilted spherical spin glass to a narrow band around a TAP critical point. The resulting sharp approximation is designed for applications in which normalized Wasserstein accuracy is insufficient, including total-variation sampling by stochastic localization.

We extend the results of \cite{FLLLWW20} to the super setting. More precisely, we define two families of coideal sub-superalgebras of quantum affine superalgebras of type $A$, called $\jmath$- and $\imath$-quantum affine superalgebras respectively, and establish Schur--Weyl duality between them and affine Hecke algebras of type $C$.

The famous algebraicity conjecture of Gross and Zagier for the values at CM points of higher Green's functions was recently proven by Bruinier, Li and Yang. Motivated by this the values of the automorphic Green's function at pairs of special points in hyperbolic $3$-space $\mathbb{H}^3$ were studied by Herrero, Imamo\={g}lu, von Pippich and Schwagenscheidt, and an analogous algebraicity result for traces of Green's functions was established. In this paper, we consider higher automorphic Green's functions $G_{s}(P,Q)$ on hyperbolic $q$-space $\mathbb{H}^q$ for $q\geq 2$ when the spectral parameter $s$ is a positive even integer, and prove an explicit finite formula for their traces over geodesic hypersurfaces in one of the variables. This formula provides a geometric interpretation of the traces of Green's functions for every choice of a point in $\mathbb{H}^q$ for the second variable. As an application, we obtain rationality results when the second variable is evaluated at a special point providing further evidence that such results for traces of Green's functions do not depend on any underlying complex structure.

Let $T_q(n,r)$ be the largest size of a vertex set inducing a triangle-free subgraph of the graph on $[q]^n$ whose edges join words at Hamming distance $r$. We study this quantity for fixed alphabet size and fixed distance, with particular emphasis on the ternary cube. Alphabet restriction transfers the binary fixed-distance theorem of Balogh, Chen, and Li to give $T_q(n,2h)=O_h(q^n/n^h)$. We extend the known distance-one formula and distance-two order to arbitrary integer alphabet sizes, obtaining $T_q(n,1)=2q^{n-1}$ and $T_q(n,2)=\Theta_q(q^n/n)$, and establish $T_3(n,4)=\Theta(3^n/n^2)$. For general fixed distance, we prove $$T_q(n,r)=\Omega_{q,r}\!\left( \frac{q^n\sqrt{\log n}}{n^{(r+\lfloor r/2\rfloor)/2}} \right)\qquad(q\ge3),$$ while we show that $T_q(n,r)=\Omega_{q,r}(q^n/n^{\lfloor(q-1)r/q\rfloor})$ holds for prime-power $q$ and $r\ge2$ via the extended BCH parity-check constructions. At distance three, we construct coordinate weights avoiding bounded-coefficient three-term relations and obtain $$T_q(n,3)\ge q^n/[n\exp(O_q(\sqrt{\log n}))].$$ Our main ternary construction combines cap partitions, multicolor triangle Ramsey colorings, and a quadratic map to obtain $$T_3(n,3)=\Omega\!\left( \frac{\log\log n}{\log n}\,3^n\right).$$ A coordinate-line counting argument gives the complementary upper bound $T_3(n,3)\le(5/27+O(n^{-1}))3^n$, together with an explicit bound for every fixed $q\ge3$. Whether the ternary density tends to zero remains open.