数学前沿

For the generalized Constantin-Lax-Majda model with a parameter $a$, we investigate novel scenarios of self-similar finite-time blowups that arise from a new setting where the smooth initial data satisfy a certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.

We extend the theory of commutative exact algebras in braided finite tensor categories (Shimizu-Yadav) to superalgebras, motivated by extensions of vertex operator (super)algebras. For a braided category $\C$ let $\B=\C^{\gr}$ be the category of $\mathbb Z/2$-graded objects with the Koszul-signed braiding. It is the underlying category of the supercategory of Creutzig-Kanade-McRae. A supercommutative superalgebra $A=A_0\oplus A_1$ in $\C$ is the same as a commutative algebra $\bA$ in $\B$, and super-local modules are the local $\bA$-modules. Hence rigidity, pivotality and ribbon statements are instances of the ordinary theorems, with the ribbon condition $\theta_{A_0}=\id$, $\theta_{A_1}=\varepsilon\,\id$, where $\varepsilon=\pm1$ is fixed by the statistics of the odd part. For $\C$ non-degenerate and $\bA$ haploid and exact we prove that the category of super-local modules is a braided finite tensor category whose M\"uger center is $\mathsf{sVect}$. It is the centralizer of a fermion in a non-degenerate category obtained by condensing in $\C\boxtimes(\text{toric code})$. If $\C$ is ribbon and $\bA$ is symmetric Frobenius it is ribbon. We construct examples from graded simple currents, treat the free fermion and the symplectic fermion, and compare with Creutzig-Kanade-McRae and Creutzig-McRae-Shimizu-Yadav.

We show that the continuous functional calculus is computable. As consequences we obtain the computable compactness of the spectrum of any computable normal element of a computably presented $\mathrm{C}^*$-algebra, the existence of effective approximate units for computably presented $\mathrm{C}^*$-algebras, and an effective version of the Spectral Theorem for compact operators on separable Hilbert spaces.

In this paper, we classify profinite groups whose closed subgroups are subnormal of defect at most two, providing profinite versions of theorems due to Heineken and to Mahdavianary. We also establish that if G is a finitely generated periodic pro-{\pi} group, where {\pi} is a finite set of odd primes, and all its procyclic subgroups are subnormal of defect at most 2, then G is a finite {\pi}-group in which every abelian subgroup is subnormal of defect at most 2.

Let $X\subset\mathbb{R}^{d_X}$ and $Y\subset\mathbb{R}^{d_Y}$ be open, where $d_X,d_Y\geq2$ and $1\leq p\leq\min\{d_X,d_Y\}$. Let $\Phi:X\times Y\to\mathbb R^p$ be a smooth mapping, and suppose that $D_x\Phi$ and $D_y\Phi$ have rank $p$. For compact sets $A\subset X$ and $B\subset Y$, we give lower bounds on $\dim_H A+\dim_H B$ ensuring that the pinned configuration set $$\Delta_\Phi^y(A)=\{\Phi(x,y):x\in A\}$$ has positive Lebesgue measure or nonempty interior for some $y\in B$. The dimensional thresholds are expressed in terms of uniform $L^2$-Sobolev estimates for the generalized Radon transforms associated with $\Phi$. Our results extend the Fourier integral operator approach developed by Greenleaf, Iosevich, and Taylor for unpinned $2$-point configuration problems to the pinned setting. We also prove a nonlinear projection theorem that remains valid even when the Peres-Schlag transversality condition fails. Under the smoothing hypotheses used for the pinned configuration theorem, we obtain lower bounds for the Hausdorff dimensions of the corresponding slices of $A$ by the level sets of $\Phi$. Applications include distances from points to hyperplanes, spheres and lines; multiparameter distance sets; and a restricted projection family introduced by Harris.

A wiring diagram is an arrangement of $m$ pseudolines, every two of which cross exactly once. Reading its crossings from left to right encodes it as a reduced word of the permutation $w_0\in S_m$ that reverses $1,\dots,m$, unique up to commutations, and a braid move $\sigma_i\sigma_{i+1}\sigma_i\leftrightarrow\sigma_{i+1}\sigma_i\sigma_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\operatorname{VC}(G)$, where $\operatorname{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding distance in the mutation graph is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge4$, and by duality in every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge2$.

We learn schedules for a continuous penalty relaxation of MAX-CUT by backpropagating through an unrolled classical solver. A state-blind schedule generalizes to instances substantially larger than those used for training. A state-adaptive bilinear schedule also generalizes out of distribution, outperforming its untrained baseline across 15 seeds ($t=4.1$). By contrast, an additive ablation with identical inputs and output heads is statistically indistinguishable from the bilinear model in distribution but performs worse than its own baseline out of distribution ($t=-11.7$). This separation appears without regularization and persists across a $30$ round weight-decay sweep: the bilinear advantage remains significant at every tested value, whereas the additive model never recovers. The results in this setting isolate how solver state enters the schedule, rather than state adaptivity alone, as the architectural distinction associated with out-of-distribution transfer.

How many ways are there to select $k$ objects from $n$ types when type $i$ is available in only $s_{i}$ copies? For equal bounds, this is de Moivre's problem of obtaining a prescribed total with $n$ dice, now a textbook application of the principle of inclusion-exclusion. For arbitrary bounds, the answer is encoded by the rank polynomial $\prod_{i=1}^{n}(1+x+\cdots+x^{s_{i}})$. Its coefficients enumerate bounded compositions and submultisets, give, after normalization, the probability mass function of a sum of independent discrete uniform random variables, and are the weight multiplicities of a tensor product of $\mathrm{SU}(2)$ representations. Most of these interpretations are classical but dispersed; here we place them in a common notation and document their provenance. A complement involution proves symmetry and boundary identities. For arbitrary bounds, lower-half first differences give both symmetric-chain starts and mixed-spin multiplicities, and the chain-length and irreducible-dimension profiles agree. Convolution gives log-concavity and the central mode. For equal bounds with fixed $s \geq 1$ and even $ns$, local expansions give central asymptotics with explicit $1/n$ and $1/n^{2}$ corrections for the coefficient and its first difference. A Poisson limit gives the first boundary crossover of these differences at scale $k \sim \lambda n^{s/(s+1)}$, with factor $e^{-\alpha\lambda^{s+1}}$ when the proportion of bounds equal to the fixed minimum $s \geq 1$ tends to $\alpha>0$. Central first differences include the $s$-Catalan numbers, themselves Littlewood-Richardson coefficients; central layers yield maximum-size constant-sum codebooks for hash-based signatures. Under the polynomial binomial convention, the full inclusion-exclusion transform vanishes, and the omitted terms sum to the negative of the number of bounded compositions.

We construct a function $W:\mathbb{R}^{3\times3}\to\mathbb{R}$ which is frame-indifferent, $W(QF)=W(F)$ for all $Q\in\mathrm{SO}(3)$, rank-one convex, but not quasiconvex. In a neighbourhood of $\mathrm{SO}(3)$ the function is an explicit polynomial of degree four in the strain $F^TF-\mathrm{Id}$; the failure of quasiconvexity occurs at $F=\mathrm{Id}$ through a third-order mechanism of \v{S}ver\'ak type, with test functions whose gradients are arbitrarily close to $\mathrm{Id}$. The polynomial is obtained from a geometrically linear example, a rank-one convex, non-quasiconvex quartic polynomial of the symmetric part of the gradient, by evaluating it at the strain $\frac12(F^TF-\mathrm{Id})$, adding a uniaxial prestress and a cubic correction, and enlarging the quartic term. The local function is extended to all of $\mathbb{R}^{3\times3}$ by adding a frame-indifferent polyconvex function which vanishes near $\mathrm{SO}(3)$. This work was carried out with the assistance of the AI model Claude Opus 5.5; the proofs of the main results have been formally verified in Lean 4.

Assuming the Riemann hypothesis, we study a distinguished self-adjoint extension of the differentiation operator on a Hilbert space defined by the Weil quadratic form localized to a finite interval. The local quadratic form is determined by data from finitely many places, without requiring the zeros of the Riemann zeta function as input. We prove that the positive eigenvalues of this extension converge to the distinct positive ordinates of the nontrivial zeros of the Riemann zeta function as the interval expands. We also prove that the associated normalized spectral entire functions converge locally uniformly on the complex plane to the canonical product over these distinct ordinates. The key step is an identity expressing the inverse of the local extension as a compression of the inverse of a global self-adjoint operator.

In this paper, we explore various equivalences of Ekeland's variational principle within the framework of group-invariant mappings. We introduce and analyze several key theorems, including the Drop theorem, the Petal theorem, Caristi-Kirk fxed-point theorem, and Takahashi's theorem, all of them within this context. Moreover, we extend the classical Drop theorem and Petal theorem to a more generalized setting. We also demonstrate the practical signifcance of these findings through numerous applications to diverse areas of mathematics. In particular, in the context of partial differential equations, we explore their implications on the solution of the Plateau problem, and in control theory. We also extend the classical Pontyargin maximum principle.

We consider a real even Hecke-Maass cusp form $\phi$ on the modular surface, with Laplace eigenvalue $1/4+T^2$. We prove that its restriction to any fixed compact segment $\beta$ of the vertical geodesic above $i$ has $\gg_{\beta,\varepsilon} T^{1/6-\varepsilon}$ sign changes, yielding the same lower bound for the number of distinct inert nodal domains meeting the segment. Building on ideas of Jang-Jung and Jung-Young, we study correlations between values at points on the vertical geodesic separated by distances of order $T^{-1}$. Using arithmetic quantum unique ergodicity and Dirichlet-polynomial estimates, we show that the correlation is negative for a suitable choice of separation. An elementary counting argument and the Iwaniec-Sarnak supremum bound then give the claimed lower bound.

Scenario trees are widely used to discretize multistage stochastic programs into forms that are tractable for numerical optimization. Yet, traditional scenario-tree construction methods are typically based on statistical similarity between the original uncertainty process and its discretization, and may fail to preserve the aspects of the uncertainty that matter most for decision-making. This paper shows that, under a fixed probability measure and deterministic feasibility constraints, the gap between the optimal value of the original multistage stochastic program and that of its scenario-tree approximation decomposes exactly into a sum of nodewise regret terms, each measuring the optimality loss at a particular node. This decomposition provides a clear target for scenario-tree construction: minimizing total regret across the tree. Motivated by this result, we propose a practical two-phase construction method for repeated stochastic optimization. In an offline phase, computationally inexpensive surrogates for decision regret are learned from data, and in the online phase, these surrogates guide scenario-tree construction. The resulting framework retains the structured optimization and constraint-handling capabilities of multistage stochastic programming while exploiting learned information about the uncertainty to make more effective use of a limited scenario-tree budget. Numerical experiments on multistage stochastic storage and production-planning problems show that, for a fixed tree size, the proposed method constructs higher-quality scenario trees and achieves better closed-loop performance than traditional scenario-tree construction methods.

We study invariant position distributions in the Smoluchowski-Kramers approximation for Langevin equations driven by rotationally symmetric $\alpha$-stable noise, with $\alpha\in(1,2)$. Under global Lipschitz continuity and dissipativity of the drift, we establish a Wasserstein-1 bound of order $m^{1/\alpha}$ between the invariant position distribution of the underdamped system and the invariant distribution of its overdamped limit, where $m>0$ is the mass parameter. The proof combines a corrected-position test function with a regularized Stein equation. Cancellation of the transport and friction terms reduces the comparison to a first-moment estimate for the stationary velocity.

A fundamental question in $p$-adic symplectic geometry and $p$-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers $\mathbb{Q}$, if we view them as $p$-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and $p$-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the $p$-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a $p$-adic theory of integrable systems and symplectic geometry.

We compute the Grothendieck group of coherent sheaves for complete split toric surfaces having a smooth 2-dimensional cone in its fan. We show that it is free abelian, of rank equal to the number of rays in the fan. We give a partial generalization of this to all complete split toric varieties.

In this paper, we study the iteration over the real and complex loci of a specific rational map, found by Grigorchuk and Zuk. This two dimensional map arises as the renormalization map which yields the density of states measure associated with the iterated monodromy group of the Basilica Julia set. It admits two natural complex currents, and we construct their real analogs and show that they are laminar. Finally, this yields some consequences on the properties of the density of states measures and its support.

We calculate the Galois group of $f = x^n + ax^k + b \in \mathbb{Z}\left[ x \right]$ over $\mathbb{Q}$ for all $1 \leq \left| ab \right| \leq 3$, $n$ sufficiently large and $k$ coprime to $n$. In particular, we prove that the Galois group of $f$ is the maximal one that its factorization type allows. We give a criterion for the Galois group of a factor of a trinomial to be symmetric, with an eye towards future applications. To the best of our knowledge, the only previously known explicit example of polynomials of degree $n$ with bounded coefficients and Galois group $S_n$ is $x^n - x - 1$, while recent results show that the probability that a random polynomial of degree $n$ with reasonably bounded coefficients has Galois group $S_n$ tends to $1$ as $n$ tends to infinity. Using our calculation, one can construct many new such explicit examples.

An $\mathrm{ISK}_4$ is a graph that is isomorphic to a subdivision of $K_4$; a \textit{diamond} is $K_4$ minus one edge; and a \textit{bowtie} consists of two triangles sharing exactly one vertex. L\'ev\^eque, Maffray, and Trotignon~(2012) conjectured that every $\mathrm{ISK}_4$-free graph is $4$-colorable. Chen, Chen, Cui, Feng, and Liu~(2021) proved this for $\{\textup{diamond},\textup{bowtie}\}$-free graphs. We prove this conjecture for diamond-free graphs. Moreover, the bound is optimal.

We propose a neural exploratory framework for variational problems over graphons, i.e., symmetric measurable functions $[0,1]^2 \to [0,1]$, that arise as limits of dense graph sequences. Such problems are central to two parts of the mathematical literature: extremal graph theory, which studies graph parameter optimisation under given restrictions, often for those graphs with a large number of vertices, and the large deviation theory of dense random graphs, which characterises the structure of rare events through constrained graphon optimisation. We represent graphons by implicit neural networks and optimise graphon objectives by gradient descent. Our design combines three ingredients: a multi-scale sinusoidal residual architecture biased toward sharp, step-like graphons; an embedded solver that enforces a single empirical density constraint and is differentiated by implicit differentiation; and symmetry-aware Monte Carlo estimators. On generalised Tur\'an problems that previously required substantial human effort, our framework rediscovers known optimal graphons without human intervention. Applied to open instances, it produces candidate optima, including a previously unreported family of extremal structures. Applied to the variational problems from the large deviation theory of Erd\H{o}s--R\'enyi random graphs, the framework produces new candidate optimal graphons across both upper- and lower-tail regimes for multiple pattern graphs; for the case of the triangle graph and the upper-tail regime, these candidates improve upon the best-known reference construction of Lubetzky and Zhao. The structures found by our framework have also led to new theorems on the optimal graphons for both families of problems.