数学前沿

Plabic tangles define geometric promotion maps between configurations and, after choosing a lift, between Grassmannians. The main conjecture in \cite{Tangles} is that when the tangle is dominant and solvable the induced algebraic promotion is cluster quasi-homomorphism. This work extends the cluster perspective in two directions. First, we initiate the study of quantum promotions. We conjecture that all promotions defined from dominant solvable tangles are naturally lifted to a \emph{quantum cluster quasi homomorphism}, and prove for all cases in the literature that the were shown to be (classical) cluster quasi-homomorphisms. Secondly, we define the notion of cluster-Lusztig quasi-homomorphisms, which are cluster quasi homomorphisms that take Lustig's dual canonical basis elements to dual canonical basis element, up to a Laurent monomial in frozen variables, with clearing and normalization chosen uniformly on each content component. We prove the unexpected result that the classical and quantum $m$-unary star promotions fall into this category, and conjecture that all promotions coming from dominant solvable tangles share the same property. We then consider promotions via non solvable cores of intersection number higher than one. We use it to study facets of amplituhedron cells, and to show that some of them do not lie in zero loci of dual canonical basis elements, disproving a conjecture by T. Lam \cite{LamGP}.

We prove that the deformation functor associated to an algebra over a sufficiently nice Koszul operad can be lifted to an operadic formal moduli problem over the Koszul dual operad. We further show that this operadic formal moduli problem corresponds to the derived center of the algebra under the operadic Lurie-Pridham correspondence. As an application, we prove that the classical Gerstenhaber algebra structure on polyvector fields of a commutative algebra agrees with the derived center of the algebra as an abelian Poisson algebra.

An additive coloring assigns positive integer labels to the vertices of a graph so that adjacent vertices have different sums of their neighbors' labels. The additive chromatic number $\eta(G)$ is the least $k$ for which labels from $[k]$ suffice. We give an elementary orientation argument proving that every finite simple graph $G$ satisfies $\eta(G) \leq \Delta(G) + 1$. This argument yields a $(\Delta(G) + 1)$-additive-coloring in polynomial time and improves upon the current best bound of $\eta(G) \leq \Delta(G)^{2} - \Delta(G) + 1$, valid for $\Delta(G) \geq 2$, from Akbari, Ghanbari, Manaviyat, and Zare.

We prove a relative fundamental lemma for the unitary symmetric variety $GL_n(k')\backslash U_{2n}$, proving a conjecture of Leslie, Xiao, and Zhang. Our proof reduces to the Lie algebra and argues over function fields in positive characteristic, using methods from Ngo-Laumon and Ngo. In particular, we reinterpret orbital integrals in terms of the cohomology of relative affine Springer fibers and globalize this using unitary versions of Hitchin moduli spaces for symmetric pairs via a product formula. As the main global input, we prove a matching of perverse sheaves between relative Hitchin moduli spaces and their corresponding endoscopic variants.

Mechanistic attribution in Earth system science requires quantifying uncertainty and communicating its implications. Here, we develop an uncertainty-aware mechanistic attribution pipeline that constructs probabilistic fingerprints from ensemble estimates, including covariance-aware separation of competing mechanisms, and an evidence ladder that systematically varies the information provided to language models (LLMs). We evaluate this framework in an idealized stochastic system and a coupled El Ni\~no model using controlled interventions and ensemble data assimilation. In the idealized system, numerical separation increases GPT-4o's intervention-label agreement from 0.80 to 0.98, but not GPT-4o-mini's. In the coupled system, both LLMs identify seven of eight fingerprint-ambiguous events when given full fingerprints with verbal interpretations; GPT-4o's false ambiguity reports decrease from 10/82 to 1/82, without consistent gains in intervention-label agreement. For eight historical ENSO events, the uncertainty-aware condition agrees with the model-inferred mechanism labels in seven cases. These findings distinguish uncertainty assessment from its communication.

Let $q=p^n$, where $p>2$ is prime. We prove that every $q$-set $U\subseteq\mathrm{AG}(2,q)$ determining exactly $(q+3)/2$ directions is affinely equivalent to the graph of $x^{(q+1)/2}$. For $n\ge2$, we also prove that a $q$-set determining at most $(q+p)/2$ directions is either an affine $\mathbb{F}_p$-linear set or affinely equivalent to this graph. The classification for $n=1$ is due to Lov\'asz and Schrijver. For $n=2$, the classification and the stated stability result follow from results of G\'acs, Lov\'asz, and Sz\H onyi and the direction theorems of Blokhuis, Ball, Brouwer, Storme, and Sz\H onyi, completed by Ball. These direction theorems establish affine $\mathbb{F}_p$-linearity for functions with fewer than $(q+3)/2$ determined directions. Our proof uses R\'edei polynomials, results of G\'acs, Lov\'asz, and Sz\H onyi on their coefficients, and an earlier polynomial comparison lemma of the author.

We classify all abstract automorphisms of the elementary and full twisted Chevalley groups of types ${}^2A_\ell$ ($\ell\geq3$), ${}^2D_\ell$ ($\ell\geq4$), ${}^2E_6$, and ${}^3D_4$ over arbitrary commutative rings, for every invariant isogeny type. The coefficient automorphism $\theta$ has order dividing that of the diagram automorphism. We assume that $2$ is invertible and, for triality, that $3$ is invertible as well. Three features distinguish the answer from the usual inner--diagonal and coefficient decomposition. The elementary groups of types ${}^2A_3$ and ${}^2A_4$ admit an additive family of exceptional automorphisms supported at $\theta$-fixed maximal ideals with residue field $\mathbb F_3$. In type ${}^2D_4$, diagram automorphisms may vary on idempotent components, subject to compatibility with the involution and the isogeny lattice. In triality, the coefficient automorphisms preserve the trace operator $r\mapsto r+\theta(r)+\theta^2(r)$ but need not commute with $\theta$. We determine which elementary automorphisms extend to the full groups and describe the remaining central factors. A nonzero exceptional unitary factor extends precisely in the simply connected type ${}^2A_3$, giving a classification for the hyperbolic group $\operatorname{SU}_4(R,\theta;Q_4)$. The proof glues uniquely determined normalization data for a finite root frame and then recovers the coefficient action from root-subgroup relations.

This paper presents a framework for robust topology optimization of Kirchhoff-Love (KL) multi-patch shells using isogeometric analysis and non-matching meshes and provides a CAD-compatible method. The material distribution is optimized via a NURBS-based level set method, enabling accurate boundary representation and seamless export to standard CAD/CAM workflows. The shape derivative is computed using the speed method, yielding a volume-based formulation that avoids boundary approximations and aligns with the isogeometric principle of coarse, high-order discretizations. This eliminates the need for local refinement, adaptive quadrature, or trimming as the descent direction for the compliance and volume objectives are obtained through a biharmonic equation. The framework is extended to robust optimization under uncertainty, accounting for stochastic loading and material properties via Monte Carlo sampling and Gaussian Random Fields. Numerical examples demonstrate the method's efficiency and robustness across single- and multi-patch shells, reproducing literature results with fewer elements than traditional FEM or refined IGA approaches. The developed methods and tools can be directly extended to problems involving stress-based optimization and risk-averse metrics. The numerical implementations are open-source and available on GitHub.

We develop an adaptive time-stepping strategy for scalar auxiliary variable (SAV) schemes combined with variable-step BDF2 discretization for gradient flows. The strategy selects step sizes based on the instantaneous energy dissipation rate and enforces monotonicity of a computable symmetric energy through a step rejection mechanism. We show that the step-size rules used in practice correspond to equidistributing a monitor function along the energy curve, connecting energy-adaptive time stepping to the classical theory of r-adaptive mesh methods. The monitor $g = |\dot{E}|$ is asymptotically optimal as it minimises the leading-order maximum symmetric-energy discrepancy for a given number of steps. A combined monitor that switches to $g = \sqrt{|\dot{E}|}$ near equilibrium caps step growth without degrading this error bound. The arclength parameter $\beta$ used in prior work is shown to be the ratio of the largest permitted step to the energy-error tolerance. The controller is paired with the SSAV--BDF2 discretization. Accepted steps satisfy monotonicity of the symmetric energy, from which we deduce a bound on the physical energy that is uniform in the number of steps, $\sup_n E(u^n) \le E(u^{0}) + \tau_1 + O(\Delta t_{\max}^2)$, and the scheme achieves second-order global accuracy under standard step-ratio restrictions. Numerical experiments on Cahn--Hilliard, Ohta--Kawasaki, Phase--Field Crystal, and functionalized Cahn--Hilliard models confirm the theory and demonstrate the method's efficiency.

We develop minimal slicing via capillary hypersurfaces to understand scalar curvature metric on manifolds bounded below with boundary. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain. Moreover, we prove a comparison and rigidity result of manifolds with Listing type scaling invariant scalar and mean curvature comparison : $R g \ge -12 g_{\mathbb{H}^{4}}$ and $\hat{H}^{2} g \ge \hat{H}_{0} g_{\mathbb{H}^{4}}$, where $\hat{H}$ is a shifted mean curvature depending on the scale of manifold. This proves the traced version of Gromov's conjecture on comparison and rigidity of hyperbolic convex domains

It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper's operator) has a gap in its spectrum with that labelling number. This answers the strong version of the so-called "Ten Martini Problem". When specialized to the particular case where the coupling constant is equal to one, it follows that the "Hofstadter butterfly" has for any quantum Hall conductance the exact number of components prescribed by the recursive scheme to build this fractal structure.

On coarse grids, fronts of fractional conservation laws can be narrower than one cell, while the flux through each face depends on the whole domain. For the forced space-fractional Burgers equation, we write the finite-volume defect exactly, with the Riesz flux acting as a nonlocal interface flux. Our proposed closure, NRC, is decoder-only, since the encoder is the exact cell-average projection and a small network only maps cell averages to a sub-cell reconstruction. The decoder adds rational atoms with learned complex poles to a trigonometric histopolant, preserves every cell average, and has closed-form advective and Riesz fluxes. We prove that the scheme conserves mass for every value of the weights and that the learned sub-grid flux has a dipole tail. On 64 cells, NRC reaches a median test error of 0.0308% over three seeds, against 0.431% for WENO5-Z and 0.153% for a black-box flux correction. Ablations show that the exact global operator, the projection and the sub-grid Riesz flux carry most of this gain, and most sub-cell fronts carry a deep learned pole within a quarter cell. NRC keeps its advantage over WENO5-Z under shifts of the order and the forcing and, when trained on several diffusivities, under the tested diffusivity shifts by factors from 0.6 to 2.5 in the median over seeds. From 32 to 128 cells, NRC stays an order of magnitude more accurate than WENO5-Z on the same grid. This gain lies in accuracy per degree of freedom and in an interpretable closure.

We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^\alpha$-flat polynomials. The theorem asserts that no $L^\alpha$-flat polynomials exist within the class of trigonometric polynomials of the form$$\sum_{m=1}^{n} a_m \cos(m\theta+\alpha_m), \qquad n \in \mathbb{N}^*,$$satisfying$$k\, n^2 \sum_{m=1}^{n} a_m^2 \;\leq\; \sum_{m=1}^{n} m^2 a_m^2,$$where $k$ is an absolute constant (alternatively expressed on the right-hand side as $K = 1/k$). We show that OpenAI's counterexample is contained within the class of hyper-lacunary sequences. Consequently, it offers only a limited refutation, as such sequences fail to satisfy an implicit assumption of Littlewood's theorem. We also discuss broader questions surrounding this counterexample and provide a brief historical overview.

We establish sharp Beckner--Onofri on the CR sphere $\mathbb{S}^{2N-1}\subset\mathbb C^N$, $N\ge2$, under vanishing holomorphic moments through degree $m\ge1$. The optimal coefficients are $\frac{1}{m+1}$. The function inequality admits a stronger form involving only the energy above degree $m$. We also prove a quantitative concentration estimate in the $1$-Wasserstein distance and show that every maximizing sequence has diverging energy and approaches the family of uniform measures on regular $(m+1)$-polygons. The proof combines subcritical minimization with rigidity of the constrained Euler--Lagrange equation, obtained from an exact radial energy identity and a Hardy space projection estimate. Finally, weighted ball automorphisms yield sharp inequalities without moment constraints, with scalar conformal, barycenter, and holomorphic Gram determinant corrections.

In this paper, we establish an affine fractional Orlicz--Sobolev inequality by replacing the Euclidean fractional Orlicz seminorm with an energy obtained by minimizing over origin-symmetric star bodies of prescribed volume. Under finite lower and upper power growth assumptions, a subcritical upper growth bound, and a monotonicity condition on $t^{n/s+2}\Phi''(t)$, the resulting inequality has the same best constant as its Euclidean counterpart. The power case recovers the affine fractional Sobolev inequality of Haddad and Ludwig with the corresponding Luxemburg normalization. To establish this inequality, we investigate the underlying geometric extremal problem. For every nonzero smooth function of compact support, the existence and uniqueness of an optimal star body are proved under the growth assumptions alone, and its radial function is characterized by an equation involving $\Phi(t)+(s/n)t\Phi'(t)$. We also establish an anisotropic rearrangement inequality, together with its equality conditions under additional strictness assumptions. Moreover, the normalized local limit is investigated. The optimized energies converge to the local energy, and the radial functions of the minimizing bodies converge uniformly on the sphere. The limiting modular is determined by $\overline\Phi(t)=\int_0^t\Phi(\tau)\,d\tau/\tau$. Finally, we compare optimization over star bodies with optimization in the affine orbit.

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups (answering a question of Genevois), Coxeter groups, weak Garside groups (in particular Deligne's groups and fundamental groups of Salvetti complexes of oriented matroids), hierarchically hyperbolic groups, and other examples. Along the way, we show that those groups satisfy the Morse local-to-global property, providing a unified proof for the whole class. In the metric setting, the same condition of non-positive curvature allows to provide a local definition (that is, in a sense, optimal) of rank one/Morse geodesics, mirroring the one using parallel Jacobi fields from Riemannian geometry; to deduce linearity of divergence from linearity on a sequence; to obtain new cases in which Morse geodesics are strongly contracting. The main new tool introduced is the generalised contraction space, a hyperbolic space that encodes the negative curvature of a given space.

We construct a bijection between $2143$-avoiding and $3421$-avoiding permutations that preserves the descent top set and the ascent top set and reverses the left-to-right order of the peaks. This bijection allows us to establish a bijection between $1243$-avoiding and $3412$-avoiding permutations with the same properties. These results refine two cases of a conjecture posed by Burstein and prove a conjecture posed by Burstein and Jones.

I present a geometric multigrid solver for the Poisson equation with continuous cubic elements in three dimensions, smoothed by a multiplicative vertex-patch Schwarz method with exact local solves. All fields are kept in persistent cell-wise storage, where every cell holds its own copy of each shared degree of freedom. The patch smoother yields a continuous solution without any exchange between cells. On the GPU, the patch update is computed in registers, one cell per thread, and the cells of a patch exchange data by warp shuffles, so shared memory is avoided almost entirely. Despite the low occupancy that this design implies, the arithmetic of the patch update runs close to the FP32 peak. Fast diagonalization, merged transforms, and a reflection symmetry of the patch then reduce the arithmetic until a sweep moves its data at $87\%$ of the bandwidth of a plain device copy. The hierarchy therefore runs in single precision inside a double-precision outer iteration. Rounding the unassembled residual destroys the cancellation between opposite-sign copies of shared degrees of freedom; exchanging the copies between cells in double precision performs this cancellation before rounding, without changing the dual vector they represent. With this \emph{rebalancing}, the cycle converges as in double precision and independently of the mesh size, the error converges at the optimal rate down to the precision of the solve, and the solver reaches over $400$ million degrees of freedom per second on a single NVIDIA A100 GPU.

Kemeny's constant of a network quantifies the expected time for a random walk to reach a target vertex chosen according to the stationary distribution, independently of the starting vertex. The purpose of this work is twofold. First, we introduce the class of slightly regular networks, whose orthogonal degree vector is an eigenvector of the Laplacian, and derive an exact expression for Kemeny's constant for this class, extending previous results beyond regular networks. Second, we address the efficient approximation of Kemeny's constant using spectral sparsification. We establish multiplicative bounds relating Kemeny's constant of a network to that of any of its spectral sparsifiers, and use these bounds to design an approximation algorithm. While standard exact approaches based on a full spectral decomposition or on the computation of a pseudoinverse have $O(n^3)$ complexity, our method has complexity $O(n^2\log^{1+o(1)}n)$ for fixed approximation parameters, and is therefore nearly quadratic in the number of vertices even for dense networks.

Self-interacting jump processes (SIJPs) describe systems with non-Markovian stochastic dynamics in which transition rates depend on empirical observables of the process, which gives rise to long-range memory and feedback. We derive the ``level-2.5'' large deviation (LD) principle governing the joint fluctuations of empirical occupation measure and the flux matrix for a broad class of SIJPs with general functional dependence on an empirical observable. The derivation is based on an exponential tilting construction and reveals a separation between a faster timescale of the microscopic dynamics and a slower timescale of the memory-driven evolution of transition rates, which is expressed through an exponentially discounted LD rate functional. Using this variational framework, we derive kinetic and thermodynamic uncertainty relations that extend classical Markovian bounds to non-Markovian systems, and illustrate their performance with simple examples.