数学前沿

We investigate $p$-adic cohomologies of log rigid analytic varieties over a $p$-adic field. For a log rigid analytic variety $X$ defined over a discretely valued field, we compute the Kummer pro-\'etale cohomology of $\mathbb{B}_{\mathrm{dR}}^+$ and $\mathbb{B}_{\mathrm{dR}}$. When $X$ is defined over $\mathbb{C}_p$, we introduce a logarithmic ${B}_{\mathrm{dR}}^+$-cohomology theory, serving as a deformation of log de Rham cohomology. Additionally, we establish the log de Rham-\'etale comparison in this setting and prove the degeneration of both the Hodge-Tate and Hodge-log de Rham spectral sequences when $X$ is proper and log smooth.

We determine the zero-forcing number of maximal biconnected outerplanar graphs. When the weak dual is a path, the zero-forcing number and the induced path-covering number are both two. Otherwise, the branching faces determine a canonical core and a family of terminal strips. We prove that an optimal forcing process can be chosen in a normal form governed by explicit vertex and face capacities on the core. After fixing forests on the terminal strips, the zero-forcing number is expressed through an optimization over core-edge selections and the minimum number of root connections that must be removed to obtain an executable orientation. The path-covering number is obtained from the same capacity optimization without the repair cost. We characterize executable orientations by blocking-circuit equations and identify the two-path obstruction with a first cohomology class of a signed contact multigraph. Local replacements and a finite comparison of boundary recurrences establish the normal form for graphs of arbitrary size. A fifty-six-state recurrence constructs a minimum zero-forcing set and a complete forcing chronology in linear time from indexed triangle adjacency. We also obtain exact replacement results for rooted blocks and construct infinite families in which the zero-forcing number exceeds the path-covering number.

We prove that every bounded truncated Toeplitz operator on a Hardy model space admits a bounded symbol if and only if its defining inner function is one-component, meaning that it has a connected sublevel set at some level in $(0,1)$. This resolves the bounded symbol conjecture of Baranov, Chalendar, Fricain, Mashreghi and Timotin and, through known equivalences, the embedding and weak factorization conjecture of Baranov, Bessonov and Kapustin. A uniform bound $R$ for the ratio of least bounded symbol norm to operator norm on the rank-one family associated with point evaluation in the disk already forces this geometry. The corresponding bound for all bounded truncated Toeplitz operators is at most $CR\log(e+R)$, where $C$ is absolute, and the logarithmic factor is optimal. In a nonzero de Branges space, the same geometric condition characterizes when every sufficiently small equivalent change of norm preserving the de Branges structure is represented exactly by reweighting the original integral with a positive density bounded above and away from zero. This characterization remains valid when the norm changes are restricted to discrete sampling norms. Outside the one-component class, every model space admits self-adjoint trace class truncated Toeplitz operators without bounded symbols, with arbitrarily small norm and singular values satisfying any prescribed positive upper bounds.

We show that a closed spin four-manifold with nonzero signature, even after being punctured along a finite graph, does not admit a complete metric with uniformly positive scalar curvature. We also show that this implies Schoen's conjecture on $L^{\infty}$-metrics with non-negative scalar curvature for these manifolds. This is in contrast to higher dimensions. A key new observation, which could be of independent interest, is that Gromov's $\mu$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.

We prove an $e$-positive formula for the chromatic symmetric function of proper circular arc graphs solving the $q=1$ case of a conjecture of Ellzey and of Alexandersson and Panova. In doing so, we provide a new proof of the $e$-positivity of unit interval graphs, which alongside Guay-Paquet's reduction gives a new proof of the Stanley--Stembridge conjecture. We define color matrices, which count proper colorings, and tableau matrices, whose entries are nonnegative rational numbers and ratios of elementary symmetric functions. We prove the two matrices are related by a single family of change of basis matrices, which become invertible after restricting to finitely many colors, and we show the chromatic symmetric function of proper circular arc graphs comes from taking the trace of these matrices. Using Hikita's tableaux, this gives an explicit formula for the chromatic symmetric function of a proper circular arc graph as a weighted sum over tableaux whose first and last $k$ vertices lie in the same columns.

Negative orientable sequences, i.e. periodic sequences with elements from a finite alphabet of size at least three in which an n-tuple or the negative of its reverse appears at most once in a period of the sequence, were introduced by Alhakim et al. in 2024. The main goal in defining them was as a means of generating orientable sequences, which have automatic position location applications, although they are potentially of interest in their own right. In this paper we develop new upper bounds on the period of negative orientable sequences which, for n>2, are significantly sharper than the previous known bound. The approach used to develop the new bounds involves examining the nodes in the subgraph of the de Bruijn graph corresponding to a negative orientable sequence, and to consider the implications of the fact that the in-degree of every vertex in this subgraph must equal the out-degree. However, despite improving the bounds, a gap remains between the largest known period for a negative orientable sequence and the corresponding bounds for every n>2.

While traditional hub capacity planning models optimize effectively for quantitative inputs, they often fail to digest qualitative business context. We propose a novel framework where a large language model (LLM) agent iteratively proposes hub capacity decisions guided by natural-language business context descriptions. The key mechanism is a chain-of-thought reasoning protocol: the LLM constructs a structured decision table that maps each contextual item to specific capacity adjustments based on the implied direction and magnitude of changes. The new capacity decision is then validated through a feedback loop with an optimization model, which provides routing-based performance metrics to guide the agent's selection. On a real-world 13-hub freight network in the southeastern US, our framework achieves a 2.8% optimality gap relative to the hidden ground-truth, a significant improvement over the 11.0% gap produced by the traditional optimization model without textual business inputs. This demonstrates that LLMs can serve as a contextual bridge, integrating qualitative business insights into Operations Research workflows.

Let $(M^n,g)$ be a smooth closed connected Riemannian manifold of dimension $n\ge3$ with positive Yamabe constant. For any prescribed finite set $\Lambda\subset M$ containing at least two points, we construct a complete metric on $M\setminus\Lambda$, conformal to $g$, with scalar curvature $n(n-1)$. At each singularity, the local geometric term in the reduced action comes from the first nonzero homogeneous metric term of degree at most $(n-2)/2$ in finite conformal normal coordinates, or from the mass of the Green function metric when all such terms vanish. We completely solve Schoen's conjecture about singular Yamabe problem.

Linear and convex quadratic programs are fundamental problems in optimization. While many methods have been developed for solving these programs in large-scale applications, to our knowledge none have attempted to solve their continuous-time gradient flow, a differential inclusion, in full generality without assumptions beyond feasibility. Recent work solved the gradient flows in two special cases, basis pursuit and the lasso, and obtained competitive algorithms. Motivated by these results, we solve the general case exactly. Using the minimal selection principle for monotone differential inclusions, we show that the trajectory of the gradient flow consists of finitely many pieces, each obtained from one nonnegative least-squares problem. Its final piece yields an optimal solution or a certificate of unboundedness. This analysis leads to two algorithms. The first algorithm is exact, terminates after finitely many steps, and applies to linear programs and projections onto polyhedra. The second algorithm, which we call minimal selection descent, discretizes the gradient flow for general convex quadratic programs. Its iterates converge to an optimal solution when one exists and otherwise certify unboundedness. We apply both algorithms to a positivity-preserving limiter for a discontinuous Galerkin scheme and to optimal control problems for the Poisson and heat equations. In the limiter, the first algorithm conserves the total mass more accurately than the state-of-the-art Douglas--Rachford splitting method. On the control problems, minimal selection descent reaches the optimal value to machine precision and is competitive in time with commercial and open-source solvers.

KL geometry accelerates restart schedules in stochastic weakly-convex optimization, but the schedule depends jointly on an unknown and potentially nonunique geometric description. We introduce CARP-KL, a certified portfolio that adapts over valid KL descriptions while using the same primitive stochastic-subgradient oracle for both optimization and candidate selection. Over declared polynomial parameter ranges, CARP-KL competes with every affordable valid description and preserves the known-description polynomial Moreau-stationarity rate up to logarithmic factors, without objective values, an exact proximal map, direct Moreau-gradient access, or a bounded-domain assumption. Controlled experiments quantify the finite-budget price of this universality. The portfolio repeatedly contains strong candidates, while certification consumes a substantial share of the oracle budget and finite-sample ranking limits their recovery. A common conservative gap bound further reduces the portfolio from $O(\log^3 T)$ to $O(\log^2 T)$ candidates with an explicit scale-mismatch penalty. The theory and diagnostics together establish certified unknown-geometry adaptation and pinpoint selection as the principal practical frontier of the current construction.

We prove that the projected stochastic subgradient method with the standard anytime stepsizes $\eta_t = \gamma/\sqrt t$ can suffer expected last-iterate optimization error of order $(\log n)/\sqrt n$ already on the real line. Combined with the classical in-expectation upper bound, this establishes the sharp worst-case order of the projected stochastic subgradient method.

We study the effect of replacing a family of independent statistical experiments by independent repetitions of their average. We give a simple, complete characterization of the $f$-divergences for which this operation increases divergence by at most a constant depending only on the generator, uniformly in the number of observations and the underlying measurable space. The characterization is governed by a local-to-global principle: a comparison restricted to two observations on a binary alphabet already determines the validity of the general inequality. It admits an explicit scalar formulation, which simplifies further for symmetric divergences. The result applies to arbitrary convex generators and requires no differentiability or symmetry assumptions. It recovers the homogenization inequality for total variation and identifies exactly the admissible members of the power-divergence family. In particular, whenever uniform homogenization fails, the obstruction is already present in a family of binary experiments with two observations. Moreover, the global comparison constant is controlled uniformly over generators with a common bound in the local inequality.

We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed

We prove that every finite almost simple subgroup that is not simple of the group of birational automorphisms of the three-dimensional complex projective space is isomorphic to one of the following: $\mathfrak{S}_5$, $\mathfrak{S}_6$, $\mathrm{PGL}_2(\mathbb{F}_7)$ or $\mathrm{PGL}_2(\mathbb{F}_9)$. We also prove that the group of birational automorphisms of the three-dimensional complex projective space contains unique (up to conjugation) subgroups isomorphic to $\mathrm{PGL}_2(\mathbb{F}_7)$ and $\mathrm{PGL}_2(\mathbb{F}_9)$.

Let a finite simple graph be given. We study symmetric matrices with this graph pattern that admit a null vector with prescribed exact support. Such a vector exists precisely when its support is a fort, meaning that no vertex outside the support has exactly one neighbor inside it. Over the complex numbers, we prove that the Zariski closure of each exact-support coefficient locus is irreducible and compute its codimension in terms of boundary vertices and a labeled forest on the connected components of the support. The corresponding forest deficiency also determines the generic restricted nullity. An incidence parametrization yields a prime decomposition of the reduced zero-forcing failure locus. For a prescribed common kernel, we describe the coefficient space through equilibrium stresses, exterior affine dependencies, and free coefficients. We then construct boundary complexes recording local solution values and equation residuals. Their gluing maps are defined over the universal coefficient ring and commute with arbitrary base change. By retaining the complete collection of support signatures associated with each seed set, we obtain a composition formula for the minimum zero-forcing number and an algorithm for its computation.

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty)$, and study spectral measures in its dual space $H'_w$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\|\mu_h\|_{H'_w} \leq \|\mu\|_{H'_w}$, where $\mu = \sum_k\delta_{\lambda_k}$ and $\mu_h = \sum_k \delta_{\lambda_{k,h}}$. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze $H_w$-adapted rational approximation of shifted symbols $\phi(x)=(x+\kappa^2)^{-\beta}$ and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise settings due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lies deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

We extend the dual complex of a simple normal crossings divisor to a homotopy type invariant $\mathcal{D}(X)$ of arbitrary complex varieties, and we prove that it is contractible when $X$ is connected and of klt type. In fact, the dual complex belongs to a broader class of invariants, which we call RC (rationally connected)-invariant functors. We prove a rigidity theorem governing the behavior of such functors with respect to rationally connected fibrations.

We construct a continuous function on $[-1,1]^2$ for which the infimum of the $L^2$-approximation error over the class of ReLU networks with two hidden layers and architecture $(2,2,1,1)$ is not attained. This counterexample shows that deep ReLU networks do not satisfy the best approximation property in general, even for continuous target functions. Our findings contrast with existing results for shallow neural networks, where ReLU is one of the few activation functions that guarantee the existence of global minima in typical regression tasks. The proof is based on the fact that the breaklines for neurons in the second hidden layer can bend, which breaks the symmetry arguments used in the one-hidden-layer case.

We study positive weak super-solutions of the Lane-Emden inequality $\mathcal{L}u := \sum_{i=1}^{n}(-\Delta_{p_i})^{s_i}u \ge u^q$ in $\mathbb{R}^N$, where each $(-\Delta_{p_i})^{s_i}$ is a fractional $p_i$-Laplacian of order $s_i\in(0,1]$ and exponent $p_i\in(1,\infty)$, and $\max_i s_ip_i<N$. Let $\kappa_i := (N-s_ip_i)/(p_i-1)$ be the decay exponent of the fundamental solution of the $i$-th operator, and let $\overline{\kappa} := \max_i\kappa_i$. We show that the Serrin exponent of $\mathcal{L}$ is $\mathfrak{q}_{\mathcal{L}} = N/\overline{\kappa}$. More precisely, for $0<q<\mathfrak{q}_{\mathcal{L}}$ the only non-negative continuous weak super-solution is $u\equiv 0$, while for every $q>\mathfrak{q}_{\mathcal{L}}$ we construct explicit positive weak super-solutions of the form $A(\lambda+|x|)^{-\kappa}$. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents $N/\kappa_i$ of the single operators, that is, the one of the operator whose fundamental solution decays fastest. No structural assumption relating the different operators is needed. The endpoint $q=\mathfrak{q}_{\mathcal{L}}$ is left open. The local order $s_i=1$ is treated together with the nonlocal orders, and all the arguments are carried out in the weak setting.