Papers

We develop a theory of logarithmic $A_{\mathrm{inf}}$-cohomology with coefficients for a class of $p$-adic log formal schemes that are “sufficiently log smooth”, where the coefficients are given by relative log BKF modules. Then we establish comparison isomorphisms with étale, de Rham, and crystalline cohomology, and also extend these results to the derived setting. As an application, we give a new proof of the $C_{\mathrm{st}}$ conjecture for semistable local systems. The proof also uses the prismatic interpretation of semistable local systems established by Du–Liu–Moon–Shimizu.

A block design is a set family, whose members are called blocks, where every pair of points in the universe is covered by exactly one block. Erdős asked the following question: is it true that if the universe is of size $n$ and each block is of size $\Omega(\sqrt{n})$, then there exists a transversal intersecting every block in $O(1)$ points? We provide a negative answer to this question. Indeed, for arbitrarily large $n$, we construct such a block designs where every transversal intersects some block in $\Omega(\frac{\log n}{\log\log n})$ points.

We consider simultaneous internal stabilization of square proper real-rational plants of arbitrary but fixed finite input-output dimension $n$, arranged as an affine right-coprime factor segment, under explicit nondegeneracy hypotheses. After a classical normalization, the remaining global spectral-cut constraint and the reconstruction constraints at finite poles and at infinity are encoded, without loss, as a finite-dimensional semialgebraic seed, using a function-level double-Cayley factorization. For input coefficients in an effectively presented real closed field, that seed yields an exact existence decision that does not enumerate controller McMillan degree; the decision engine is classical quantifier elimination. The same encoding, now ranging over rational radii, admissible seeds, and residual unimodular and rational Schur parameters, yields a surjective atlas of all proper real-rational common stabilizers of the segment. The results concern this structured class and are compatible with the rational undecidability of simultaneous stabilization of three general plants.

We study the loopy polynomial L_G, a multivariate graph invariant arising from bizonotopal graphical algebras and defined by deletion and loopy contraction, in which the contracted edge is retained as a loop. It determines the Tutte polynomial and admits a spanning-forest expansion recording external activity componentwise. It also determines Stanley's chromatic symmetric function for every graph; the degree sequence, induced edge count profile, and independence polynomial for loopless graphs; and the clique and matching polynomials for simple graphs. Separating component size from external activity gives a refinement that is equivalent to the extended U-polynomial and the extended polychromate. The ordinary U-polynomial, Tutte's universal $V$-function, and Stanley's Tutte symmetric function are specializations of this refinement. We conjecture that L_G and U_G have the same distinguishing power on simple graphs, and verify this through eleven vertices. We exhibit five-vertex loopless multigraphs with equal U-polynomials but distinct loopy and extended U-polynomials, solving a problem of Merino and Noble. We also disprove Loebl's conjecture for chordal graphs and its restriction to split graphs, with counterexamples on eleven and fourteen vertices, respectively. For the score polytope P_G of the external bizonotopal algebra, we give a geometric realization of loopy deletion-contraction. The resulting forest-indexed geometric parking complexes partition the lattice points of P_G and are piecewise integral-affinely parametrized by boxes whose interval lengths are the component indices in the forest expansion.

We investigate distributional properties of two random tessellation models on the $d$-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For \(d=2\), we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.

Euclidean sphere-constrained optimization problems represent perhaps the most basic class of nonconvex-constrained problems for which there exist efficient optimization algorithms for specific problem instances, such as the eigenvalue problems and learning problems arising in the recent literature. However, when and why we can efficiently optimize on the sphere using simple first-order methods is poorly understood. We introduce Riemannian Gradient Alignment (RGA) as a unifying structural condition enabling fast convergence. RGA is a parameterized directional error bound requiring a tangent vector to sufficiently align with a target solution vector. We prove that Riemannian Gradient Descent with diminishing step sizes converges linearly when RGA holds with order $r=1$, and at rate $O(k^{-1/(2r-2)})$ for $r>1$. The result applies even to tangent vector fields that are not gradients of any objective. Our main technical contribution is a family of derivative-based sufficient conditions for RGA. We prove these conditions for a range of learning problems, including multi-instance learning/max pooling, Gaussian halfspaces, single-index and generalized linear models, phase retrieval, complete orthogonal dictionary learning, and (sparse) PCA. The provided examples span distributional families that include standard Gaussian, conditionally Gaussian, and more general structured distributions. The provided guarantees cover objectives that are generally not (geodesically) convex, may be nonsmooth, and may arise from discrete distributions. These results establish tractability of problem classes where no prior error bound conditions were known, as well as unify, strengthen, and generalize several prior results.

We study the equidistribution of small points over finitely generated fields, in connection with a conjecture of Yuan–Zhang. We first give a counterexample showing that numerical smallness, defined using Moriwaki heights for all polarizations, does not imply equidistribution at every valuation. We then prove that numerically small points do equidistribute at every fully transcendental valuation. In particular, when \(F\) is the function field of a curve \(B/\mathbb{Q}\), these valuations correspond to points of types \(2\), \(3\), and \(4\) in the Berkovich analytification of \(B_{\mathbb{Q}_p}\).

In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud's monograph Distribution modulo one and Diophantine approximation, which goes back to Mendès France. The problem states that the spectrum of the sequence $(\xi\alpha^n)_{n\ge1}$, that is, the set of irrational $\theta\in(0,1)$ for which $(\xi\alpha^n-n\theta)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi\ne0$ and $\alpha>1$. More precisely, we prove that for every real $\alpha>1$ there are $2^{\aleph_0}$ real numbers $\xi>0$ for which the spectrum of $(\xi\alpha^n)_{n\ge1}$ contains one and the same uncountable set.

We establish that a JB$^*$-triple $E$ is commutative if and only if its numerical index $n(E)$ is one. As a consequence, $n(E) = 1$ implies $n(E^*) = n(E^{**}) = 1$. This settles a question left open in the literature since 2008.

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension, a homological invariant that extends the classical projective dimension. In this paper, we investigate this invariant from the perspective of Gorenstein homological algebra. First, we show that the quasi-projective dimension coincides with the Gorenstein projective dimension under certain conditions. Second, we introduce and study the quasi-Gorenstein projective dimension as a Gorenstein analogue of the quasi-projective dimension.

Jarnicki and Pflug constructed a bounded balanced domain of holomorphy in $\mathbb C^n$, $n\ge 3$, with continuous Minkowski function which is not Kobayashi complete, and explicitly asked whether the same phenomenon can occur in dimension two. We give such a domain in $\mathbb C^2$.

We exhibit an explicit pair $(A,B)$ of integral symmetric $9\times9$ matrices, defining a nonsingular pair of quadratic forms over $\mathbb{Q}$, such that no member of the rational pencil $\lambda q_A+\mu q_B$ has Witt index $4$ over $\mathbb{Q}$. This refutes a conjecture from \cite{Que16a}, which predicted that every nonsingular pair in $n$ variables generates a pencil containing a form of Witt index $\lceil (n-1)/2\rceil$. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over $\mathbb{Q}_2$. The proof is finite and elementary: a parity argument on $\det(\lambda A+\mu B)$, a congruence lemma reducing $\mathbb{P}^1(\mathbb{Q}_2)$ to the twelve classes of $\mathbb{P}^1(\mathbb{Z}/8)$, and a verification at each class, in which the anisotropy verdict is certified in two independent ways. All scripts are provided in the GitHub repository.

Hilbert-space embeddings enable inference for populations of persistence diagrams, but separation between individual diagrams need not survive population averaging. We develop a framework for inference on population mean embeddings, with particular attention to the additive landmark representations PLACE and PALACE. Treating each diagram as one independent observation, we apply Hilbert-space limit theory to obtain covariance estimators, two-sample tests, and confidence balls under suitable moment conditions, without requiring a lower-distortion bound. For additive embeddings, we identify the population mean as an embedding of the mean counting measure and show that geometric separation of these measures alone cannot guarantee uniform testing power. We then introduce a model with latent template diagrams, missing features, and location perturbations. Under common or feature-specific prevalence conditions, a diagram-level lower-distortion certificate yields explicit lower bounds on population mean separation. These margins provide finite-sample uniform power guarantees, and an additional information-divergence comparison gives matching sample-complexity bounds over restricted scale ranges. Confidence sets yield lower bounds on transport separation of population mean measures and exclusion guarantees for specified structured alternatives. We also quantify how orthogonal truncation changes the certified signal and the approximation allowance needed for confidence sets targeting the full embedding, relating sample size, retained coordinates, and template separation. Simulations examine calibration, power, and coverage, and an analysis of resting-state connectivity from the Autism Brain Imaging Data Exchange illustrates the procedures.

Extrusion-based 3D concrete printing builds structural elements layer upon layer without formwork, so freshly deposited material must carry the growing weight of the structure while it is still gaining strength and stiffness. The largest number of layers that can be printed before failure, the buildability, is therefore central to process design. Its prediction rests above all on how accurately the constitutive model represents fresh concrete. Existing finite element frameworks pair gradual pre-failure hardening only with smooth yield surfaces, which leaves the Mohr-Coulomb surface that best describes fresh concrete restricted to perfect plasticity. This work combines a smoothed Mohr-Coulomb yield surface with nonlinear isotropic hardening of the cohesion, formulated in an updated Lagrangian setting with an activation field for layer deposition. Failure is identified from a tangent-stiffness eigenvalue criterion that matches the analytical self-weight buckling load of a slender wall and is insensitive to geometric imperfection. The framework reproduces the measured collapse of a printed straight wall within one layer and predicts the collapse of a hollow cylinder closer to experiment than existing frameworks. A study over six diameters and six hardening rates shows buildability peaking at intermediate diameters and rising with the hardening rate. Recording how much of its strength the material has used at failure, slender cylinders fail with much still in reserve while intermediate ones use nearly all of it. The two failure modes are therefore identified as the limits of one process rather than separate failure mechanisms.

We study mixed multiple orthogonal polynomials associated with rank-one matrices of measures and their recurrence relations. On the step line we identify the corresponding banded recurrence matrix through the Gauss–Borel factorization of the moment matrix. Elementary Christoffel transformations lead to a factorization of this recurrence matrix into lower and upper bidiagonal factors. For the mixed Piñeiro system we derive contour-integral and generalized hypergeometric representations and formulas for the recurrence and factorization coefficients.

A group (G,*) is natural if its group operation is uniquely determined by a metric structure (G,d) on G in the following sense: every group structure (G,.) for which all right translations are isometries of (G,d) must be isomorphic to (G,*). Examples included all connected Lie groups or all groups generated by involutions (Sutherland). The orientation rigidity theorem of Leemann and de la Salle allows to upgrade the non-abelian structure theorem: every non-abelian group of cardinality not larger than the continuum that is not generalized dicyclic is natural. A consequence is that all non-metabelian Lie groups, all simple group of cardinality not larger then the continuum, all homeomorphism-, diffeomorphism -or symplectomorphism groups of manifolds, or automorphism groups probability spaces or non-abelian crystallographic groups are natural. Also the abelian structure theorem is extended: affine rigidity of Jarosz, together with Mazur-Ulam's theorem implies that the additive group of every real or complex vector spaces is natural and that all connected abelian Banach Lie groups are natural. A theorem of Babai implies that every Boolean group is natural for every cardinality. Undecided is whether C_p^k is natural for cardinals k>c and odd prime p, and whether the additive group of any field F is natural, or whether the cardinality assumption in the non-abelian structure theorem is needed. A major question is whether G^2=G implies that G is natural. This is already open for abelian groups: does 2G=G imply that G is natural?

In these lecture notes, we review the recent results around the chain rule for the generalised KPZ equation and the geometric KPZ equation. We introduce the main ideas of Regularity Structures via a B-series type expansion and the two main combinatorial sets for encoding this expansion: decorated trees and multi-indices. Then, we explain how the chain rule was first understood via integration by parts and diagrammatic computations. For the generalised KPZ equation, a scalar-valued equation, we use multi-indices to derive a characterisation in terms of iterations of covariant derivatives. The same characterisation for decorated trees is more involved and requires the use of operadic and homological tools. We summarise the main arguments of this proof.

This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of $L$-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proof of the optimality of the optimized gradient method (OGM) and establish its uniqueness among all fixed-step first-order methods. For the alternative measure of final squared gradient norm (relative to initial suboptimality), we prove the OGM-G method is optimal and uniquely so among fixed-step first-order methods. Finally, for the setting measuring the final squared gradient norm (relative to the initial squared distance to a minimizer), we show the recently proposed Lemniscate method is optimal and uniquely so. Our proofs rely on algebraic reductions for lower bound arguments rather than traditional information-theoretic bounds, which were previously only able to establish OGM's optimality but not uniqueness.

We establish a weak solution theory and a singular constitutive limit for a non variational Cahn Hilliard evolution on compact Riemannian manifolds, possibly with boundary. The model combines the classical double well chemical potential with an active correction concentrated in diffuse transition layers and sensitive to the sign of the Laplace Beltrami operator, together with a monotone anchoring mechanism toward a prescribed reference state. The active correction breaks the passive gradient flow structure and introduces a nonlinear dependence on second derivatives not identified by the natural weak compactness estimates. For square integrable initial data and sufficiently integrable reference data, we construct weak solutions in arbitrary dimension. A Galerkin level one sided comparison argument, using monotonicity of the classifier and anchoring law together with biharmonic coercivity, yields strong convergence of the approximate Laplacians and hence strong second order compactness. This identifies the nonlinear active term in an ordinary weak formulation, without a second derivative defect. In two dimensions we prove uniqueness and continuous dependence. The estimates depend only on the bound and monotonicity of the classifier, not on its slope, and are therefore uniform for increasingly steep arctangent classifiers. Their singular limit is governed by the maximal monotone sign graph acting on the Laplace Beltrami operator. In arbitrary dimension we obtain subsequential strong second order convergence to a weak solution of the resulting differential inclusion. In two dimensions the limiting state is unique, so the entire steep classifier family converges; uniqueness of the constitutive multiplier on the zero Laplacian set is not asserted. This is a constitutive steepness limit at fixed diffuse interface thickness, rather than a sharp interface limit.

We study Chow-quotient compactifications of the moduli space of degree-$d$ rational self-maps of the projective line. We compute the degree of every three-dimensional orbit cycle in the Clebsch–Gordan compactification in terms of the main component and the hole depths. As consequences, we obtain the generic orbit class and a rational map on Chow quotients induced by iteration. We also introduce a marked Chow quotient designed to record collections of rescaling limits, describe it by decorated trees, and extend the fixed-point multiplier map to the normalization of the unmarked Chow quotient.

Full-parameter fine-tuning of large language models has substantial memory costs because backpropagation stores activations and gradients. Zeroth-order optimization avoids this by estimating update directions from loss evaluations, but existing methods require tuning a sensitive learning rate for each model and task. We propose the momentum $p$-dimensional subspace trust-region method (MpSub). At each iteration, MpSub searches within a $p$-dimensional subspace: one direction preserves historical momentum from the most recent accepted step, while the remaining directions explore via fresh random sampling. The subspace gradient is estimated by central differences, a trial step is computed from a linear trust-region model, and the trust-region radius adapts according to the agreement between predicted and observed loss reduction, eliminating the learning rate. For LLM fine-tuning, evaluations within an iteration share a minibatch, and directions are regenerated in place from seeds, using forward passes alone. For smooth deterministic objectives under unorthogonalized Gaussian directions, we bound the finite-difference error, quantify gradient energy captured by the subspace, and prove that $\lim_{k\to\infty} \|\nabla f(x_k)\|_2 = 0$ almost surely under a safeguarded radius update. Under a matched budget of 8,400 training-objective forward passes, we fine-tune OPT-125M and OPT-350M on CommitmentBank. With the same preset parameters at both model sizes, MpSub attains mean test accuracies of 0.673 and 0.690 over three seeds, matching tuned MeZO (0.685) without any learning-rate search.

The paper is concerned with the numerical approximation of some two-dimensional asymptotic models for the propagation of internal waves in a two-layer system in the form of pde's for the interfacial deviation and the velocity. After the study of well-posedness of the periodic initial-value problem, the convergence of a semidiscrete approximation based on a spectral Fourier-Galerkin method is analyzed. The resulting semidiscrete system is then integrated numerically in time with the implicit midpoint rule. Some properties of the fully discrete scheme are described and its performance is illustrated with some numerical experiments.

We study the critical biharmonic equation \[ \Delta^2 u=u^{\frac{n+4}{n-4}}, \] on a complete, connected, and non-compact Riemannian manifold \((M^n,g)\) of dimension \(n\geq 5\) with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space \(\mathbb{R}^n\), and the solution is given by \[ u(x)= \frac{ \left[\lambda^2 n(n-4)(n^2-4)\right]^{\frac{n-4}{8}} } {\left(1+\lambda |x-x_0|^2\right)^{\frac{n-4}{2}}}, \qquad \lambda>0,\quad x_0\in\mathbb{R}^n . \]

Let \(N\ge2\) and let \(p\) be its least prime divisor. We prove that every nonzero polynomial with nonnegative real coefficients divisible by \(\Phi_N\) has support diameter at least \((p-1)N/p\). Equality holds precisely for positive scalar multiples of monomial shifts of the \(p\)-term geometric sum \(\sum_{j=0}^{p-1} X^{jN/p}\), thereby proving a conjecture of Steinberger. The proof turns cyclotomic divisibility into the vanishing of the first \(p-1\) Fourier moments of a positive measure on the circle and then applies a classical extremal trigonometric polynomial. As a consequence, we establish the Coven–Meyerowitz diameter bound under their tiling conditions and determine its equality cases. Longer initial intervals of vanishing Fourier coefficients yield stronger diameter bounds, including an explicit refinement in terms of the prime-power divisor sets. The extremal trigonometric polynomial also yields a quantitative concentration estimate for measures and cyclotomic multiples with near-minimal support diameter.

We compute the Wedderburn decomposition of a finite twisted skew group algebra over a finite abelian group $G$ and a finite field $K$. To do so, we prove that this twisted skew group algebra is a homomorphic image of a finite group algebra. We then characterise its Shoda pairs, as well as propose an algorithm for computing a complete set of them.

Semilinear evolution equations are solved by means of the variation of constants formula under the action of smoothing linear operators and a finite number of nonlinearities operating between chosen Banach spaces of a given family. Admissible spaces of initial conditions are determined for the existence and uniqueness of a suitable notion of solution. Its maximal time of existence, short time and blow up time profiles, global extendibility and regularity are analyzed. For extrapolated fractional power scale of Banach spaces the fulfillment of the corresponding Cauchy problem is shown. Chosen applications to the modified viscous Cahn-Hilliard equation in R^N, the evolution of cell population of varying genotype and the semilinear Schrödinger equation are presented.

We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $\Gamma$ for which $$H^1_{\mathrm{cusp}}(\Gamma\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.

We prove that the $Z$-polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class. We also prove that, for each fixed positive corank, the $Z$-polynomials of uniform matroids in consecutive ranks strictly interlace. Our approach is to prove real-rootedness of the corresponding $\gamma$-polynomials for sparse paving matroids and their strict interlacing for uniform matroids, and then transfer both properties to the corresponding $Z$-polynomials.

We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of $\mathbb{H}_\tau^d$. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

For a crossed module $\chi: G \to H$, we introduce the notion of $\chi$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $\chi = id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $\chi: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $\chi: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $\chi$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $\chi$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $\chi$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $\chi$-crossed ribbon or admit an orthogonal $G$-decomposition.