Papers

We develop a unified description of convolution on the sphere by lifting to the rotation group. A number of definitions of spherical convolution have been proposed, differing in whether the kernel is axisymmetric or directional, whether the output is on the sphere or the rotation group, whether the signals carry spin, and in their computational cost. No unified treatment exists. Regarding the sphere as the homogeneous space $S^2 \cong \text{SO(3)}/\text{SO(2)}$, we lift a spin $s$ field to the rotation group, convolve on $\text{SO(3)}$, and project back to the sphere through the adjoint of the lift for a potentially different spin $s'$. We recover every rotation-equivariant convolution considered previously through constraints on the kernel, spin choices and inclusion or omission of the lift and projection. For maps between fields of fixed input and output spin, we prove that directional kernels, output on the sphere and equivariance cannot be achieved simultaneously. We make explicit that spin fields are sections of associated line bundles, linking the physics and bundle literatures, and express the spin rotation phase as the character of the twist of the section. We present harmonic-space algorithms for the exact and efficient computation of lifted spherical convolutions of band-limited fields by appealing to sampling theorems on the sphere and rotation group. All operators are implemented in the open-source s2conv code, and numerical experiments confirm the reductions and the equivariance of the operators to machine precision. Finally, we illustrate the operators on fields measured on the Earth and on the celestial sky.

In this paper, we explore mixed Buchsbaum–Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum–Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form $IF$ for $\mathfrak m$-primary ideals $I$. The general converse problem in dimension at least two is stated at the end.

We study the logarithmic Schrödinger equation with an arbitrary positive semidefinite quadratic potential and a periodic potential in the unconfined directions. The main variational difficulty is to turn a constrained minimizer into a critical point when the energy is extended-valued and not of class $C^1$ on the natural Hilbert space. We establish an abstract principle for convex lower semicontinuous energies with a smooth perturbation, allowing a scaling law for the full energy without requiring homogeneity of its convex part. Combined with a profile decomposition under integer translations, this principle yields a positive ground state for every rank of the quadratic form. When the periodic potential vanishes, we prove a sharp anisotropic logarithmic Sobolev inequality, determine its optimal constant and all equality cases, and classify positive ground states as explicit anisotropic Gaussons, unique up to translations along the kernel of the quadratic form. When the periodic component is constant, we also establish existence and classify all normalized ground states at every prescribed positive mass, with explicit formulas for the minimum energy and the associated Lagrange multiplier.

Current AI predicts materials by speeding up energy minimisation and training on large databases. However, scaling data generation prevents manual analysis by domain experts. If novelty is evaluated by chemical compositions or traditional representations in Crystallographic Information Files, or standard structure matchers based on manually chosen thresholds, one can easily generate millions of “new” compositions or files without discovering truly novel materials. This paper uses a hierarchy of crystal invariants to continuously trace and quantify structural novelty based on the new mathematical definition of a crystal structure and asymptotic guarantees for invariant descriptors. The invariant-based analysis through cascade computations on a modest computer exposes the unexpected scale of duplication in the world's largest materials databases, shown here in the high-profile case of GNoME datasets. To prevent current “embarrassment of over-prediction”, we propose necessary conditions for next-generation AI that should be automatically enforced before claiming any new materials.

We introduce a general class of interacting reinforced stochastic systems evolving on a network. The dynamics are governed by two distinct interaction structures: a sampling network, which determines how information is acquired, and a reinforcement network, which determines how the sampled information influences future evolution. Processes generate Bernoulli observations through preferential, de-preferential, or mixed sampling, while observations are transmitted through an edge-dependent affine reinforcement mechanism. We formulate the dynamics as a stochastic approximation scheme and study their asymptotic behaviour. We establish sufficient conditions for almost sure convergence to a deterministic limit and, when the sampling and reinforcement graphs coincide, derive graph-theoretic criteria involving odd cycles, mixed-sampling vertices, and subcritical reinforcement. Under additional structural assumptions, we establish almost sure convergence and partial synchronization in a random-limit regime. In the deterministic regime, we also prove a law of the iterated logarithm and a central limit theorem under suitable spectral and regularity conditions. Finally, we show how the framework encompasses interacting urn models with edge-dependent reinforcement, urns with memory, and random walks with memory.

We study a continuum model of frustrated magnetism that supports particle-like field configurations with nontrivial Hopf index. The energy functional combines competing second and first order exchange interactions with a stabilizing external magnetic field and gives rise to a variational problem for maps from three-dimensional space into the two-sphere. We prove the existence of energy minimizers in infinitely many nontrivial homotopy classes. We further establish the existence of a unit-charge Hopfion in the regime of strong external fields. Finally, we analyze the corresponding large-field limit, identify the natural concentration scale, and prove strong convergence of suitably rescaled minimizers to a minimizer of a limiting variational problem.

Quantum low-density parity-check (qLDPC) decoding requires reliable corrections and low latency over long syndrome histories. We extend BP-SF, an oscillation-guided belief-propagation (BP) decoder with independent syndrome-flip trials, by introducing windowed and streaming operations. Our CUDA implementation combines disordered-memory retries, sliding windows, correlation-aware graphs, and layered scheduling. Circuit-level simulations test fixed-basis memory and complete logical Pauli-frame recovery. On the [[144,12,12]] bivariate bicycle code at physical error rate p = 0.002, parallel BP-SF reduces 99.9th-percentile call latency by a factor of 13.15 relative to an eight-chain Relay-BP GPU baseline on 100,000 matched samples for which every tested method's initial BP stage fails within 80 iterations. Windowed BP-SF maintains a flat logical error rate and nearly constant per-round compute time through 512 rounds, reaching a 17.1-fold speedup over monolithic decoding. In 512-round full-frame streaming decoding with 2.97-ms syndrome arrivals, layered BP-SF on a transformed joint graph with 20 retries reduces 99.9th-percentile correction-release age from 166.29 to 36.25 ms relative to the joint graph with flooding updates and 100 retries. These results connect parallel BP post-processing to lower decoding costs and earlier correction release in streaming quantum memory.

We propose a new class of cyclic (noncyclic) operator type condensing mappings and establish conditions that guarantee the existence of best proximity points (pairs) using measure of noncompactness. The results obtained are further applied to investigate the optimum solution of an infinite system of differential equations in $c_0$ and $\ell_1$.

Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated $p$-valuation on $G_{x,\frac{1}{p-1}+}$ for any point $x$ in the semisimple Bruhat-Tits building of $G$, given a mild assumption on the maximal torus of $G$ and that $p>2$. We also compute Lazard Lie algebras defined over $\mathbb Z_p$ and the polynomial ring $\mathbb F_p[\varepsilon]$ associated to the saturated group $G_{x,\frac{1}{p-1}+}$, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over $\mathbb F_p[\varepsilon]$.

We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-$\sigma$ KdV charge inserted with chemical potential $\alpha_\sigma$. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the $c=1/2$ free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-$5$ deformation for the first few charges evaluated on low-lying states.

In this paper, we study biholomorphic mappings $F: \mathbb{B} \to X$ of the form $F(z) = f(z)z$ for $z \in \mathbb{B}$, where $f: \mathbb{B} \to \mathbb{C}$ is a holomorphic function satisfying $f(0) = 1$. Our main contributions are as follows: We establish sharp estimates for the coefficients of starlike functions of order $\alpha$ as well as quasi-convex functions of type $B$ and order $\alpha$ by employing the notion of majorization. The results established here directly extend the results due to Robertson \cite{Robertson-1936} to the broader setting of complex Banach spaces.

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a $C^{1,\beta}$-manifold. This extends the analysis in \cite{Ca24} to dimensions $\geq 3$ and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones $\beta$-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.

Two graphs are degree similar if their adjacency matrices and their degree matrices are simultaneously similar, a notion recently introduced by Godsil and Sun. We extend it to signed graphs in two ways, degree similarity and net-degree similarity, according to whether the degree or the net-degree matrix is used. We prove that signed trees are determined by their degree similarity class, but not by their net-degree similarity class. We then develop constructions of (net-)degree similar signed graphs, based on a signed version of Godsil–McKay switching, on partial transposition and on block extensions. As a byproduct, we obtain infinite families both of pairs of net-degree similar unicyclic signed graphs with non-isomorphic underlying graphs, and of unicyclic signed graphs, not necessarily regular, which are not determined by their degree similarity class. This settles the net-degree version of a problem of Godsil and Sun on unicyclic graphs.

Using spectral and polynomial methods, we show several sharp spectral bounds for the clique and chromatic numbers of graph powers that only use spectral information from the original graph. In particular, we strengthen earlier spectral-radius estimates to obtain an asymptotically sharp colouring bound for general graph powers, and prove a sharp bound for powers of line graphs with a complete characterisation of equality. We also develop polynomial methods yielding upper and lower bounds for clique numbers of graph powers. Regarding the connection between the clique and coloring parameters of graph powers, we propose a spectral analogue of Wegner's conjecture, determine the extremal values for graph squares, and establish the conjecture for all powers when the spectral-radius bound is sufficiently small. Finally, we connect our spectral polynomial method with coding theory: we prove that for distance-regular graphs, optimisation over the adjacency algebra recovers Delsarte's linear programming bound for anticodes.

We compare three heights attached to a curve of genus two: the weighted height of its Igusa–Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron–Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron–Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\inv}\le h_{\text{GIT}} + \frac{1}{2} \log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron–Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\times E$.

We construct a smooth embedded annulus in $R^3$ with negative Gauss curvature in its interior whose boundary consists of two strictly convex curves in planes tangent to the annulus, and which contains two closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from the periodic solution of a Riccati equation.

The network simplex method is a classical algorithm for minimum cost flow problems formulated as linear programs, that has attracted recent interest from the optimal transport community. Indeed, discrete optimal transport is a subclass of minimum cost flow for which network simplex techniques compute exact solutions while leveraging the inherent problem sparsity. However, extending these approaches to other, nonlinear optimal transport formulations has yet to be thoroughly investigated. In this paper, we adapt the network simplex algorithm to unbalanced optimal transport. Our proposed framework is applicable to various unbalanced formulations, and can be implemented explicitly in the most common cases, including semi-unbalanced optimal transport. The resulting algorithms are memory-efficient, and perform competitively with existing approaches, as shown by our experiments on a range of transport and barycenter calculations.

Given a bivariate copula $C$, the bivariate copula $\Psi(C)$ is defined as the Markov product of $C^T$ and $C$. In this paper we consider the diagonal section of $\Psi(C)$ under the additional assumption that its value at a single point is known. We determine the exact upper and lower bounds for the diagonal section of $\Psi(C)$, given that $\Psi(C)(a,a)=d \in [a^2,a]$, where $a \in [0,1]$ is a fixed point. Furthermore, we describe some additional properties of the transformation $\Psi$, such as its behavior under ordinal sums and the survival operation. We also study checkerboard copulas in the range of $\Psi$ and matrices associated to them.

We prove a dimension-free Markov–Bernstein-type inequality for polynomial mappings $(f_1,f_2)\colon\mathbb R^n\to\mathbb R^2$ of degree at most $d$. For product probability measures whose one-dimensional marginals have densities bounded by $1$, we obtain a lower bound for the integral of the Gram determinant of $\nabla f_1$ and $\nabla f_2$ in terms of the Gram determinant of the $d$-th derivative tensors $D^df_1$ and $D^df_2$, with a constant depending only on $d$. For pairs in a fixed Wiener chaos, the resulting Gaussian inequality answers a question of Nourdin, Nualart, and Poly concerning a lower bound for the expectation of the Malliavin determinant in terms of the determinant of the covariance matrix.

In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L_p$ for $1 < p < \infty$, $p \neq 2$. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space $X$ with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to $X$. To this end, we introduce an ordinal index that detects whether $X$ embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to $L_1$ and $H^1$, these subspaces fail to be complemented.

Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in \({\mathbb C}^n\), \(n\geq 3\), whose Bergman spaces are finite-dimensional and whose Bergman metrics have constant positive holomorphic sectional curvature \(2\). In this note, we prove a Sandwich Principle for the Hayashida-Kamimoto domains that allows us to construct, inside every prescribed sandwich, uncountably many pairwise biholomorphically inequivalent smooth non-Reinhardt domains whose Bergman metrics also have constant positive holomorphic sectional curvature \(2\). We also classify biholomorphisms between domains sandwiched by the domains of Hayashida-Kamimoto with the help of an affine rigidity theorem in the spirit of Cartan rigidity, which should be of independent interest. For explicit angular perturbation families, we determine the full automorphism group and obtain, in particular, an abundance of examples with trivial automorphism group.

We answer a question of Chai about the rigidity of Tate-linear subvarieties of $\mathcal{A}_g$ over $\overline{\mathbb{F}}_p$: we show that that a smooth subvariety of a central leaf of $\mathcal{A}_g$ which is Tate-linear at one point is Tate-linear at every point. This is achieved by a careful analysis of the $p$-adic monodromy groups of Tate-linear varieties, building on recent work of D'Addezio and van Hoften.

An $n$-vertex graph $G$ is $(\rho, d)$-dense if for all $U \subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, Rödl, and Schacht conjectured that, for any fixed graph $F$, the number of homomorphic copies of $F$ in a given $n$-vertex $(o(1),d)$-dense graph is at least $(1-o(1))d^{e(F)}n^{v(F)}$. Despite considerable effort, only a few graphs are known to satisfy the conjecture. We find a new broad class of graphs that satisfy the conjecture, including cographs.

Abstract polytopes and hypertopes are thin and residually-connected incidence geometries or, equivalently, chamber systems. Their degree of regularity is understood by looking at the action of their automorphism group on the set of their chambers. When this action has only one orbit, they are called regular. In this article, we are interested in polytopes and hypertopes with exactly two orbits on their chambers. Let $I = \{1,2,\ldots,n\}$. For any $J\subsetneq I$ with $|I| \geq 3$, we show that there exists a class $2_J$ two-orbit hypertope whose diagram is any graph on $|I|$ vertices, with edges all labeled by $\infty$. In order to do so, we first establish a general framework for constructing hypertopes as quotient of chamber systems, and then exploit the machinery of acylindrically hyperbolic groups for showing that this framework can be used to create two-orbit polytopes and hypertopes of class $2_J$. Note that all polytopes and hypertopes constructed in this paper are infinite.

Standing waves $e^{i\omega t}u(x)$ of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein–Gordon equations, have a profile $u$ that solves a nonlinear elliptic ODE of the form $\Delta u+f(u)=0$. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation, on a large class of spherically symmetric manifolds and nonlinearities $f$. Under mild assumptions, we prove uniqueness of such bound states for the model nonlinearity $f(u)=|u|^{p-1}u-u$. Both existence and uniqueness results apply to some manifolds with a singularity at the origin.

Interior-point (IP) methods are powerful second-order optimization algorithms but can make slow progress during early iterations. We propose Warm-IP, a hybrid framework that combines a path-following variant of the alternating direction method of multipliers (ADMM) with a primal–dual IP method. The ADMM phase generates a warm start that remains close to the central path, enabling the subsequent IP phase to rapidly refine the solution to high accuracy. We evaluate Warm-IP on a diverse collection of real-world and synthetic quadratic programming (QP) instances and compare it with leading commercial solvers (MOSEK and Gurobi) and open-source solvers (Clarabel, PIQP, and OSQP). We also curate and release a new benchmark of large-scale QPs from cancer radiotherapy treatment planning, built from real patient data and provided in canonical QP form, with instances containing hundreds of thousands of constraints. Warm-IP substantially outperforms existing solvers on large-scale problems. On the large-scale radiotherapy benchmark comprising 60 lung cancer cases, for example, Warm-IP solves an instance in 47 seconds on average with no failures, while the strongest competitors, MOSEK and Gurobi, average 55 seconds with failures on 23% of the cases and roughly 240 seconds with no failures, respectively. To support reproducibility and future benchmarking, we publicly release the implementation and benchmark data through a \href{https://github.com/Radiotherapy-Optimization/Warm-IP}{GitHub repository}. These results demonstrate that appropriately warm-started IP methods can combine the rapid initial progress of first-order methods with the high-accuracy convergence of second-order methods.

Gaussian process solvers for linear partial differential equations (PDEs) return posteriors rather than point estimates, but they condition only on equality information: the PDE residual, boundary data, and measurements. Many solutions of interest also obey inequality constraints, including nonnegativity, monotonicity, and absence of arbitrage, which equality conditioning cannot enforce. We prove that conditioning a Gaussian process prior jointly on linear equality observations and a finite-dimensional inequality event is equivalent to conditioning sequentially in terms of Borel measures on the solution space. The physics-informed posterior is therefore a valid prior for the inequality update at virtual points. The combined law is non-Gaussian yet computable through truncated-Gaussian moments, and neither the PDE conditioning nor the inequality update requires modification. We evaluated the construction on a Poisson benchmark, option-price surfaces, and air-quality fields. Under incomplete boundary data, a monotonicity constraint lowered the error of a two-dimensional Poisson reconstruction by $63\%$. On a real S&P 500 option chain thinned to $20$ retained quotes, no-arbitrage constraints reduced held-out extrapolation error by $22\%$ and cut the calendar-arbitrage rate from $10\%$ to $1\%$. On four real air-quality snapshots, nonnegativity reduced a Gaussian moment-based sub-zero-tail proxy by $74\%$ at the sparsest monitor count. Changes in mean accuracy and constraint diagnostics differed across data regimes. The probability of the noisy virtual constraint event under the equality posterior measures agreement with the specified bounds before the update. The construction turns admissibility from a post-processing step into conditioning information for probabilistic PDE solvers.

Let $f$, $g\colon A\to B$ be two A$_{\infty}$-morphisms between two A$_{\infty}$-algebras (linear over a commutative ring $\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\infty}$-algebra $B\otimes R$, where $R:=\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\times B$ in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A$_{\infty}$-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A$_{\infty}$-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A$_{\infty}$-algebras with (non strict) A$_{\infty}$-morphisms, and we explain why weakly equivalent A$_{\infty}$-morphisms need not be homotopic.

We complete the proof that every compact 3-manifold group is linear. Specifically, we construct a faithful representation in GL5(Q[u]) for a finite-index subgroup of every closed non-geometric graph-manifold group.