Papers

The crystalline differential operators on a smooth variety X give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist X'. In some cases, this Azumaya algebra splits when restricted to finite covers of X'. In this short note, we show that, whenever X has a non-closed global one-form, there is a degree one cover of X' on which the Azumaya algebra does not split, answering a question of Sasha Petrov.

Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss-Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second-order optimization methods. In this work, we introduce IGA-ODIL, a spline-based residual minimization framework combining ideas from Optimizing DIscrete Loss (ODIL), robust variational residual minimization, and Isogeometric Analysis (IGA). Instead of neural-network parameterizations of PINNs, the unknown solution is represented by smooth B-spline basis functions, leading to sparse structured Jacobians and efficient Gauss–Newton optimization. We also derive robust residual formulations based on weighted Gram operators, making the loss function related with the true error. The resulting systems inherit locality, sparsity, and approximation-theoretic properties of classical finite element and isogeometric methods while preserving the residual-learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection-dominated advection–diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen–Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders-of-magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.

Entropy-regularized optimal transport is widely used to compare and align data distributions, but Sinkhorn can converge slowly at small regularization. We introduce Acc-Sinkhorn, a two-state Hessian-driven Nesterov iteration built on the dual mirror geometry of the row-reduced Sinkhorn map. The method combines momentum acceleration with an inexpensive descent safeguard, sta- ble log-domain evaluation, and a prescribed homotopy schedule requiring neither line search nor curvature estimation. We prove global linear convergence of the safeguarded method, rule out con- secutive restarts, and establish local accelerated convergence for fixed parameters without restarts through spectral analysis. Numerical comparisons with six methods on synthetic problems, color transfer, and bilingual word alignment show runtime speedups of up to about 26 over Sinkhorn, with additional rendering-accuracy comparisons for color transfer.

We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study.

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).

For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $\sigma_k$, $1\le k\le b-1$. Our results complement earlier results, which apply only to $\sigma_k$ with $k\ge b$ and depend on the geometry near the boundary, by showing how the interior geometry influences the low eigenvalues. Our result also yields lower bounds for the low Steklov eigenvalues in the setting of pinched negatively curved manifolds, thus recovering similar results in that context through an alternative proof. The proof of the main result is based on the trace inequality relating the Steklov eigenvalue to the Neumann eigenvalues of the connected subdomains of the manifold containing a boundary collar. The geometric coefficient appearing in this inequality is given by an explicit formula in terms of a quantity that can be interpreted as the electrical resistance of the boundary collar.

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with Pólya–Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.

We investigate transient clustering dynamics in nonlocal aggregation-diffusion systems from an energetic perspective. Starting from a stochastic interacting particle system, we study the associated macroscopic McKean-Vlasov equation on the torus and exploit its Wasserstein gradient-flow structure to analyse the thermodynamic competition between interaction-driven aggregation and entropy-driven diffusion. Through numerical experiments for locally attractive interaction kernels, we identify alternating aggregation- and diffusion-dominated transient regimes along trajectories converging to fixed equilibria. These dynamics can be interpreted as a form of non-monotone clustering behaviour. Moreover, we demonstrate that clustering observables, such as the density peak height, are only partially coupled to the underlying energetic mechanisms and therefore do not uniquely characterise the relevant macroscopic transport dynamics. Our results highlight the role of the variational structure not only for equilibrium analysis, but also as a framework for understanding transient clustering phenomena in interacting particle systems.

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating bounds (from above and below) from one to the other. As an application, we provide new upper and lower bounds for the eigenvalues of the quantum dot Dirac operators, which depend only on geometric quantities of the underlying domain. In particular, we obtain some Faber-Krahn type inequalities for convex thin domains.

We prove an inversion theorem for recursive formulas satisfied by certain families of converging power series in two variables. These power series are indexed by the Harder-Narasimhan types of principal $G$-bundles of degree $d \in \pi_1 G$ on a smooth projective curve $X$, where $G$ is a connected complex reductive group. As an application, we obtain a closed formula for the Hodge-Poincaré series of moduli stacks of semistable principal $G$-bundles of degree $d$. We also compute the variation of Hodge structure of the moduli stack of all principal $G$-bundles over $X$, as a function of the period matrix of that curve.

In this paper, we present HyperPrecision, a Mathematica package for high-precision numerical evaluation of general Horn-type multivariate hypergeometric functions and their Laurent expansions in a small parameter $\epsilon$. Such functions appear widely in physics and mathematics, with applications ranging from quantum field theory and string theory to number theory and statistics. Their high-precision numerical evaluation, however, remains challenging, since their defining series converge only in restricted domains and analytic continuation beyond these domains is, in general, non-trivial. HyperPrecision addresses this problem by automatically constructing the Pfaffian system of partial differential equations for a given hypergeometric function and restricting it to a one-dimensional contour in the space of variables connecting the starting to the target point. The resulting ordinary differential equation is then solved by the Frobenius method, with boundary conditions fixed analytically by the defining series. We illustrate the use of the package by evaluating commonly occurring multivariate hypergeometric functions, including the Appell $F_1$, $F_2$, $F_3$, and $F_4$ functions, the Horn $G$- and $H$-series, and the Lauricella $F_A$, $F_B$, $F_C$, and $F_D$ functions, as well as by considering applications to angular integrals, Feynman integrals, and cosmological and holographic correlators.

We construct the causal fermion system for globally hyperbolic spacetimes starting in the framework of algebraic quantum field theory. The fermionic projector is identified with the one-particle density operator of a quasi-free Hadamard state. The ultraviolet regularization is built into the fermionic projector via a chart-independent $i\varepsilon$-regularization scheme. The continuum limit analysis is developed in globally hyperbolic spacetimes. It is shown that the Euler-Lagrange equations of the causal action principle are satisfied in this setup if and only if the coupled Einstein-Dirac equations hold.

Recent literature shows that hypocoercivity properties of linear evolution equations (in particular their exponential decay and the sharp short time decay of their propagator norm) carry over to their discretization via the midpoint rule. This note discusses this connection for the (other) $\theta$-methods, i.e.\ for $\theta\ne\frac12$. It is shown that any implicit discretization with $\theta\in (\frac12,1]$ (pertaining to a hypocoercive continuous-time evolution equation) is contractive, and not only hypocontractive – in contrast to the midpoint rule. For a coercive continuous-time evolution equation, a discretization with $\theta\in [0,\frac12)$ is contractive for time steps small enough.

We study Lusin-measurable functions with values in locally convex spaces. In particular, the behavior of pointwise limits of sequences of Lusin-measurable functions and exhibit pathological phenomena arising in the nonmetrizable setting. Moreover, we establish approximation and density results for $L^p$-spaces constructed with this notion of measurability, including the density of simple functions in Hausdorff locally convex spaces and convergence results obtained through dyadic approximations.

The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space into an $L_2$ space. Furthermore, that process is meant to be used for specific operators, where natural spectral embeddings or equivalences may be found. That process has previously been considered in arXiv:2411.06281 and in arXiv:2511.18189, where it has been introduced through nonstandard techniques. Our contribution aims to be the reformulation of the theory through classical analysis arguments, without the use of nonstandard techniques nor ultraproducts.

We investigate the random close packing density, $\phi_\textrm{RCP}$, of polydisperse hard sphere systems using a theoretical framework based on the equilibrium model of crowding. We derive a closed-form solution for $\phi_\textrm{RCP}$ in terms of the moments of the diameter distribution, enabling an analytical exploration of the effects of polydispersity ($\delta$) and skewness ($S$) on packing density. For a binary mixture, it is possible to explore a broader range of dependence of $\phi_\textrm{RCP}$ on $\delta$ for a given $S$ or on $S$ for a given $\delta$. We show that the dependencies of $\phi_\textrm{RCP}$ on skewness for a variety of continuous distributions collapse onto a theoretical master curve obtained for the binary mixture case. By correcting the theory so that it obeys known exact limiting behaviours for extreme size asymmetry, our analytical predictions not only agree with previously obtained numerical results, but also predict previously unexplored regions of the $\phi_\textrm{RCP}$ parameter space.

Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but with three given matrices $A,B,C$ and $\lVert A - BXC\rVert$ in place of $\lVert A - X\rVert$. We extend previous studies of the latter problem in three directions: incorporating an affine term, replacing matrix product by Kronecker product in various manners, and generalizing Frobenius norm to any orthogonally invariant norm. We will solve several of these in closed form. For the rest, we develop an iterative algorithm that works for any Schatten norm, proving that it converges to a global minimizer regardless of the initial point. In addition, the algorithm relies purely on numerical linear algebra, and notably does not compute any explicit gradients or subgradients. Along the way, we will also show that there is no Mirsky-type theorem for rank constrained generalized matrix nearness problems.

We study the Dirac equation in the Reissner-Nordström geometry in horizon-penetrating coordinates up to the Cauchy horizon. A mass decomposition theorem is proved, which gives a covariant representation of the spacetime inner product that naturally involves the fermionic signature operator and the fermionic flux operator. We compute their spectra and show that both are bounded symmetric operators on the solution space $\mathcal{H}_m$ of the massive Dirac equation. The corresponding fermionic projector state is constructed and shown to satisfy the Hadamard condition. Lastly, we give some physical interpretations of the fermionic flux operator.

Given a non-zero polynomial $P(x)$, we study Fuchsian differential operators of the form $L=\partial_x^2-u(x)$ such that for all $\lambda\in\mathbb{C}$ the operator $L+\lambda P(x)$ is monodromy free. We prove that all such operators are obtained from populations of critical points of ${\widehat{\mathfrak{sl}}_2}$ master functions. Moreover, we show that the reproduction procedure of critical points corresponds to a Darboux transformation of operator $P^{-1}(x)L$. As a result, we obtain a classification of all operators $L$ with such properties in the case of $P(x)=x^k$.

We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group $V$. Our methods also show that the infimal exponential growth rates of $\mathrm{Aut}(F_{2^{n+2}})$ and of $\mathrm{EL}_{2^{n+2}}(R)$, for every finitely generated ring $R$, tend to $1$. As an application, we obtain the first example of an acylindrically hyperbolic group, and the first example of a Kazhdan group, of non-uniform exponential growth.

Score-based diffusion models have demonstrated remarkable empirical success in learning high-dimensional distributions, particularly those exhibiting low-dimensional and multi-modal structures. However, theoretical understanding of their statistical efficiency remains limited. Existing theories typically rely on strong regularity assumptions, such as uniformly bounded densities or globally smooth score functions, which fail to capture such intrinsic structures. In this work, we study the sample complexity of diffusion models for learning distributions supported on a union of low-dimensional subspaces. Assuming that the data distribution within each subspace is subgaussian, we show that diffusion models require at most $\widetilde{O}(\varepsilon^{-k \vee 2})$ samples to achieve $\varepsilon$ error in 1-Wasserstein distance, where $k$ is the intrinsic dimension. This near-optimal convergence rate depends only on the intrinsic dimension and significantly improves upon prior theoretical guarantees that suffer from the curse of dimensionality. Notably, our analysis applies to a broad collection of distributions without imposing smoothness, bounded-density, or log-concavity assumptions. Overall, our results show that diffusion models can statistically adapt to intrinsic low-dimensional structure while naturally accommodating multi-modal data, offering a rigorous theoretical justification for their success in complex high-dimensional learning tasks.

We show that every stable UCT Kirchberg algebra has a principal étale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our methods. In particular, we obtain a principal étale groupoid model for the Cuntz algebra $\mathcal{O}_\infty$.

Holographic coherent X-ray imaging enables nanoscale imaging of biological cells and tissues, rendering both phase and absorption contrast, i.e. real and imaginary parts of the refractive index. Unlike the standard model, which assumes a perfectly coherent incident beam, we consider partial coherence characterized by a known covariance operator. In addition, we assume time-resolved intensity measurements, granting access not only to expected intensities but also to their correlations. We investigate the information content of these correlations and analytically demonstrate that, under a symmetry-breaking condition on the sample and the illumination area, both phase and absorption contrast can be uniquely recovered in both the full and the linearized models. A key challenge in numerical reconstruction is the substantial increase in data dimensionality caused by computing intensity correlations during preprocessing. We propose a novel approach that leverages a low-rank assumption on the incident beam covariance operator, bypassing explicit correlation computation while still exploiting its full information. Numerical experiments demonstrate its feasibility, yielding accurate simultaneous reconstructions of phase and absorption contrast.

Consider, on the space of marked groups, the map $\mathrm{Res}_{\mathcal{C}}$ which associates to a marked group its greatest residually-$\mathcal{C}$ quotient, for different sets $\mathcal{C}$ of groups. Except for trivial cases, this map is discontinuous. We use the Weihrauch lattice to quantify how discontinuous it is. We show that equational noetherianity of $\mathcal{C}$ and whether the set of residually-$\mathcal{C}$ groups is a quasivariety both can be characterized in terms of the position of $\mathrm{Res}_{\mathcal{C}}$ within the Weihrauch lattice. We give exact classifications of $\mathrm{Res}_{\mathcal{C}}$, for $\mathcal{C}$ one of: the set of finite groups, of nilpotent groups, of $k$-nilpotent groups, $k\ge1$, of finitely presentable groups, of LEF groups, of torsion free groups.

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $\mu$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $\lambda_c(\mu)$ for the one-dimensional model. In particular, we prove that $\lambda_c(\mu)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.

We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic setting, including recent advancements, and then present a generalization of the strategy to linear hyperbolic multiscale problems. We address the limitations of earlier constructions for the wave equation, which only achieve second-order convergence in space, independent of the chosen polynomial degree. Building on the methodology of enriched corrections recently developed for parabolic multiscale problems, we motivate and propose an enriched higher-order LOD method for the wave equation. The enriched corrections exhibit exponential decay and can be computed on patches. Under minimal assumptions on the coefficient and standard well-preparedness conditions on the data, we derive a priori error estimates that achieve optimal high-order convergence rates, thereby overcoming the previously observed saturation of the convergence rate. With the fifth-order Rosenbrock-Wanner (ROW) time integrator, we conduct a series of numerical examples to verify our theoretical results. We provide examples showing the optimal spatial convergence of the method including the localization errors for different polynomial orders. We also present examples showing the optimal convergence rates of the time discretization.

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\mathcal{H}(b)$, and its norm in $\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $\phi$ via the formula \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2. \] However, the formula can break down for some other $f\in\mathcal{H}(b)$. In this article we extend the scope of the formula to all $f\in H^2$ for which the right-hand side is finite, provided that either $\phi\in H^2$ or $\phi$ is rational. If merely $\phi\in H^p$ for some $p\in(0,2]$, then the formula still holds provided that, in addition, $\sum_{m\ge0}m^{2/p-1}|\hat{f}(m)|^2<\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in\mathcal{H}(b)$.

A central question in high-dimensional statistics is to understand statistical–computational gaps: regimes in which recovering a hidden signal is information-theoretically possible but conjectured to be computationally intractable. The low-degree framework offers a concrete way to study this gap by restricting attention to estimators that are polynomials of degree at most $D$ in the observed data. In this paper, we study low-degree estimation in planted dense subhypergraph, sparse tensor PCA, and tensor PCA with a general prior. For the planted dense subhypergraph model on $n$ vertices, we identify two regimes depending on whether the planted set is larger or smaller than $\sqrt{n}$. Above this scale, we identify a sharp threshold for low-degree estimation. Below this scale, we establish hardness in the regimes predicted by prior work, thereby resolving a question of Schramm and Wein (2022) and Sohn and Wein (2025). For sparse tensor PCA, we identify an analogous sharp phase transition. For tensor PCA with a general prior, we prove a low-degree estimation lower bound at the critical signal scale, matching the degree–signal tradeoff suggested by prior work. Our lower bounds apply to degree $D=n^{\delta}$, where $n$ is the dimension and $\delta>0$ is a constant, and we complement them with corresponding low-degree upper bounds. In addition, for planted dense subhypergraph and sparse tensor PCA above the $\sqrt{n}$ scale, we convert our upper bounds into polynomial-time algorithms that achieve almost exact recovery above the sharp threshold, yielding polynomial-time algorithms succeeding up to this threshold. Our proofs extend the framework of Sohn and Wein (2025) through a conditional variant that yields the correct signal-to-noise ratio in settings where the unconditional approach is insufficient.

We show that random linear codes of fixed dimension $d$ over sufficiently large prime fields have nearly optimal output list size for list recovery. Fix an error fraction $0<\alpha<1$ and $0<\varepsilon<1$. For all sufficiently large block lengths $n$, with probability at least $1-\varepsilon$, the following holds simultaneously for every input list size $\ell\le 2^{\delta n/\log n}$, where $\delta>0$ depends only on $\alpha,\varepsilon$, and $d$. Given any list of $\ell$ symbols at each coordinate, at most $(1+\varepsilon)\ell/(1-\alpha)$ codewords agree with the lists in at least $(1-\alpha)n$ coordinates. Our proof adapts the colored-graph and determinant arguments of Matousek, Privetivy, and Skovron for reconstructing planar point sets from their projections. We also prove that the range of input list sizes is nearly optimal. There is a constant $c=c(\alpha,\varepsilon)>0$ such that, for every linear code of dimension at least two over a sufficiently large prime field $\mathbb{F}_p$ and every $2^{cn}\le\ell\le p$, there are lists of size $\ell$ and a set of more than $(1+\varepsilon)\ell/(1-\alpha)$ codewords, each of which agrees with those lists in every coordinate. The main results of this paper were discovered by ChatGPT 5.5 Sol using harness components from the UCLA Moonshot Harness Project, in response to a less optimal result found by the authors. See the end of the introduction for a discussion.

In this paper, we answer negatively to a question posed in the context of the 2025 Oberwolfach Mini-Workshop “The Yang-Baxter Equation and Representations of Braid Groups” regarding the existence of split extensions classifiers in the category of skew braces. To this end, we show that the variety of skew braces is not action accessible by investigating the categorical notion of centraliser in the sense of Huq. In light of their intrinsic relationship with skew braces, an analogous result is proven for the variety of post-Lie algebras over a fixed field.