Papers

We develop an approximate risk minimization framework for shrinkage-thresholding estimation in normal mean problems. In the canonical multivariate normal mean model, we introduce a general functional class of estimators that contains classical shrinkage and thresholding behavior, including James-Stein-type and lasso-type rules. We express quadratic risk as a functional over this class, derive optimality conditions for both oracle risk and data-driven approximate risk minimization, and construct a feasible approximate risk criterion from the observed data when the oracle risk is unavailable. The resulting estimator, NOMAD, is obtained by minimizing this approximate risk over the proposed class. For the canonical model, we develop an approximate risk minimization theory that includes optimizer characterization, sieve-based consistency under regularity conditions, and approximate-risk inequalities relative to benchmark procedures in the admissible class. We then extend the framework to multivariate normal mean estimation with correlated observations, develop both MLE-based and conditional MLE-based constructions, and establish consistency results under regularity conditions. We further apply the framework to linear regression and derive an equivalent penalized regression representation in which the shrinkage-thresholding map induces a data-adaptive penalty, recovering ridge-type and lasso-type behavior as special cases or limiting forms. The results provide a unified risk-based framework for shrinkage, thresholding, and regularization across canonical and correlated normal mean estimation and linear regression.

In this paper we analyze finite element approximations of ground states of the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling. In this regime, the healing length and vortex core size are of order $\eps \ll 1$, while the effective confinement potential may degenerate as the angular velocity approaches a critical value. In this setting, we analyze the $\eps$-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution. More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state. In particular, we prove that the absolute $H^1$-error behaves asymptotically like $h/\eps^2$. However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement $h \lesssim \eps$. The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface. With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose-Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.

We study the Kohmoto model, a family of discrete Schrödinger operators with Sturmian potentials depending on a frequency and a coupling constant. We prove that, for all non-vanishing coupling constants, all spectral bands admit a hierarchical structure. This structure offers a variety of applications, including a detailed description of the Kohmoto butterfly and a central step towards the resolution of the dry ten Martini problem for Sturmian Hamiltonians, which we carry out in a subsequent work.

We study the proof theory of Sacchetti's modal logics, a family of logics generalizing Gödel–Löb provability logic by replacing transitivity with n-transitivity. We make three main contributions. First, we solve an open problem of Iwata by providing an effective cut elimination procedure for Sacchetti's logics. Second, building on this result, we introduce a new non-wellfounded sequent calculus for this family of logics with an improved subformula property. Third, using this calculus together with interpolation templates, we prove that Sacchetti's logics have the uniform Lyndon interpolation property, substantially strengthening previous interpolation results for these logics.

We study irreducible real pure spinors on pseudo-Riemannian manifolds of neutral signature using the theory of real spinorial forms. We prove that the square of such a spinor is a decomposable differential form of middle degree satisfying a natural duality condition. In signature $(4,4)$, we show that non-pure spinors correspond to $\mathrm{Spin}_0(4,3)$-structures, yielding an intrinsic algebraic characterization of these structures. In addition, we characterize real pure spinors parallel with respect to metric connections with torsion in terms of an equivalent differential system for their squares. As an application, we study left-invariant supersymmetric solutions of the NS-NS supergravity system on certain four-dimensional Lie groups.

The Petrie symmetric function $G(k,n)$, introduced by Grinberg, is defined as the sum of monomial symmetric functions $m_\lambda$ indexed by partitions $\lambda\vdash n$ satisfying $\lambda_1<k$. This article demonstrates that the Schur positivity pattern of $\nabla^r G(k,n)$ for all $r\geq 1$ depends exclusively on whether $k$ divides $n$, thus answering an open problem of Bergeron noted in Grinberg's work.

We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (“Fourier shells”) and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.

In this note, the geometric set-up, the rank two bundle, the local HYM ansatz, and the global gluing construction are the same as in the preceding work \cite{Fu}. The new point is an exponential estimate for the radial ordinary differential equation obtained near each branch point. If $u_\epsilon$ denotes the local radial solution and $\frac12\ln r$ the singular limiting solution, then for every integer $k\ge0$, there exist positive constants $C_k$ and $c_k$ such that \[ \big\| u_\epsilon - \frac12\ln r \big\|_{C^k([r_0,2r_0])} \le C_k e^{-c_k/\epsilon}. \] Consequently, all results of the preceding paper can be refined.

Let $T$ be a torus with the complexification $T^{\mathbb{C}}$ and $(X, ds^{2})$ a compact Kähler Hamiltonian $T$-manifold with the moment map $\Phi$ such that $T^{\mathbb{C}}$ acts on $X$ holomorphically. For each $\alpha$ in the moment body $\Phi(X)$, the Kähler quotient $X_{\alpha}=\Phi^{-1}(\alpha)/T$ is a reduced normal complex analytic space admitting a unique Kähler structure $\kappa_{\alpha}$ induced from $ds^{2}$. Inspired by the theory of variation of Geometric Invariant Theory, when $\alpha$ moves from a subpolytope (a connected component of the set of regular values of $\Phi$) to another one in the interior of $\Phi(X)$, we show that the quotient $X_{\alpha}$ undergoes a bimeromorphic transformation, and this enables us to compare the Kähler classes of the different quotients. In particular, as applications, we prove that each nondegenerate singular Kähler quotient has a partial and rational desingularisation which is obtained by shifting the moment map; moreover, we obtain a formula on the Riemann–Roch numbers of singular Kähler quotients.

Branch-and-cut interleaves branching with cutting-plane generation. How the two operations share the work of proving a bound is a basic theoretical question. We study an abstract model in which a tree certifies a target bound $Z$. Each branch node improves the bound by $\ell$ on one child and by $r$ on the other, where $0<\ell\le r$. The $i$th cut along a root-to-node path improves it by $c_i\ge0$, with cumulative improvement $C_k=\sum_{i=1}^k c_i$. Asymmetric branching enters through the rate $\lambda^{\star}>0$ defined by $e^{-\lambda^{\star}\ell}+e^{-\lambda^{\star}r}=1$. We establish uniform two-sided bounds of order $e^{\lambda^{\star}Z}$ on the minimal leaf count of pure branching trees. We then identify $\log k$ as the sharp threshold scale for the power of cutting. For cut schedules with extended limit $\gamma=\lim_{k\to\infty}C_k/\log k\in[0,\infty]$, minimal-size trees obey a trichotomy. If $\gamma=\infty$, cuts prove asymptotically all of the target. If $0\le\gamma<\infty$, the limiting fraction of the bound proved by cuts is $\gamma\lambda^{\star}/(1+\gamma\lambda^{\star})$. If $\gamma=0$, branch-and-cut has the same exponential size rate as pure branch-and-bound. This resolves open questions raised by Kazachkov, Le Bodic, and Sankaranarayanan on minimal-size trees under harmonically-worsening cuts, and generalizes their results to asymmetric branching and to all cut schedules in the model with this logarithmic limit. Finally, we show that branch-and-cut attains polynomial size in terms of $Z$ if and only if polynomially many cuts reduce the residual bound to $O(\log Z)$.

A graph with Laplacian matrix $L$ is called Laplacian integral if the eigenvalues of $L$ are all integers, and it is called $\{-1,0,1\}$-diagonalizable if $L$ has a full set of eigenvectors with entries from $\{-1,0,1\}$. We herein develop a structure theorem for both Laplacian integral graphs and $\{-1,0,1\}$-diagonalizable graphs of prime order, and combine it with some novel computational techniques to characterize all such graphs for orders larger than was previously possible. For example, we enumerate all Laplacian integral and $\{-1,0,1\}$-diagonalizable graphs of order $13$ or less, all $\{-1,0,1\}$-diagonalizable graphs of prime order $23$ or less, all regular integral graphs of order $15$ or less, and all regular $\{-1,0,1\}$-diagonalizable graphs of prime order $53$ or less. As an immediate byproduct of our work, we show that the $S_{n,n}$ conjecture for Laplacian integral graphs is true when $n = 12$, thus making $n = 16$ the smallest open case; additionally, we disprove two related conjectures regarding Laplacian spectra. We also establish an exponential lower bound on the number of connected $\{-1,0,1\}$-diagonalizable graphs of order $n$, thus beating the previously best-known (subexponential) lower bound. Finally, we show that every bipartite $\{-1,0,1\}$-diagonalizable graph is regular (a fact that fails to generalize to Laplacian integral graphs).

This is a Séminaire Bourbaki survey on spectral gaps of random hyperbolic surfaces. The first part of the text is a brief survey on the geometry and spectra of random hyperbolic surfaces. After this, we discuss the results by Hide-Magee, Anantharaman-Monk and Hide-Macera-Thomas on near optimal spectral gaps for random surfaces.

We study the estimation of bilinear forms from noisy, partially observed multilayer data. The signal follows a Tucker2 model, with shared unit and time factors across tensor layers and slice-specific cores. The missingness pattern is structured and motivated by staggered adoption designs, which are common in causal inference and related applications. We first analyze the four-block missingness pattern, the basic building block for general staggered adoption, and propose a spectral algorithm that pools information across layers and targets the functional directly. We prove a non-asymptotic mean squared error bound that exhibits a phase transition in the number of layers, showing when pooling improves estimation, and match it with a local minimax lower bound up to constants when ranks and logarithmic factors are treated as constant-order quantities. We then extend the construction to general staggered adoption designs via an anchored four-block reduction, and derive analogous theoretical guarantees. Finally, we validate our theoretical findings using synthetic and real-world data on Castle Doctrine laws and COVID-19 policies

In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the “endgame” regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based deflation. We introduce a new “Arclength Endgame” for corank-1 systems which combines the idea of the classical pseudo-arclength continuation method with the estimation of the Puiseux series of the curve. We formally prove that it has an R-superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian. In contrast, previous homotopy techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system, or need additional nondegeneracy assumptions involving higher-order derivatives of the system at the singular root. A key step in our approach (as well as in the standard power series endgame) is estimating the ratio $k_1/c$ of the Puiseux series of the curve. We present a new method for that, and discuss its advantages over previous methods.

In non-convex markets, a competitive equilibrium may fail to exist. This turns out to be an important issue in real-world non-convex auction markets, such as electricity markets, as it complicates pricing and requires the auctioneer to resort to out-of-market discriminatory side payments to sustain an equilibrium. We investigate whether the introduction of convex financial trading induces a smoothing effect, mitigating the issues arising from non-convexities. We develop a two-stage non-convex market model (a forward market followed by a spot market) in which convex financial traders participate in the forward market. Our model predicts that financial trading reduces the magnitude of side payments required to support the cleared allocation. To test the prediction of our model, we examine the introduction of a transaction fee on financial traders in 2020 by PJM, the US's largest electricity market. We show that the substantial decline in financial trading volume caused by this policy coincided with a significant increase in side payments, in line with our theoretical predictions.

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$. We consider gradient descent $x_{n+1} = x_n - \tau_n \nabla f(x_n)$, where the stepsize $\tau_n = \tau \cdot e^{rn}$ is exponentially growing (with $\tau > 0$ and $0 < r \ll 1$). This diverges for almost all initial values. We show that restarting the algorithm whenever $\|x_{n+1} - x_n\| \geq e^r\|x_n - x_{n-1}\|$ has good properties: it works very well in practice; we determine the limiting convergence rate in the case of convergence to a non-degenerate local minimum: it improves on classic gradient descent even though computational cost is comparable. The precise choice of $0 < r \ll 1$ does not matter much and the method is virtually independent of an initial stepsize $\tau$ that is too small: while the convergence rate for gradient descent decays linearly as $\tau \rightarrow 0$, it decays as $1/\log(1/\tau)$ in this modified version; numerical examples illustrate the results.

In this paper, we show that there is a countable Noetherian complete lattice $L$ and an order-compatible $d$-topology $\tau$ on $L$ such that $(L, \tau)$ is not well-filtered, and there exist a dcpo $P$ and an order-compatible well-filtered topology $\tau$ on $P$ but the Scott topology $\sigma (P)$ is not well-filtered. For such poset $P$ and topology $\tau$, let $Y=(P, \tau)$ and $X = 1$ (the topological space with single point), then the function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is not well-filtered. These results answer three open problems concerning the well-filteredness of $T_0$-spaces.

We study local well-posedness of the initial value problem for a class of fourth-order nonlinear dispersive systems on the one-dimensional torus. The main difficulty comes from the loss of derivatives in the nonlinear terms. By introducing suitable structural conditions that compensate for derivative loss through cancellation mechanisms, and constructing modified energies via gauge-type transformations combined with a diagonalization procedure, we establish local well-posedness in high-regularity Sobolev spaces. The assumptions in the present result relax those in previous works, at the expense of requiring higher regularity. In particular, our structural conditions extend the range of admissible nonlinearities beyond the scalar case and provide a unified framework for treating multi-component systems.

The use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.

We study the infinite-width Gaussian-process limit of random neural networks through the lens of tensor programs, and we provide a quantitative convergence theory in Wasserstein distance. Our main result gives explicit finite-width error bounds, of order inverse square-root of the widths between finite-network executions and their Gaussian-process limits. The framework is architecture-agnostic and covers feed-forward models together with weight-sharing schemes relevant for recurrent and transformer-type architectures.

Due to significant progress in physical layer (PHY) technologies, future 6G networks are expected to feature much denser antenna deployments. Cell-free MIMO network designs are one of the most promising candidates to enable denser networks. While significant effort has been put into its PHY research, the challenges it brings to networking disciplines remain relatively unexplored. In this work, we propose various fronthaul network designs for cell-free MIMO. Our results show that while tree topologies may suffice for small-scale deployments, they become infeasible as the number of antennas increases. In contrast, with growing network size, the proposed Clos topology performs almost as well as the optimal topology.

We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number $N$ of training pairs, the number $n$ of input observations, and the output resolution $m$. The condition is derived from a coupled error analysis that interprets the surrogate as a reconstruction from approximate data. This yields a decomposition of the total error into reconstruction and learning contributions that can be analyzed independently. As a consequence, we obtain quantitative scaling laws describing how $N$, $n$, and $m$ must be coupled to guarantee convergence and to balance offline learning and online reconstruction errors. The resulting estimates extend previous analyses of kernel-based operator learning. We further introduce a physics-informed extension that incorporates knowledge of the underlying PDE at evaluation time. Rather than encoding constraints directly into the kernel, we augment the online reconstruction step by penalizing PDE residuals at collocation points. The method requires no retraining for new inputs. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the proposed physics-informed reconstruction strategy.

The present paper is an extended version of the lecture notes of a course given by the first author at the summer school on Formulas of Siegel and Weil (Bielefeld, September 2025). We survey three perspectives on the Siegel-Weil formula: classical, geometric, and arithmetic. We first recall the Siegel-Weil formula for elliptic theta series arising from positive definite lattices. We next discuss the higher genus case for lattices of arbitrary signature from an adelic viewpoint. After introducing orthogonal Shimura varieties, we present the geometric Siegel-Weil formula, in which the generating series of volumes of special cycles is shown to be an Eisenstein series. We conclude with the (partly conjectural) arithmetic Siegel-Weil formula, relating degrees of special cycles in arithmetic Chow groups to central derivatives of Eisenstein series.

Harmonic immersions of surfaces are minimal, while biharmonic maps form a fourth-order extension of harmonic-map theory. Because every harmonic map is automatically biharmonic, the basic existence problem is to find proper biharmonic maps, namely biharmonic maps that are not harmonic. This paper asks a more geometric question: when can a fixed nondegenerate surface in three-dimensional anti-de Sitter space be made proper biharmonic by changing only the conformal metric on its domain? Equivalently, how much of biharmonicity is determined by the immersed surface, and how much can be created by conformal reparametrization? Writing the induced metric as $g=\lambda^2\bar g$ and introducing the weighted mean curvature $u=\lambda^2H$, we first reduce the map equation to a normal scalar equation coupled to a tangential first-order constraint. The resulting system reveals a sharp rigidity–existence dichotomy. A nonminimal spacelike constant-mean-curvature solution must have constant dilation and is locally the totally umbilical hyperbolic plane of curvature $-2/L^2$. Once the constant-mean-curvature assumption is removed, however, there is an open set of local analytic solutions for which both $H$ and $\lambda$ vary. A moving-frame invariant then identifies the ambient one-parameter symmetry and separates elliptic, hyperbolic, and index-three parabolic orbit types. In the parabolic class the geometric system reduces to a scalar third-order analytic equation, from which we reconstruct explicit local spacelike and real-principal timelike families in null coordinates. The paper therefore locates the rigid branch, proves that the rigidity can be escaped, and gives an explicit mechanism for producing the resulting non-CMC surfaces.

In \cite{HH}, H. Huang-X.Huang introduced the generalized Reifenberg condition which describes the non-increasing property of numbers of almost Euclidean factors of geodesic balls and gave a transformation theorem under this condition. In this note, we study how almost Euclidean factors of concentric geodesic balls interact with harmonic splitting maps under a lower Ricci curvature bound and prove a dimension gap for harmonic functions of almost linear growth and then derive a transformation theorem under a non-decreasing property compared with the non-increasing property above. We also construct complete manifolds with nonnegative Ricci curvature whose balls are uniformly almost $1$-Euclidean at every scale, but for which no single harmonic function can be transformed into a splitting function at all smaller scales which give a negative answer to the existence question for all-scale splitting functions in \cite{HH}.

This paper considers kernel-based approximation methods for nonlinear partial differential equations. To this end, the problem is formulated as an optimal-recovery generalized interpolation problem, that is, as an optimization problem in an RKHS with nonlinear functional constraints. This formulation provides the basis for a convergence analysis carried out directly in the RKHS and extends existing results by relaxing the uniqueness assumption on the PDE solution. In the nonunique case, the limiting object is characterized as a minimum-norm solution. Furthermore, a residual-greedy strategy for adaptive collocation point selection is proposed, and convergence of the resulting sequence of generalized interpolants is established. Numerical experiments for a stationary nonlinear heat equation illustrate the method and indicate that residual-greedy point selection can lead to markedly smaller PDE residuals than point sets selected according to fill-distance criteria.

We prove equality conditions for the Ahlswede–Daykin (AD) inequality and the Fortuin–Kasteleyn–Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Björner's and Fishburn's inequalities for linear extensions of finite posets, the Lam–Postnikov–Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede–Daykin–Schur (ADS) inequality recently introduced in Chan–Chen–Pak–Soskin (2026), which is an AD type extension of the LPP inequality.

The Jensen-Shannon divergence (JSD) is a symmetric and bounded measure of the dissimilarity of two probability distributions, which has become a standard tool in statistics, information theory, and machine learning. We complement the understanding of its mathematical properties by presenting an analysis of the amount of data that is needed to distinguish between two distributions, given the value of JSD between them. We find the number of independent and identically distributed samples that suffice for a classifier to determine which of two distributions generated observed data at a desired error rate, for two complementary classifiers: we show that for the log-likelihood-ratio classifier, a sample size that grows as the inverse JSD is sufficient, whereas for a majority-vote classifier assembled from independent single-sample decisions, the sufficient size grows as the squared inverse JSD. These distinct scalings offer operational readings of JSD values and their translation into distinguishability in different contexts.

Based on the newly introduced concept of strong Lyapunov functions for rough differential equations \cite{ducjost25}, we study a tamed numerical scheme to approximate the solutions of the continuous system. We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system. As a result, we prove the convergence of the tamed scheme in the $L^1$ sense. For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.

We study the Cauchy problem for the three-dimensional compressible Navier–Stokes equations with eddy diffusion, an anisotropic dissipative mechanism that arises naturally in geophysical fluid dynamics (cf.~\cite{Jabin-Bresch-2018,Temam-Ziane-2004}). In contrast to the classical compressible Navier–Stokes system, the momentum equation here carries no full vertical Laplacian: the velocity is diffused only in the horizontal directions, and the sole vertical regularization it receives is the partial one transmitted through the compressible mode $\operatorname{div}\mathbf{u}$. This degeneracy invalidates the standard parabolic energy framework as well as the classical high–low frequency Green-function bounds. We prove that the constant non-vacuum equilibrium $(\bar{\rho},0)$ is globally nonlinearly stable against small Sobolev perturbations: global classical solutions exist in $H^{N}(\mathbb{R}^{3})$ for every $N\ge 3$, and the density and velocity relax to equilibrium with explicit, genuinely anisotropic decay rates. The mechanism behind the result is a hidden dissipation produced by the pressure–divergence coupling between $\nabla\rho$ and $\operatorname{div}\mathbf{u}$, which compensates for the missing vertical smoothing of the density and the compressible part of the velocity; the solenoidal part of the velocity, by contrast, is governed by a purely horizontal heat flow and therefore decays only at the two-dimensional rate. The analysis rests on a refined anisotropic spectral decomposition of the Green matrix, a div–curl treatment of the velocity, and time-weighted nonlinear energy estimates tailored to the degenerate dissipation. To the best of our knowledge, this is the first global stability and large-time behavior result for the three-dimensional compressible Navier–Stokes equations with eddy diffusion in the whole space.