Papers

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.

We construct linearly twisted spectral triples on $C^\ast$-crossed products $\mathfrak{A}\rtimes_\alpha\mathbb{Z}^d$, starting from a spectral triple $(\mathfrak{A},\mathcal{H},D)$ and an action $\alpha$ admitting an adapted weight. The weight compensates for the growth of the base Dirac operator under the dynamics while preserving the canonical differential structure in the $\mathbb{Z}^d$-directions. The construction takes place directly on the original crossed product and does not require the action to be equicontinuous. We prove that, whenever the base spectral triple is metric, the resulting twisted spectral triple is $\kappa$-metric for every $\kappa>0$: its associated Monge–Kantorovich metric metrizes the weak* topology on the state space of the crossed product. Our framework applies, in particular, to every action preserving the domain of the Lipschitz seminorm of the base triple. In this case, an exponential weight may be chosen so that the induced metric geometry restricts exactly to the original metric geometry on the base algebra. In the equicontinuous case, the weight may be chosen constant, the twist disappears, and the standard crossed-product spectral triple is recovered. We also establish a general finite-mode criterion for convergence in the spectral propinquity. As an application, we prove joint continuity, with respect to the deformation parameter and suitable fluctuations of the flat metric, for the twisted spectral triples associated with crossed products of quantum $2$-tori by fixed automorphisms in $\operatorname{SL}(2,\mathbb{Z})$, including Anosov automorphisms.

We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in $\mathcal E_{\mathrm{loc}}$ whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in $\mathbb C^2$ whose singularities satisfy local inequalities involving $\alpha\log\|f\|$ and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.

This paper establishes the first general structural solution to the problem of state estimation for nonlinear systems driven by unknown inputs. Building upon nonlinear unknown-input observability theory, we show that every such system admits a structurally equivalent representation, referred to as the UID-induced normal form. The proposed representation decomposes the information carried by the unknown inputs into two complementary components: unknown-input directions that are structurally decoupled from the observable dynamics and observable quantities that completely represent the unknown-input information affecting the observable dynamics. As a consequence, the UID-induced normal form provides a unified structural solution to unknown-input decoupling and unknown-input reconstruction, without requiring any model or stochastic assumption on the unknown inputs. The practical significance of the proposed framework is demonstrated through a previously unexplored minimal Structure-from-Motion configuration. The proposed representation enables recursive state estimation from only three point features and a single-axis gyroscope, allowing the recovery of the three-dimensional structure and camera motion up to an unknown global scale factor. Experiments on real-world data validate the proposed framework and demonstrate the feasibility of this minimal sensing configuration.

Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\Xi$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min(d,n-d)\ge\lambda n$ for some fixed $\lambda\in(0,1/2]$. We show that there exists $c > 0$, depending only on $\lambda$, such that \[ \mathbf P_{A,\Xi}\{\operatorname{rank}(A\circ\Xi)\le n-k\}\le e^{-c nk},\qquad 1\le k\le n. \] As an ingredient in the proof of the rank bound, we use the case $\kappa=0$ of the following quantitative smallest singular value estimate: \[ \mathbf P_{A,\Xi}\{s_n(A\circ\Xi)\le\kappa\} \le C\kappa\sqrt n+e^{-c' n}, \qquad \kappa\ge0, \] where $C,c'>0$ depend only on $\lambda$.

We prove that a locally maximal plurisubharmonic function on a domain in $\bC^n$, $n\geq2$, is maximal if it has locally analytic singularities with a two-sided locally bounded remainder. Using McAdam's theorem, we show that local maximality forces the analytic spread of each defining ideal to be less than $n$. We use normalized blow-ups and Siu's analyticity theorem to compare singularities through generic divisorial Lelong numbers. We also obtain a criterion for maximality in terms of maximality outside the pole set and the analytic spread of the local defining ideals.

Let $(M^n,g)$ be a complete connected noncompact smooth Riemannian manifold without boundary, with $n\ge3$ and positive scalar curvature. We prove that $M$ admits a complete smooth metric of uniformly positive scalar curvature if $$ \liminf_{d_g(o,x)\to\infty}d_g(o,x)^2R_g(x)>\frac{n-1}{n} $$ for a fixed point $o\in M$. This extends earlier existence results for orientable manifolds, due to Chen in dimension three and to the authors in dimensions $4\le n\le7$, to every dimension $n\ge3$ without an orientability assumption.

For every integer $r\geq9$, we construct a smooth Griffiths-positive Hermitian metric on $\mathcal O_{\mathbb P^r}(1)^{\oplus r}$ whose top Chern form is pointwise negative on a nonempty open set. This gives counterexamples on compact projective manifolds to Griffiths' conjecture on the positivity of Chern–Weil forms.

We consider monoenergetic particles propagating in three spatial dimensions and undergoing elastic scattering off an external medium. For an arbitrary rotationally invariant scattering kernel, we derive an exact expression for the frequency- and wave-number-dependent diffusivity, thereby resumming the hydrodynamic gradient expansion to all orders. The result is an infinite continued fraction, with each successive level determined by the relaxation rate of a higher angular harmonic of the collision operator. Thus, while ordinary diffusion probes only the first angular harmonic, the resummed constitutive relation retains information about the full scattering kernel. We show how this microscopic information manifests itself in the analytic structure of the diffusivity and in the response to a localized external source. In particular, deviations from Fick's law decay exponentially with distance, at a rate determined by the nearest complex-wavenumber singularity of the inverse diffusivity. Thus, exponentially small departures from hydrodynamics retain detailed information about the underlying scattering process.

We consider a family of non-local and non-convex functionals depending on three real parameters $(\gamma,p,\lambda)$, which was introduced by H. Brezis, A. Seeger, J. Van Schaftingen and P. L. Yung. In the case $\gamma<-p$ we prove a lower bound for the Gamma-liminf of these functionals as $\lambda\to 0^+$ and a Poincaré-type inequality. We also show that such inequality cannot hold if $\gamma>0$, and we establish a modified version in this case. These results answer some open questions posed by these authors and by H. M. Nguyen.

Entropy-based selection objectives suffer from a fundamental degeneracy: minimizing Shannon entropy $H(p_A)$ rewards confident selection regardless of whether the selected candidate is informative. We address this limitation with the residual entropy $D = H(p_A) - H(p_\beta)$, where $p_\beta$ is induced by candidate trust weights. We prove the exact identity $D = -\mathrm{KL}(p_A\Vert p_\beta) - \Delta$, where $\Delta$ measures whether the score-induced distribution and trust profile favor the same candidates. Boundary cases establish basic safety: under uniform trust, $D\leq0$ automatically, so an equal-trust, non-starving state is never penalized, while at any one-hot limit, $D\to0$ regardless of the selected candidate. For the intermediate regime where selection occurs, we prove that $D\leq0$ when candidate ordering by trust agrees pairwise with ordering by informativeness, and derive a tighter certificate based on the leading candidate's margin over its competitors. These results are independent of the candidate-scoring function and apply to both stationary and dynamically changing information. Experiments with a gradient-based mixture-of-experts router confirm that the ordering conditions can hold during real optimization and show that correct ordering improves downstream performance when candidates are non-interchangeable and selections are used directly rather than averaged. Beyond routing, margin-based reweighting matches or outperforms fixed-strength baselines in a class-imbalance task, while informative selection in a production video-prediction system reduces MSE by approximately 20$\%$ and transfers to a related species. Residual entropy, therefore, provides a safety criterion for selection and a usable signal for deciding when that selection is informative.

In the general probabilistic model of information theory, the signaling dimension of a given communication protocol (or, equivalently, of its state space) is the smallest possible complexity of a classical communication protocol capable of perfectly simulating the given one in terms of input-output correlations. The signaling dimensions of three-, four-, and five-dimensional cross-polytopes have been known to be 3. We give logarithmic lower and upper bounds for the signaling dimensions of higher dimensional cross-polytopes, pinning down the value up to an additive gap of 2. The lower bound shows that the signaling dimension goes to infinity, in contrast to the case of balls and hypercubes.

Focusing on scalar conservation laws, a TVD theory is developed for the fully discrete, single-stage Active Flux method. The classical choice of degrees of freedom is considered, yielding a third-order method. Starting from an adaptation of the concept of (discrete) total variation to Active Flux, a corresponding adaptation of Harten's lemma is achieved. Finally, the high-order method is endowed with parameters chosen such that they can be efficiently computed to fulfill the conditions of the lemma.

This paper revisits the distributed learning problem for training a multinomial logistic regression model with the Federated Averaging ($\texttt{FedAvg}$) algorithm. We concentrate on a scenario with arbitrarily large stepsizes and heterogeneous update rules where the devices may perform a different number of local updates in each round. We show that, with linearly separable data, $\texttt{FedAvg}$ is stable with any stepsizes and the objective values converge to zero at the rate of ${\cal O}(1/R)$, where $R$ is the number of communication rounds. Our result also demonstrates that the effects of device heterogeneity vanish asymptotically. For sufficiently large $R$, the objective values decrease monotonically and is bounded by ${\cal O}( 1 / (R T_{\rm avg}))$, where $T_{\rm avg}$ is the average number of local update steps per communication round across devices. Numerical experiments support our findings.

An important role in the theory of conflict-avoiding codes is played by a quantity involving the multiplicative order of $4$ modulo $d$, with $d$ running over all positive divisors of an odd integer $n$. We establish results on its distribution as $n$ varies over the odd prime numbers, respectively odd integers, using techniques from the study of Artin's primitive root conjecture. We also point out some other interpretations of this quantity, with one of them making a connection with the factorization of $X^n-1$ into irreducibles over the finite field $\mathbb F_2$.

We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles $\theta_\pm$, the angle $\theta$ between the skew lines, and the initial data $\gamma_0$, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.

We prove that the Hodge polygon of the generic fiber of a proper smooth $p$-adic formal scheme over $\mathcal O_C$ lies on or above the generalized Hodge–Witt polygon of its special fiber.

We study continuous Brascamp–Lieb inequalities associated with isotropic measures on the sphere $S^{n-1}$ and on the Grassmannian $G(n,k)$. We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp–Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp–Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on $G(n,k)$. We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp–Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp–Lieb inequality.

The concept of counterfactual explanations (CEs) has recently gained increasing attention for deriving explanations of mathematical optimization problems. For a given problem, a counterfactual is an ideally minimal change in the mutable problem parameters such that an optimal solution of the changed problem exists fulfilling some desired criterion. It was shown that so-called weak CEs can be calculated by solving a computationally demanding bilevel optimization problem, which is $\Sigma_2^p$-complete if decision variables and mutable parameters are restricted to integers. Hence, the known exact solution methods are expected to fail on practically relevant problem sizes. In this work we derive a fast heuristic algorithm by approximating the feasible region of the CE problem with a decision tree for which CEs can be calculated more quickly. To this end we analyze the mathematical structure of the feasible region and develop a hand-crafted decision tree which applies splits imitating this structure. Additionally, we develop several techniques to increase the performance of our method. The practicability of our methods is supported by computational experiments on the knapsack problem and the facility location problem.

It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules $J$ over a commutative noetherian ring $A$, we show that K-injectivity of $J$ can be detected locally: if $(A,\mathfrak{m})$ is a noetherian local ring, then $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(\widehat{A},J)$ is K-injective over $\widehat{A}$, the $\mathfrak{m}$-adic completion of $A$; and for an arbitrary commutative noetherian ring $A$, $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(A_{\mathfrak{m}},J)$ is K-injective over $A_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ in the singular locus of $A$. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.

The minimum solid angle of a simplex is shown to be a shape measure in any dimension. While this has been known in dimensions two and three, the higher-dimensional generalization is new. We prove inequalities between the minimum solid angle of a non-degenerate simplex and its fullness. Up to dimension-dependent constants, the fullness is bounded from above by the minimum solid angle and from below by the square of the minimum solid angle. The exponents are optimal for three and higher dimensions. This work therefore generalizes the minimum angle condition. We also review several simplex shape measures and their relationships.

This paper studies online quantile regression for large-scale and streaming data using Stochastic SubGradient Descent (SSGD) with constant learning rates. Classical offline inference for quantile regression is computationally and memory intensive. Existing works of online inference for quantile regression provide only asymptotic guarantees and typically require sub-exponential tail conditions for distribution theory. To bridge these gaps, we introduce new techniques to prove a quenched central limit theorem (CLT) and finite-sample Gaussian approximation for SSGD under a finite-moment assumption. We further show that Ruppert-Polyak averaging with a constant learning rate has a non-vanishing bias and fails to satisfy CLT centering at the population target. Hence we propose suffix averaging to address this issue and establish its finite-sample Gaussian approximation. Based on these results, we provide an efficient online inference method for quantile regression that avoids covariance estimation. Numerical experiments show that our method achieves desirable empirical coverage rates and competitive performance compared to other inference methods. We also apply our approach to U.S. wage data to demonstrate its practical effectiveness.

Xu asked whether the Hoare power space of a co-sober space, and in particular of a $T_2$-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable $T_{1}$ space $X$ whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space $M$ onto $X$. Direct and inverse images then make the Scott space $\Sigma\mathcal{O}(X)$ a retract of $\Sigma\mathcal{O}(M)$, so $\Sigma\mathcal{O}(M)$ is not sober. Finally, we show that, for every $T_{0}$-space $Y$, the Hoare power space $\PH(Y)$ is co-sober if and only if the Scott space $\Sigma\mathcal{O}(Y)$ is sober. Consequently, $\PH(M)$ is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.

We study exact support recovery under $(\epsilon,\delta)$-differential privacy in sparse high-dimensional linear regression. We introduce Saturated Propose-test-release, a general mechanism that privately releases the output of a discrete selector with probability one once its stability certificate reaches a finite threshold. Exploiting the coordinatewise geometry of the LASSO, we construct a computable support-stability score. The resulting computationally efficient Saturated LASSO satisfies worst-case $(\epsilon,\delta)$-differential privacy and achieves exact support recovery with high probability under explicit regularity and beta-min conditions. Maximizing sparsity-indexed certificates yields an adaptive procedure requiring no sparsity knowledge and having exactly the same finite-sample exact-recovery risk as the oracle fixed-sparsity procedure under common public tuning parameters. We also establish a minimax lower bound explicitly tracking $\delta$: under its recovery conditions, Saturated LASSO is minimax optimal up to logarithmic factors in $n$ and $1/\delta$ uniformly over $0 < \delta \leq \epsilon/16$; in the broad moderate-$\delta$ regime, it further matches the lower-bound $\delta$-dependence. A complementary information-theoretic construction with known sparsity attains the lower-bound rates up to constant factors under independent Gaussian design, at exponential computational cost. Simulations and a semi-synthetic study using public American Community Survey covariates illustrate the numerical performance of the proposed procedures.

We study rigidity for the weighted $p$-Laplace equation on smooth bounded domains in dimensions three and higher. We show that a smooth positive weight sufficiently close to $1$ is identically $1$ if the derivative of its Dirichlet-to-Neumann map at one affine boundary value agrees with that for the unit weight. We also prove local rigidity at every smooth positive reference weight which is constant in one direction. The reference weight need not be close to $1$, and the unknown weight is only required to be sufficiently close to it, without a directional restriction. No convexity, analyticity, or agreement near the boundary is assumed. The proof combines linearization at a solution without critical points with recent rigidity results for the anisotropic Calderón problem. An identity for the linearized conductivity tensors removes the remaining diffeomorphism. This last step does not require smallness of either weight.

We construct a feasible two-step estimator for the elasticity parameter in small-noise CEV-type diffusions. Under joint small-noise and high-frequency asymptotics, we establish its consistency and an $\varepsilon^{-1}$ central limit theorem with an explicit asymptotic variance. We adapt a Lamperti-based transformation strategy to elasticity inference for a CEV process with an attainable, absorbing $0$ boundary. Under a local-to-CEV scaling, an auxiliary CKLS process and a Girsanov change of measure yield a strictly positive CIR-type benchmark in which the elasticity is recast as a linear-drift parameter. A preliminary elasticity estimator based on local realized variance supplies the unknown parameter in the state mapping, yielding a feasible LSE-type estimator. The asymptotic theory combines control of the CKLS-to-CEV path-replacement error, total-variation closeness between the original and changed measures, and asymptotic negligibility of the preliminary plug-in error, thereby transferring the auxiliary CIR limit theory to the feasible estimator under the original measure.

We extend the theory of Baer, Rickart, and their local, graded, and $*$-analogues for Steinberg algebras. First, we prove that positive definiteness passes from the coefficient field to the Steinberg algebra, and that this, together with graded von Neumann regularity, characterized by Steinberg and van Wyk, yields graded locally Rickart $*$-Steinberg algebras. Then we show that local Baer behavior forces every compact open subset of the unit space to be extremally disconnected. Consequently, when compact open subspaces of the unit space are metrizable, local Baer behavior forces the groupoid to be discrete. In the discrete case, we use the orbit-by-orbit decomposition into finitary matrix algebras over isotropy group algebras to convert annihilator conditions into explicit matrix conditions. For groupoids whose isotropy is trivial or infinite cyclic, these ingredients yield complete local and unital characterizations. In particular, local Baer and graded local Baer conditions are equivalent to discreteness, while the ungraded local Baer $*$-condition additionally requires that every orbit with nontrivial isotropy be a singleton. We then apply the theory to boundary path groupoids. Using the Steinberg algebra model, we obtain the Rickart, Baer, and Baer $*$ characterizations of Leavitt path algebras, including the local and graded variants.

We study kernel functions induced by a linear time-invariant (LTI) stochastic differential equation (SDE), an approach that embeds prior knowledge about the function to be estimated directly into the state-space description of the SDE. This kernel-design perspective is relevant across the many settings where kernels play a central role, including kernel methods in statistics, Gaussian process (GP) regression in machine learning, and kernel-based regularization methods in system identification. The resulting class of SDE-induced kernels includes several kernels of practical interest, such as Matérn kernels with half-integer smoothness, spline kernels, and related kernels used in system identification. In contrast to existing linear-time GP regression methods for these kernels, which are typically based on Kalman filtering and Rauch–Tung–Striebel smoothing on the latent state, we work directly with the SSS representation of the observed covariance matrix. This yields linear-complexity recursive algorithms for the matrix operations required in GP regression, including factorization, linear solves, log determinants, posterior covariance, log-marginal-likelihood gradients, and posterior sampling; the same generator-level recursions extend, with only minor modifications, to the analogous computations in kernel-based regularization methods for system identification. Compared with previous uses of SSS representations in GP regression, we derive efficient recursive algorithms for analytic marginal-likelihood gradients in three application-relevant cases, and we develop new recursions for posterior computations and Matheron-type sampling. Numerical experiments verify the accuracy of the proposed recursions and illustrate posterior inference and sampling for representative SDE-induced kernels.

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ of algebraic degree $d$ and let $f_{\infty}$ be the induced endomorphism of the infinity hyperplane $H_{\infty}$. Suppose that for every periodic point $x_0\in H_{\infty}(\mathbb{C})$ of period $n_0$, the geometric mean of the multiplicities $e_{f_{\infty}}(x_0),\dots,e_{f_{\infty}}(f_{\infty}^{n_0-1}(x_0))$ is strictly less than $d$. Then we show that the dynamical Mordell–Lang conjecture for $f$ holds for curves in $\mathbb{A}_{\mathbb{C}}^N$.

We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of $n$ sites, we show that the Kubo–Martin–Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error $\epsilon$ in time $O(\log^4(n)+\log(1/\epsilon))$. This improves the previous $O(n+\log(1/\epsilon))$ bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate $\Omega(\log^{-4}(n))$.