Papers

Simulating large-scale free-surface water by coupling a localized 3D fluid solver to a cheaper 2D surface model has long faced a mismatch in wave dynamics: efficient 2D wave models used in graphics are typically either linear or non-dispersive. These models are fast, simple, and accurate for calm, small-amplitude seas, but coupling them with strongly nonlinear 3D solvers produces visible reflections and artifacts at the 2D–3D interface. We address this problem by introducing a nonlinear and dispersive 2D wave model based on the canonical Zakharov formulation. Its Hamiltonian structure, in which the surface elevation and surface potential form a canonical pair ($\eta$, $\psi$) governed by the wave energy, enables a canonically consistent two-way coupling scheme, allowing information to pass smoothly across the 2D–3D interface. Our 2D solver reduces mean wave-height error by 1.7–5$\times$ over SWE, BEM, and Airy baselines while running more than $10^3\times$ faster than BEM; it achieves greater nonlinear accuracy and coupling fidelity than SWE and Airy, with minor losses in speed and stability. Coupling it with a 3D Navier–Stokes solver yields a full system that suppresses visible seam artifacts across a range of experiments, including dispersion-matching and Kelvin-wake tests, and runs over 4$\times$ faster than a pure GPU NB-FLIP simulation on the same domain.

We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of Lüchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $\delta>2$, $\gamma<1-\frac{1}{\delta}$, and $\alpha<1-\gamma$.

Most competitive 4-bit LLM research pipelines now open the same way: apply a linear, function-preserving transform (rotation, scaling, permutation, non-orthogonal affine) so the outlier mass sits more favorably against the group scales, and only then round. Yet we are aware of no survey dedicated to this transform stage, and its literature is quietly re-deriving an older theory. We identify and formalize the principle that organizes it, the Great Inversion: allocation-flexible coding rewards energy concentration, whereas the grouped shared-scale quantization a deployed matrix instruction performs rewards within-group flattening. Classical transform coding (1963: decorrelate, allocate bits, quantize) spends different bits per coordinate at a fixed total rate; for a Gaussian source at high rate the Karhunen-Loeve transform's concentration minimizes distortion. A deployed operand tile instead carries one absolute-maximum scale per group and equal bits everywhere, with no allocation; on a uniform grid that objective rewards flattening, approached by Hadamard incoherence. We prove that opposition under within-group majorization: the prescriptions point in opposite directions, each backed by a proof against its own objective, and for a generic spectrum no optimality guarantee transfers. A second axis is the number format: the non-uniform FP4 grid makes flattening buy less, MXFP4's power-of-two block scale still rewards a rotation confined to that block, and NVFP4's mantissa-carrying scale largely removes that pull, so the target pole depends jointly on allocation regime and format. We survey 200 works to a June 2026 cutoff; classify 43 transform methods by structure, data-awareness, searched-versus-constructed, and runtime cost; record, where reported, how they compose with GPTQ rounding; distill a first-choice guide by deployment regime; and close with the open problems it exposes.

Collective dynamics arise in a wide range of physical, biological, and engineering applications. Examples include cell migration, swarm robotics, social dynamics, and animal behavior. A defining characteristic of these systems is the emergence of large-scale coordination from local interactions among agents; a fundamental question is thus to understand the local interactions that give rise to the observed emergent dynamics. We are interested in methods for learning interactions generally, which can describe a wide class of physical systems exhibiting collective dynamics defined by an interaction kernel, without a priori assumptions on the analytical form of this kernel (i.e. it is nonparametric). The advantage of this kernel-based approach is that it incorporates the underlying physics of the model (i.e. collective dynamics), which more general equation-learning approaches may ignore, potentially limiting their effectiveness for model accuracy and predictions. In this work, we extend existing variational learning approaches to collective systems with both interaction kernels and environmental/intra-agent forces. The proposed framework simultaneously infers the interaction kernel non-parametrically while learning the environmental force using either semi-parametric or fully nonparametric representations. The methodology is validated on several benchmark models exhibiting synchronization, alignment, attraction-repulsion, and external environmental forces. We also introduce a model-selection procedure based on our nonparametric learning framework to identify models that optimally explain a given set of trajectory observations. By exploiting the feature-identification capability of the learned models, the proposed procedure can distinguish among different collective dynamics frameworks and recover mechanistic interaction mechanisms directly from trajectory data.

Accelerating variable renewable energy integration introduces substantial operational challenges to modern power grids. Frequent curtailment of surplus renewable generation highlights the pressing need for long-duration energy storage capable of shifting generation to high-demand periods while providing essential ancillary services and grid inertia. Socio-environmental constraints increasingly limit conventional hydropower development; however, closed-loop pumped-storage hydropower (PSH) presents a viable alternative because its off-river reservoirs avoid natural watercourses. This paper introduces HERA-S, a computational modeling framework developed by PSR (supported by EDF, CTG, Brookfield, and Rio Light) to streamline and standardize regional PSH site prospecting. HERA-S automates spatial screening, dam optimization, cost estimation, and socio-environmental impact scoring. Applied to a case study in Rio de Janeiro, Brazil, the framework employs an integer programming model to optimize dam-fill and excavation mass balances against inter-reservoir distance, significantly improving pre-feasibility planning agility.

In this paper, we address the dynamic construction problem for Malfatti circles with respect to a fixed triangle, using the LC and CC modules. We show that the configuration of the Malfatti circles constitutes a stable fixed point and provide lower bounds for the main parameters of the LC and CC modules. We now remark that all mathematical proofs in this paper were generated by ChatGPT.

We study a linear filtering problem in which the signal process is described by a time-fractional linear stochastic differential equation driven by Brownian motion. We derive a stochastic integral equation for the conditional mean alongside a Riccati–Volterra type integral equation for the mean-square error function. As a core application, we introduce a time-fractional state-estimation framework for modelling learning trajectories in children with developmental dyscalculia.

Waveform inversion seeks to estimate the wave speed of a heterogeneous, inaccessible medium, from time-resolved measurements of the waves at user controlled sensors. We consider this inverse problem for acoustic waves and an active array of source/receiver sensors that emit probing signals and measure the generated pressure waves. The forward map, from the wave speed to the measurements, is nonlinear and oscillatory. The oscillations cause cycle skipping, the main impediment to using the standard, nonlinear least-squares data fitting formulation, known as full waveform inversion (FWI). A recently introduced alternative waveform inversion approach computes from the measurements an algebraic surrogate of the wave operator, a reduced order model (ROM) matrix, which is then used to estimate the wave speed. The mapping from the measurements to the ROM is nonlinear, but well understood. It is computed efficiently, in a non-iterative manner. The nonlinear mapping from the ROM to the wave speed is less understood, and its approximation involves time-consuming optimization. Our goal in this paper is to use a neural network to map the ROM matrix to a nearby one, that has a simpler and explicit dependence on the wave speed. This simplifies and reduces the computational cost of the ROM-based waveform inversion. We introduce the methodology, called ROMNet, and test it with numerical simulations, using two training data sets: The first set consists of random media with variations of the wave speed modeled by a superposition of Gaussians with random amplitudes and standard deviations. The second is the publicly available GeoFWI dataset introduced for benchmarking FWI using deep learning. We compare the performance of ROMNet with the direct ROM-based inversion and with two representative deep learning approaches to FWI: “Fourier-DeepONet" and “InversionNet".

For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(\sigma)=|E(G)\cap E(\sigma(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In 1977 Erdős asked whether $f(n,4)=2n-4$, observing that $K_{2,n-2}$ gives the upper bound. We prove that, for all sufficiently large $n$, \[ f(n,4)=2n-4. \] Equivalently, every sufficiently large $n$-vertex graph with at most $2n-5$ edges has a relabelling with at most three common edges. Our proof is inspired by the recent work of Fang and Hou on the Erdős–Mullin five-edge intersection problem and builds on their core–buffer and absorption framework. The main additional ingredients are a growing high-degree core $C$ satisfying \[ |C|\Delta(G-C)=o(n), \] and a rigidity analysis of the equality case in the relevant first-moment estimate. This analysis shows that the only core–buffer configuration forcing four local common edges is of $K_{2,|C|}$ type; the strict bound $e(G)\leq2n-5$ then supplies a defect which breaks this configuration.

We study Frankl's union-closed sets conjecture through the height of the inclusion poset. Working in the equivalent empty-set-free formulation, where one seeks an element contained in strictly more than half of the members, we prove the conjecture for every finite union-closed family of height at most four. Equivalently, the usual at-least-half formulation holds for every union-closed family containing the empty set and having height at most five. We also develop a structural theory for the next unresolved case. Assuming a smallest empty-set-free counterexample of height at most five, we show that it has even cardinality $2t$, at least three critical elements of frequency $t$, and satisfies the minimal-counterexample bound $t \geq 2n-1$. Every critical element determines a coatom of the form $U \setminus \{x\}$, while every critical pair satisfies a dichotomy between a full double-avoidance top and a large avoidance fiber admitting a three-layer trace normal form. Coordinate deletion further yields an exact matching-defect obstruction. Finally, introducing the minimum number of join-irreducible members required to cover all critical elements, we exclude the five-cover case and show that this critical join-cover number is either three or four. These results substantially constrain any possible height-five counterexample while leaving the remaining transfer problem explicit.

We present a high-order entropy-stable framework for the compressible Euler and Navier-Stokes equations on moving domains. The space-time mapping describing the domain motion is recast in an arbitrary Lagrangian Eulerian (ALE) formulation, in which the physical inviscid fluxes and the contributions induced by mesh motion are treated in a unified manner. At the continuous level, we prove that the proposed moving-wall boundary conditions are entropy conservative for the Euler equations and entropy stable for the Navier-Stokes equations. The no-slip condition is formulated in terms of the velocity relative to the moving wall, yielding a bounded inviscid contribution to the entropy balance, while the viscous terms contribute only entropy dissipation. Using diagonal norm summation-by-parts (SBP) operators together with appropriate numerical fluxes, these properties are extended to the semi-discrete formulation, resulting in nonlinear stability in the $L^2$ sense. The accuracy, robustness, and scalability of the proposed method are demonstrated in practice through an extensive set of numerical experiments, ranging from canonical two-dimensional verification cases to large-scale turbulent and supersonic simulations involving moving boundaries and fluid-structure interaction. The results confirm the suitability of high-order entropy-stable schemes for complex moving-domain problems across a broad range of flow regimes and multiphysics applications. Because the analysis relies on the SBP property rather than on a particular discretization, the framework naturally extends to a broad class of methods based on diagonal-norm SBP operators, including finite volume, finite element, and flux reconstruction schemes.

Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_\pi(x)=\mathbb{E}_\pi[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(\mu,\nu)=\inf_{\eta\preceq_{\mathrm{cx}}\nu}\mathrm{IGW}^2(\mu,\eta)$. Here $\eta\preceq_{\mathrm{cx}}\nu$ means that $\nu$ is a mean-preserving spread of $\eta$. Thus wIGW searches for an intermediate target geometry that can be refined into the prescribed target law without changing conditional means. Under finite second moments, minimizers exist and martingale gluing recovers an optimal coupling. With ridge regularization, moment duality gives an $A$-$B$ min-max problem whose inner step is weak optimal transport with a quadratic cost parameterized by $A$ and $B$; the outer problem optimizes these matrices. For finitely supported measures, we give an iterative algorithm. Under a quantitative ridge condition, the reduced problem is convex–concave, and the projected outer iteration satisfies an explicit contraction bound for inexact inner solves. Point cloud and graph feature refinement experiments illustrate how mean-preserving target refinements can have zero cost. A paired peripheral blood mononuclear cell (PBMC) multiome study evaluates atlas based cell type transfer through RNA/ATAC alignment in cell to cell and prototype to cell settings, with the prototype to cell setting representing the one-to-many case.

In this chapter, we review and discuss the regularity properties of spatially adaptive regularisation weight functions used in variational image reconstruction. Incorporating such weights into classical model-based regularisers, such as Total Variation (TV) and Total Generalised Variation (TGV), allows the regularisation strength to vary across the image and adapt to local image content. When appropriately estimated, these weights can thus significantly improve edge and detail preservation in the reconstructions. We review the existing theoretical literature on this topic for different regularity classes, including constant, continuous, and piecewise constant functions. Our discussion is motivated by recent work on hybrid image reconstruction methods that combine model-based regularisation with deep neural networks to learn highly adaptive regularisation weights. In particular, we discuss how the structural properties of these weights influence the reconstruction from both theoretical and practical perspectives. Through representative examples in image denoising and magnetic resonance imaging (MRI) reconstruction, we demonstrate that the learned weights are often of low regularity and can adapt not only to the image structure but also to the specific noise realisation. We conclude by highlighting several directions for future research on this topic.

With the aid of the torsion-theoretical approach, several properties/characterizations of hereditary QF-$3^{+}$ rings are found. These characterizations are formulated in terms of the category of projective modules, the category of injective projective modules, and also in terms of the maximal left/right ring of quotients by Utumi. Moreover, it is shown that, for a hereditary QF-$3^{+}$ ring, the `largest stable submodule' radical (on the category of left/right modules) is permutable with injective envelopes. Besides, it is shown that there is a bimorphism (in the category of associative rings with identity) from such a ring to a semisimple left/right Artinian ring.

This manuscript focuses on the topological behaviour of algebraic structures in $C_p(X)$. Closure of ideals in $C_p(X)$ is determined and are used to characterise various topological properties of the underlying Tychonoff space $X$. These include compact space, locally compact space, locally pseudocompact space, $P$-space, almost $P$-space and normal space. Moreover, it has been established that a space $X$ is totally separated space if and only if the set of all units in $C(X)$ is dense in $C_p(X)$. A subring of $C(X)$ is found to be dense in $C_p(X)$ when and only when it separates points.

In this article, we contribute to the spectral analysis of graph associahedra by providing a lower bound for the second largest eigenvalue of $\mathcal{A}(G)$. Furthermore, using equitable partitions, we analyze the spectrum of the stellohedron $\mathcal{A}(K_{1,n})$. Specifically, we prove the existence of an eigenvalue in each interval $(n-i, n-i+1]$ for $1 \leq i \leq 5$, establish the presence of an eigenvalue with high multiplicity in $(n - \frac{3}{n} + \frac{2}{n^2-n}, n)$, and identify two additional small eigenvalues.

We study a nonlocal evolution problem driven by the fractional $1$-Laplacian with a Carathéodory nonlinearity satisfying subcritical growth conditions. We develop a potential well framework to investigate the existence and qualitative behavior of solutions at different initial energy levels. By combining a modified potential well method with Galerkin approximations, we establish the global existence of weak and strong solutions under appropriate conditions on the initial energy and the sign of the associated Nehari functional. To address the singular nature of the fractional $1$-Laplacian, we approximate the problem by a family of fractional $p$-Laplacian equations, derive estimates uniform with respect to $p>1$, and pass to the limit as $p\to 1^{+}$. Furthermore, in the low-dimensional regime $N<2s$, we employ a subdifferential approach to establish the local existence of strong solutions and investigate their qualitative behavior.

Large language model multi-agent systems (LLM-MAS) rely on inter-agent communication to solve complex reasoning tasks, yet rigorous guarantees relating consensus performance to computational resources remain limited. Here, we present a novel way to characterize collective belief dynamics as a discrete-time switched system in which communication topologies have distinct consensus-contraction rates and token costs. By augmenting the belief dynamics with the remaining computational budget, we define a consensus safe set that jointly captures agreement and resource feasibility. We derive explicit bounds on consensus time and token expenditure and construct a consensus-budget certificate region guaranteeing finite-time convergence without resource exhaustion. We further establish conditions under which adaptive topology switching achieves a trade-off between convergence speed and communication cost relative to fixed-topology strategies. Numerical experiments and live LLM-MAS deployments show the predicted consensus-cost trade-offs, demonstrating how control-theoretic certificates can enable resource-aware coordination in AI systems.

Understanding how (multimodal) large language models perform on physics problems requires benchmarks that reflect the difficulty and breadth of expert-level physical reasoning. Existing physics benchmarks remain limited in the following two important ways: (1) short of high-difficulty datasets, and (2) lack of comprehensive coverage of visual forms, knowledge points, and step-by-step solution processes. As a result, model performance on current datasets may not be fully representative of their ability to solve complex physics problems. To address these issues, we present PhysElite, a large-scale bilingual multimodal benchmark for Olympiad-level physics reasoning. PhysElite contains 11,586 Olympiad-tier problems. For each problem, we provide corresponding visual diagrams, step-by-step bilingual Chinese-English solution derivations, and the final answer. We benchmark 18 open-source and closed-source MLLMs, and find that even the strongest model reaches only 33.7% answer accuracy. We additionally conduct step-level process evaluation to diagnose where models fail in the reasoning chain. Our datasets are released at https://huggingface.co/datasets/physelite/PhysElite.

The proliferation of distributed energy resources (DERs) in distribution grids enables the active coordination of these assets to reduce costs and enable cleaner operations. Realizing this potential requires solving multiphase AC optimal power flow (AC-OPF) quickly across varying loads, DER availabilities, and topology reconfigurations, at much greater speed and scale than conventional nonlinear solvers. Learning-based surrogates can offer millisecond inference, yet existing methods target largely balanced transmission systems and do not scale to the multiphase, unbalanced, and reconfigurable nature of distribution feeders at utility scale. We present the Penalty + Sequential Linearized Feasibility Seeking (SLFS) algorithm, a self-supervised learning framework for multiphase distribution AC-OPF under switch-induced topology changes. Penalty+SLFS requires no labeled optimal solutions and trains directly from the AC-OPF objective and constraints through a differentiable fixed-point power flow solver, avoiding expensive label generation and admitting robust training procedures. Topology changes are handled efficiently using Sherman-Morrison-Woodbury updates of the admittance-matrix inverse, while an M-step Jacobian approximation accelerates differentiation through the power flow solver. At inference, SLFS repairs any infeasible predictions, providing feasibility guarantees with low computational overhead. On IEEE feeders ranging from 13 to 8,500 nodes, Penalty+SLFS achieves negligible optimality gaps and near-zero constraint violations, delivers up to three orders of magnitude speedups over IPOPT, and remains robust under large distributional shifts, demonstrating a viable path toward real-time, topology-aware AC-OPF for large-scale distribution grids.

In this note, we establish a connection between denoising diffusion models and large temperature dynamic Schrödinger bridges computed with respect to certain reference processes. In particular, we show that in the large temperature regime (i.e. $T \to +\infty$), the Schrödinger bridge behaves like a forward time diffusion process over the time interval $[0,T/2]$, and approximately like the time reversal of a diffusion process over $[T/2,T]$. This construction provides a process-level characterization of the gradient flow approximation developed by Clerc et al. (2023).

A complete mapping of a semigroup $S$ is a bijection $\alpha\colon S\to S$ such that the map $\theta\colon S\to S$ defined by $x\theta=x\cdot x\alpha$ is also a bijection. Equivalently, it determines a transversal of the multiplication table of $S$. Complete mappings connect group theory, Latin squares, and cryptography, and their existence for finite groups was characterized by the resolution of the Hall–Paige conjecture. In this paper, we develop the corresponding theory for finite semigroups. We prove that every finite semigroup admitting a complete mapping is regular and that the problem reduces to principal factors. We classify the existence of a complete mapping in Rees matrix semigroups without zero, give a Hall-type criterion for Rees $0$-matrix semigroups over groups with complete mappings, and prove sufficient conditions for Rees $0$-matrix semigroups whose maximal subgroups do not have complete mappings. As the main application of the Rees $0$-matrix analysis, we show that $T_n$ has a complete mapping if and only if $n=1$ or $n\geq 4$. Equivalently, $T_n$ has a complete mapping if and only if the same holds for $S_n$. We prove that the full linear monoid of a finite-dimensional vector space has a complete mapping except in dimension $1$ over a field of odd order and in dimension $2$ over $\mathbb F_2$. We also prove that the partition monoid $\mathcal P_n$ has a complete mapping if and only if $n=1$ or $n\ge4$, and that every finite aperiodic regular $*$-semigroup has a complete mapping. As a consequence, the planar partition, Motzkin and Jones monoids have complete mappings. The paper concludes with open problems.

We present a natively multiscale operator learning method for the surrogate modeling of (numerical solvers for) multiscale partial differential equations (PDEs). The primary novelty of our method lies in a novel multiscale kernel frame function approximation technique. Leveraging this new kernel frame technique, we cast the operator learning problem as one of learning frame coefficients of output functions as a function of frame coefficients of input functions. The generalization step then automatically allows for a multiscale decomposition of the output functions. Our method is applicable to both tensor-product grids and point clouds. We present interpolation proofs, error estimates, and numerical convergence rates for our frame approximation. We the demonstrate the applicability of our method for the surrogate modeling of inherently multiscale PDEs. The new multiscale frame kernel method is significantly more accurate than popular neural operators on challenging problems from the literature, while simultaneously admitting an a posteriori multiscale decomposition upon generalization.

The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schrödinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schrödinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schrödinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schrödinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.

Related to a bounded-diameter bipartite analogue of the Henderson–Ryser conjecture, DeBiasio, Kamel, McCourt, and Sheats proved that the vertices of every $2$-colored complete bipartite graph can be covered by two monochromatic subgraphs, each of diameter at most four. We improve this bound on the diameter to the best possible value of three.

Given a set $V$ of $n$ vertices and an integer $k\ge1$, our goal is to maximize the number of edges in graphs $G_1, G_2, \ldots, G_k$, defined on $V$, under the constraint that the union of all graphs, thought of as a multi-graph, does not contain a rainbow copy of the path $P_5$ on $5$ vertices, that is, a copy of $P_5$ with each of its four edges belonging to a different $G_i$. We consider two versions of the problem, in which, respectively, $\sum_i e(G_i)$ and $\min_i e(G_i)$ is maximized. In the former case, we determine the maximum precisely for all $k\le n-1$ (and also for $P_4$). In the latter, we obtain an asymptotic value for $k\in\{5,6,9\}$ and formulate a very plausible conjecture for all other values of $k$. We also solve the problem for $k=4$, but under an additional assumption of completeness.

We introduce a matrix divisor function $\tau_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $\tau_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove an essentially sharp uniform upper bound for $\tau_n(T,M)$, for an arbitrary non-singular matrix $M$.

A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton–Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of $$-\Delta u+|Du|^\gamma=f$$ in $\mathbb T^d$, where $d\geq 2$, $\gamma>2$, and $f\in L^{q_c}(\mathbb T^d)$ with $q_c=d(\gamma-1)/\gamma$. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of $L^{q_c}$, we rule out this concentration and prove maximal $L^{q_c}$ regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local $L^{\gamma q_c}$-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton–Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.

Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ is supercritical in the sense of continuum percolation, then the first homology group $H_1(M)$ of $M$ can be correctly inferred from $G_M(n;r)$ with high probability as $n \to \infty$. Specifically, one can obtain $H_1(M)$ by taking the cycle space of $G_M(n;r)$ and quotienting out all the cycles of metric diameter $O(r|\log r|)$ (or of graph diameter $O(|\log r|)$). Our method of estimating $H_1(M)$ exploits a coarse-topological fact about supercritical percolation, as opposed to usual methods, which examine the topology of neighborhoods of the point cloud. Whereas previous methods use combinatorial models which require $O(n \log n)$ edges, our method only requires $O(n)$ edges. We also show that, in all phases of the thermodynamic regime, if one instead takes the quotient by cycles of metric diameter $o(r|\log r|)$, with high probability, one will not recover $H_1(M)$. Thus $\Theta(r|\log r|)$ is the “right scale.” On the way, we show that an arbitrary compact $d$-dimensional Riemannian manifold has a first homological percolation threshold in the sense of Bobrowski and Skraba \cite{BS2020} which coincides with the continuum percolation threshold on $\R^d$, a result previously only known for the flat torus. This strongly suggests that our results are optimal, in the sense that $H_1(M)$ cannot be inferred from $G_M(n;r)$ in the subcritical thermodynamic regime. All results hold for homology with arbitrary coefficients.

The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|n\xi\|\,\|n\zeta\|=0 $$ for all real $\xi,\zeta$. It is proven to hold for almost every pair $(\xi,\zeta)$. Schleischitz, however, has recently disproved the full statement and showed that the set of counterexamples contains a dense $G_\delta$ set. We prove that a set of counterexample pairs with the first coordinate being a badly approximable number has Hausdorff dimension at least $3/2$. We further show that the set of badly approximable numbers $\xi$ for which there exists $\zeta$ such that $(\xi,\zeta)$ is a counterexample to ULC has full Hausdorff dimension. This contrasts with the classical Littlewood conjecture, for which the set of possible counterexamples is known to have Hausdorff dimension $0$.