Papers

We consider Cantor-type sets of Hausdorff dimension zero, consisting of all numbers whose base-2 expansion can have a 1 only at positions belonging to a given sparse set (local count at least log k in every interval of length k). We prove that the measure induced by independent, non-identically distributed Bernoulli digits assigns full mass to numbers that are normal in every odd base. The proof extends Schmidt's 1960 method to this Hausdorff zero-dimensional setting, and we provide an explicit algorithmic construction of such numbers – yielding the first known examples of numbers deterministic in base~2 yet normal in all odd bases. This work supports our broader conjecture that given determinism in one base, normality in all multiplicatively independent bases is prevalent.

This paper investigates over-the-air federated learning (AirFL) in wireless systems where the access point is equipped with a multi-waveguide pinching antenna system (PASS). We adopt the widely studied learning-oriented AirFL formulation, which seeks to maximize the number of selected devices while keeping the aggregation distortion below a prescribed threshold. The resulting joint optimization of device selection, receive beamforming, and pinching-antenna placement is highly nonconvex due to the intricate coupling among these system variables. To address this challenge, we develop AirPASS, an alternating optimization framework with two main components: a homotopy-Riemannian margin-consolidation method for device selection and receive beamforming under a fixed PASS configuration, and a homotopy-assisted geometry optimization method for updating the pinching-antenna positions under fixed selected devices and beamformer.

We resolve a conjecture of de Saint Germain: all Y-friezes arise from $\mathrm{SL}_2$-friezes. Additionally, we give a threefold characterisation of those $\mathrm{SL}_2$-friezes that give rise to the same Y-frieze: an elementary characterisation using quiddities, a combinatorial characterisation using triangulated polygons, and a geometrical characterisation using chains of horocycles.

Power System State Estimation (PSSE) is commonly formulated as a nonconvex weighted least-squares (WLS) problem, making global optimality difficult both to attain and to certify. Recent work has shown that, when voltage magnitudes are known to sufficient accuracy, the remaining angle estimation subproblem can be reduced to phase synchronization and solved effectively using spectral initialization and spectral certification. This paper explains why these spectral methods succeed. We prove that their behavior is governed by the measurement error, normalized against the usual observability margin from the classical literature. Below a fixed threshold, spectral initialization recovers the true voltage angles to first-order accuracy, and the WLS estimator is unique up to a global phase. Moreover, a zero-duality-gap spectral certificate verifies recovery of this unique WLS estimate. In the noiseless observable regime, spectral initialization exactly recovers the true angles and certification is exact without local refinement. Numerical experiments on standard benchmark systems support the theory and show that the spectral methods remain effective beyond the conservative regime covered by the guarantees.

We study fixed-tolerance low-rank approximation in the matrix-free setting, where a matrix or linear operator $\mathbf{A}$ is accessible only through matrix-vector products and its rank must be determined adaptively to meet a prescribed error tolerance. We introduce a family of adaptive, matrix-free randomized QB algorithms. A randomized error indicator estimates the residual norm – in either the Frobenius or the spectral norm – directly from a random sketch, remaining accurate down to machine precision. A matrix-free rank-pruning step decouples the computational block size from the final rank, so that large, BLAS-3-friendly blocks can be used without over-estimating the rank, and an adjoint-free variant returns the orthonormal basis using only the forward operator. Across test matrices with diverse singular-value decays, the proposed methods attain ranks close to the truncated-SVD optimum while meeting the prescribed tolerance with high probability.

This paper provides the first near-optimal lower bounds for one-bit compressed sensing of approximately sparse signals lying in a scaled $\ell_1$ ball, which is a commonly adopted relaxation of the exactly $k$-sparse assumption. In prior works, the best known upper bounds on uniform Euclidean error are of order $\widetilde{O}((k/m)^{1/3})$, where $m$ is the number of measurements. Under sub-Gaussian matrices, we establish nearly matching lower bounds for both the canonical one-bit compressed sensing model and the uniformly dithered model. Our argument is to first embed a small Euclidean ball into the signal set, which is straightforward for the dithered model but relies on a lifting map for the canonical model, and then construct two signals in this small ball that are separated in Euclidean distance by at least $(k/m)^{1/3}$ (up to logarithmic factor) but are indistinguishable from the binary measurements. Moreover, our argument extends to approximately sparse signals that live in a properly scaled $\ell_q$ ball $(q\in [0,1])$, yielding a lower bound $\widetilde{\Omega}((k/m)^{\frac{2-q}{2+q}})$ that smoothly bridges the cases of exact sparsity ($q=0$) and $\ell_1$ sparsity ($q=1$). Finally, we discuss the extensions of our lower bounds to sub-Weibull matrices, adversarial bit flipping, matrix recovery, and characterize the transition to the non-sparse case.

We lift a product rule in the Cartan subalgebra of quantum $\mathfrak{sl}_2$ to a filtration of the plethystic representation $\Delta^{(n,m)}\mathrm{Sym}^d E$ of the affine group scheme of the algebraic group $\mathrm{SL}_2$, where $E$ is the natural representation and $\Delta^{(n,m)}$ the Weyl functor. This is a significant step towards a categorification of quantum $\mathfrak{sl}_2$. Our filtration is an addition to a growing family of field-independent isomorphisms of $\mathrm{SL}_2$ representations that include Hermite reciprocity and the Wronskian isomorphism. It is the first such field-independent result requiring multiple filtration layers. It is proved by combinatorial techniques using the authors' symmetric functions model for Weyl modules.

We are concerned with the properties of Ornstein-Uhlenbeck evolution operators acting on functions defined in a Hilbert space and p-integrable with respect to a suitable Gaussian measure. These operators provide solutions to the infinite dimensional, non-autonomous backward Kolmogorov equation. The first part of the paper focuses on some general regularization properties, while the second part carries on a deep analysis of the asymptotic behaviour in the periodic case. In particular, we identify the optimal convergence rate of the Ornstein-Uhlenbeck operator and we give an optimality criterion depending only on the drift term of the Kolmogorov equation.

This paper proposes Variational Inference-based Bayesian Estimation with Sobol screening (VIBES), a two-stage scalable framework for Bayesian uncertainty quantification (UQ). The proposed approach combines Sobol global sensitivity analysis (GSA) for screening and dimensionality reduction, followed by variational inference (VI) for UQ of kinetic, design/operational, and economic parameters. In the first stage, Sobol GSA is performed to identify dominant variables and parameters governing uncertainty in process outputs. In the second stage, Bayesian inference is performed only on the reduced dimensional space using VI, thus reducing computational burden and enhancing scalability. The framework is demonstrated on a process for bioadhesive production through base-catalyzed depolymerization of kraft lignin and subsequent crosslinking with isolated soy protein. A Python-Aspen interface is developed for automated simulation and parameter estimation, enabling Bayesian calibration through stochastic gradient-based optimization and automatic-differentiation. The methodology is generic and readily generalizable to other biomass conversion pathways. The results show that application of VIBES consistently reduces predictive uncertainty bounds across all model outputs by more than 80%, even when only the reduced-space input variables and parameters are optimized during Bayesian estimation. The framework can be potentially applied for scalable, uncertainty-aware decision-making in high-dimensional, complex chemical process systems.

This paper studies linear-quadratic mean field games on compact Riemannian manifolds with a hybrid interaction topology. The network structure is a superposition of a deterministic graph for local geometric connectivity and a stochastic directed graph for non-local interactions. The global graph is constructed via random sampling based on a continuous kernel $K$. The out-degree of each node scales as $\Theta(\log N)$ or as $\Theta(N)$ to represent a sparse or dense network, respectively. In the infinite-population limit, the continuum system is governed by a coupled system of forward-backward partial differential equations, where the dynamics of the expected state incorporate the integral operator corresponding to the non-local sampling. The existence of a Nash equilibrium is established for this limit system. Furthermore, the approximation error is analyzed using operator concentration inequalities and analytic semigroup theory. Non-asymptotic high-probability error bounds between the finite-population empirical state and the continuum limit are derived. The convergence rates differ depending on the two topological regimes. Under the dense regime, the tracking error exhibits a polynomial decay rate dependent on the manifold dimension and Sobolev regularity, while under the sparse regime, the error decays at a rate of $\mathcal{O}((\log N)^{-1/2})$.

The Ding-Iohara-Miki (DIM) algebra (quantum toroidal algebra of $\widehat{gl_1}$) is related to a wide class of quantum many-particle integrable systems, a typical one being the Ruijsenaars trigonometric system with eigenfunctions that are a triad formed by the Noumi-Shiraishi power series, the Macdonald polynomials, and the Baker-Akhiezer multivariable function. Other integrable systems of this type are obtained from the Ruijsenaars system by twisting. At the same time, the Ruijsenaars Hamiltonians are directly related to the Hamiltonians of another quantum integrable system, the Cherednik DAHA Hamiltonians of type $A$ (and their twisted versions in the twisted case), due to the correspondence between the DIM algebra and the spherical DAHA. The eigenfunctions of the DAHA Hamiltonians are non-symmetric Macdonald polynomials. Similarly, there is a class of integrable DAHA Hamiltonians of type $C^\vee C$, the spherical version of which, in turn, allows one to generate integrable Koornwinder Hamiltonians. The eigenfunctions of these two integrable systems are, respectively, non-symmetric and symmetric Koornwinder polynomials, which are our main interest in this paper. Here we consider the cases of both type $A$ and type $C^\vee C$ systems, since they are sufficiently similar, and point out important distinctions between them.

We study graded locally finite and graded just infinite Steinberg algebras of ample Hausdorff groupoids. For gradings by discrete groups induced by a cocycle, we characterize finite-dimensional homogeneous components in terms of finite cocycle fibres. Moreover, we obtain general criteria ensuring that, once one homogeneous component is infinite-dimensional, all of them are. Under suitable isotropy hypotheses, we show that for locally finite Steinberg algebras all irreducible representations act by finite-rank operators. We also obtain necessary conditions for Steinberg algebras to be just infinite. In addition, we give a complete characterization of graded just infinite Steinberg algebras in terms of finite invariant reductions and identify conditions under which graded just infiniteness is equivalent to graded simplicity. We apply these results to Steinberg algebras of Deaconu–Renault groupoids and to subshift algebras. For Deaconu–Renault groupoids, we obtain dynamical criteria for graded just infiniteness. For subshift algebras, we show that there are no nontrivial finite-dimensional examples, we characterize when all homogeneous components are infinite-dimensional, and in the finite-alphabet case, we prove that graded just infiniteness, graded simplicity, minimality of the associated groupoid, and hyper-cofinality of the subshift are equivalent.

This paper concerns the $H$-convergence of nonlinear nonlocal monotone operators defined through the Riesz fractional gradient and divergence. We show that the $H$-convergence in this nonlocal framework is equivalent to the $H$-convergence of the corresponding local one. As a consequence, we obtain a $H$-compactness result for a suitable class of nonlocal monotone operators. We then study the $\Gamma$-convergence of nonlocal energy functionals associated with the subclass of conservative monotone operators, proving that it is equivalent to the $\Gamma$-convergence of the corresponding local energies. A key ingredient is a new uniqueness result for the integral representation of both local and nonlocal functionals. As a by-product, we obtain the $\Gamma$-compactness of the class of nonlocal energies under consideration. Finally, we show the equivalence between the $H$-convergence of nonlocal conservative monotone operators and the $\Gamma$-convergence of the associated energy functionals.

Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

To a log-resolution of a complex pair $(X,\I)$ we associate the weights $\lambda_E=(a_E+1)/\nu_E$ of the prime divisors $E$ of the total transform of $\I$, where $\nu_E$ is the vanishing order of $\I$ and $a_E$ the Jacobian (Mather discrepancy) order along $E$. On Jacobian-adapted resolutions, $\min_E\lambda_E$ is the volume-growth exponent of $K_\I=\sum_j|f_j|^2$ for the $2n$-dimensional Hausdorff measure: the log canonical threshold for smooth $X$, Ishii's Mather one for singular $X$, with a logarithmic correction given by a resonance multiplicity. The main object is the divisorial persistent homology $\DPH$: the persistent homology of the dual complex filtered by the subcomplexes on divisors of weight at most $\alpha$. It records incidences of divisors, which the threshold ignores. We prove that $\DPH$ is invariant under blow-ups along smooth centres in normal crossing with the boundary, for adapted resolutions; $A_2$ shows adaptedness is necessary. Independence of the resolution holds for complex algebraic pairs (weak factorization) and normal surface singularities, and is a hypothesis for non-algebraic analytic pairs of dimension $\ge3$. When $V(\I)$ is complete, $\DPH$ equals the weight-zero part of the cohomology of the union of divisors of weight at most $\alpha$ (Deligne). For $A_n$ the minimal resolution is adapted iff $n$ is odd, and $\DPH_0$ has one finite bar of length $(n-1)/2$, so within this family it detects $n$ (detection, not classification). We compare with Nicaise–Xu and de Fernex–Kollár–Xu.

In this article, we address the problem of estimating a forcing parameter in a stochastic differential equation inspired by a model that describes instantaneous turbulent kinetic energy. The stochastic differential equation we analyze is of the nonlinear McKean-Vlasov type, where the drift term depends on a power of the expected value of the solution, which also introduces nonlinearity in an algebraic sense. We propose an estimation algorithm based on the Expectation-Maximization framework and show the consistency of our method. We illustrate our findings through numerical experiments.

We give a new proof of the celebrated theorem of Knutson and Tao that the spectra of triples $A, B, A+B$ of Hermitian matrices exactly correspond to positions of boundary rays of honeycombs. Most importantly, our proof gives new insights into why honeycombs are related to Hermitian matrices in the first place. Our proof is axiomatic: We distill four essential properties shared by honeycombs and spectra of Hermitian triples, and show that any two objects sharing these four properties must be equivalent. In this way, we argue that honeycombs are `model organisms' for Hermitian triples: they are families of objects satisfying the same defining properties, but in more obvious ways.

We extend the linear programming-based algorithm of De Rosa et al~\cite{derKhaWan24} for K-means clustering to two important clustering paradigms: fair K-means clustering and spectral clustering. For fair K-means clustering, we show that widely used notions of group fairness can be incorporated into the partition-matrix formulation of K-means clustering through a linear number of linear inequalities. For spectral clustering, we consider a linear programming relaxation of the minimum ratio-cut problem that fits naturally within the same framework. We complement these formulations with problem-specific initialization and rounding procedures and evaluate the resulting algorithms on a large collection of real-world data sets. Denoting by $n$ the number of data points, our computational results demonstrate that the proposed approach solves $90\%$ of benchmark instances with $n \leq 3000$ to within $1\%$ optimality in at most three hours. This in turn demonstrates the remarkable strength of the proposed LP relaxations in both applications. Moreover, for more than $56\%$ of the instances, the proposed algorithm finds better solutions than those produced by popular fair Lloyd-type and spectral clustering heuristics.

The paper is devoted to the study of the connection between the numerical dispersion arising in FDTD modeling of electromagnetic signal propagation in nondispersive homogeneous media optically different from vacuum and the Courant number in the 2D case. The main results are formulated in the form of four statements, as well as a number of corollaries and remarks that determine the nature of the numerical dispersion, the optimal value of the Courant number and the limitations of the method. It is proved that the optimal choice of the Courant number eliminates the numerical dispersion and extends the capabilities of the developed numerical algorithm to media, which refractive index lesser than refractive index of vacuum, as well as media with negative refraction.

An $\infty$-harmonic function is a viscosity solution of $\nabla^2 u(\nabla u,\nabla u)=0$, or equivalently, an absolute minimizer of $\|\nabla u\|_{L^\infty}$. We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$-harmonic functions in domains of $\mathbb{R}^2$ are $C^{1,1/3}$. 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p\to\infty$ limit of the classical duality between $p$-harmonic and $q$-harmonic functions in $\mathbb{R}^2$, where $\frac1p+\frac1q=1$.

Learning from positive data models a scenario in which a learner receives longer and longer initial segments of an enumeration of an unknown set and repeatedly outputs a conjecture about it. Classical identification in the limit requires the learner to eventually stabilise to a hypothesis naming the target exactly, while anomalous learning relaxes this requirement by allowing finitely many errors in the final hypothesis. We develop a natural generalisation for computably enumerable (c.e.) sets in which correctness is determined by an arbitrary equivalence relation on their indices that respects equality of the sets they enumerate. We call the resulting framework tolerant learning. Exact learning and finite-anomaly learning arise as special cases. By comparing tolerance relations according to the families they make learnable, we obtain a new degree structure, the tolerant learning hierarchy. We extend the classical methods of locking sequences and tell-tales to this setting and derive a combinatorial characterisation of those tolerance relations that are universal (that is, under which every family of nonempty computably enumerable sets is learnable). We then show that the hierarchy is a bounded lower semilattice but not an upper semilattice, is atomless and coatomless, and contains chains and antichains of size continuum. We also analyse natural benchmark relations, including c.e.-index analogues of standard equivalence relations from descriptive set theory, and establish substantial incomparability among their learning powers. Finally, we separate non-effective from computable tolerant learning by exhibiting a universal relation that is not computably universal. These results show that semantic tolerance gives rise to a genuine degree theory at the interface of algorithmic learning and the theory of equivalence relations.

We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank–Daubechies–Freeman–Freeman and confirming the conjectured characterization communicated to us by those authors. We provide two different proofs of this fact, both based on a decomposition of the $\ell^2$-coefficients of each random variable. The first is a compactness proof, which makes use of the infinite divisibility of limit laws of tail sums. The second is a quantitative proof, which substitutes the compactness step with an explicit dichotomy based on anticoncentration estimates of Sperner type. This latter proof was partially LLM generated based on the ideas in the first proof and a considerable amount of guidance by the authors. An autoformalization of our main result in Lean 4 is also provided, following the ideas in the quantitative proof.

We introduce a notion of K-polystability for compact Kähler holomorphic Poisson manifolds. On the one hand, this notion of stability is well-adapted to constructions of moduli spaces. For instance, when the underlying manifold is K-polystable with reductive reduced automorphism group, Poisson K-stability is equivalent to geometric invariant theoretic stability in the space of Poisson bivectors, but there also exist K-unstable varieties that become stable after incorporating a Poisson structure. On the other hand, the Poisson K-stability condition interacts well with generalized Käher metrics – the background geometry of (2,2) supersymmetric string theory. In particular, we conjecture that Poisson K-polystability characterizes the existence of constant scalar curvature symplectic generalized Kähler structures with a sufficiently small Poisson tensor – a natural extension of the Yau–Tian–Donaldson (YTD) conjecture. Our main result is a proof of the existence part of this “semiclassical YTD conjecture” for Poisson structures on Kähler–Einstein Fano manifolds, using infinite-dimensional momentum map techniques. In this way, we obtain the existence of many new examples of symplectic generalized Kähler structure of constant scalar curvature, and prove the conjecture completely in the case of the projective plane.

We completely resolve the algebraic Montgomery - Yang problem, a conjecture of Kollar stating that every rational homology projective plane with quotient singularities and a simply-connected smooth locus has at most three singular points. The crux of our proof is a new lattice theoretic constraint, obtained by combining Donaldson's diagonalization theorem with the distinguished spin^c structure on the smooth locus whose determinant line bundle is the canonical bundle. Together with the orbifold Bogomolov - Miyaoka - Yau inequality, this constraint rules out all remaining cases in the problem and completes the proof.

We show that an enriched $\infty$-operad is completely determined by its category of right modules together with a `marking' of the representable modules. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of colored $\mathcal{V}$-enriched $\infty$-operads and a certain full subcategory of the category of presentably symmetric monoidal $\mathcal{V}$-module $\infty$-categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-operad theory to the theory of presentably symmetric monoidal $\infty$-categories. As an application, we describe a notion of univalence (or Rezk-completeness) for enriched $\infty$-operads, and directly construct an equivalence between univalent $\mathcal{S}$-enriched $\infty$-operads in our sense and Lurie's model of $\infty$-operads. We study envelopes and categories of algebras for enriched $\infty$-operads and show that, in the $\mathcal{S}$-enriched case, the resulting notions agree in both models.

Wilson loop operators in Chern-Simons theory have revealed profound links between quantum field theory, the fractional quantum Hall effect, topology, conformal field theory, and string theory. In Chern-Simons theories with charge conjugation symmetry, we construct a new class of observables: codimension-two monodromy defects around which fields return to themselves up to charge conjugation. Whereas Wilson loops are labeled by integrable representations of an untwisted affine Lie algebra, monodromy defects are labeled by those of the corresponding twisted affine algebra. The modular and fusion data of these two algebras determine the exact correlation functions of Wilson lines and monodromy defects, which together furnish a $\mathbb{Z}_2$-crossed braided tensor category. The spectrum of line defects in Chern-Simons theory thus gives a physical realization of every algebra in Kac's classification of affine Lie algebras: untwisted for Wilson loops, twisted for monodromy defects. We also determine the exact 't Hooft expansion of monodromy defects in $SU(N)_k$ Chern-Simons theory and identify their holographic duals in topological string theory. The insertion of the lightest monodromy defect has a striking effect: it replaces the resolved conifold background of the Gopakumar-Vafa duality by a specific orientifold of the resolved conifold, transmuting the dual theory of oriented strings into one of unoriented strings. Each excited monodromy defect is then realized as a collection of branes in the orientifold background, with the brane content determined by the representation of the twisted affine algebra that labels the defect.

We construct Hilbert-space isomorphisms identifying the spaces of radiation fields on the event horizon and null infinity that are induced by the forward scattering map on the Schwarzschild exterior for fixed spherical harmonic mode solutions to the linear scalar wave equation arising from Cauchy data sets at $t=0$ which are regular at the event horizon, in the sense that the local energy at the event horizon is finite. We show that fixed spherical harmonic mode solutions with no radiation on the event horizon must decay exponentially in time, at a rate that is determined by the surface gravity. On the other hand, we construct examples of fixed spherical harmonic mode solutions with no radiation on null infinity and which have polynomial decay along the event horizon. Finally, we construct examples of polynomially decaying solutions to the linear scalar wave equation which are regular at the event horizon, have unbounded support in spherical harmonic modes, and induce no radiation on the event horizon.

Determinantal point processes (DPPs) have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets. One of the key strengths of DPP based samplers is that they can be deployed over very general spaces, in contrast to more classical sampling methods beyond i.i.d. which tend to be tailored to particular well-structured settings, principally Euclidean spaces. In this work, we establish explicit rate guarantees for determinantal sampling in spaces that extend far beyond known Euclidean setups, focussing on spectral kernels obtained from eigenspaces of naturally associated Markov diffusion operators, including in particular the Laplacian on manifolds and nearest-neighbour type graphs on data point clouds and the discrete Schrodinger operator on large scale networks. This covers, in particular, the settings of Riemannian manifolds and discrete networks, as well as the problem of batch sampling from discrete datasets. Overall, our approach achieves guarantees comparable to $\big(\text{sample size}\big)^{-\frac{1}{2}-\frac{1}{2d_{\text{int}}}}$ , which match with the rates for well-known determinantal samplers on Euclidean spaces of comparable dimension. In terms of techniques, we connect to the celebrated Weyl's Law for spectra on general spaces, and leverage tools from the theory of Markov diffusion operators and Dirichlet forms, which could be of independent interest in this area. Our approach is applicable to a very wide plethora of ambient spaces beyond Euclidean as soon as these ingredients are available, including e.g. fractals, leading to guarantees that will be governed by the intrinsic structural constants of such spaces.

Standard bases of modules over algebras of formal power series play the same role as Gröbner bases of modules over polynomial algebras. In this article, we first prove the analogue of the diamond lemma for modules over formal power series; it characterises standard bases in terms of unique remainders and standard representations. Then, using standard representations, we provide a method to construct a standard basis of the module of syzygies of a standard basis. This construction can be applied inductively to obtain a free resolution, similar to the Schreyer resolution, for finitely presented modules over formal power series.

Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate integral binary quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of $x$. We develop a method for answering this question for certain cubic and quartic polynomials $P$, as well as for certain polynomials of the form $P(x)=R(Q(x))$, where $R(t)$ and $Q(x)$ are polynomials of degrees $3$ and $2$, respectively. Applying this method with $F(y,z)=y^2+z^2$, $R(t)=t^3-4$ and $Q(x)=x^2$, we conclude that $x^6-4$ is a sum of two squares infinitely often. In turn, this implies that the equation $y^2+x^3y+z^2+1=0$ has infinitely many integer solutions. Prior to this work, it was the shortest equation for which it was unknown whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations for which the finiteness problem remains open. All main results of this paper have been formalized in Lean using Aristotle.