Papers

We construct an explicit countable group \(G_*=N(R)\rtimes A\) whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here \(A\) is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and \(R\) is an explicitly defined quotient of the group algebra \(\mathbb F_2[A]\). The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in \(\mathfrak B_4\mathfrak B_2\). This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.

Let $X$ be a connected graph and let $\lambda_1(X)$ denote the spectral radius of its adjacency matrix. An edge $e$ of $X$ is sustainable if it can be deleted and replaced by a non-edge so that the resulting graph is connected and its spectral radius does not decrease; $X$ is sustainable if every edge is. We study this edgewise notion of spectral robustness for trees, where each replacement is an elementary exchange between trees on the same vertex set. We establish basic properties of sustainability and exhibit infinite families of sustainable and non-sustainable trees. Our main results concern trees with a vertex of large degree: if $T$ is a tree on $n$ vertices with $\Delta(T)\geq n/2$, then every vertex of maximum degree is a principal vertex, that is, a vertex at which the Perron vector is maximal. Building on this, we prove that every tree with $\Delta(T)\geq n/2+1$ is non-sustainable, and we classify the sustainable trees in the boundary case $\Delta(T)=n/2$: for $n\geq 4$, the only such trees are the double broom $B(2,n/2,n/2-1)$ and one exceptional tree on eight vertices. We conclude with open problems on the generalized tree-shift order and the asymptotic proportion of sustainable trees.

Does randomization improve the worst-case first-order complexity of nonconvex optimization when the objective is very smooth? We prove that, for every fixed finite integer $p\ge3$, finding an $\eps$-stationary point requires $\Omega_p(\eps^{-5/3})$ exact value-and-gradient queries, even for $C^\infty$ objectives with initial gap at most one and simultaneous global Lipschitz budgets of one for derivatives of orders $1,\ldots,p$. Together with the known deterministic upper bound, this determines the accuracy exponent up to a single logarithmic factor. The lower bound holds for unrestricted adaptive randomized algorithms, unbounded query locations, and outputs that need not have been queried. One algorithm-independent hard distribution suffices in dimension $O_p(\eps^{-47/3})$. The construction combines a block-chain potential with an exact quartic core, repeated quadratic spectra, and two static Gaussian tensor perturbations. A scheduled Haar argument controls the entire adaptive query span. An innovation-only Gaussian comparison then bounds exact transcript information without assuming finite precision or a minimum query separation. This establishes the $5/3$ randomized accuracy barrier under every prescribed finite smoothness order, extending the previously matched third-order case.

We study various symmetry conditions related to the Auslander-Reiten formula for finite-dimensional algebras. In particular, we define a new class of algebras which we call $E$-symmetric. A theorem by Derksen-Weyman-Zelevinksy implies that all Jacobian algebras are $E$-symmetric. We prove the converse for the class of monomial algebras.

Anomalous diffusion in heterogeneous media is ubiquitous across physical and biological systems, where spatially varying microstructures give rise to distinct transport regimes. Variable-order nonlocal operators provide a natural modeling framework for describing heterogeneity and transitions between diffusion regimes. However, the position-dependent long-range interactions inherent in these operators impose prohibitive memory costs and make numerical methods developed for constant-order operators computationally impractical. Consequently, accurate and computationally efficient numerical methods for variable-order nonlocal operators remain scarce. In this paper, we propose accurate finite difference discretizations for the variable-order fractional Laplacian $(-\Delta)^{\alpha(\mathbf{x})/2}$ and develop a fast Fourier transform (FFT)-accelerated scheme for their efficient implementation. The computational and memory costs are $O\!\big(MN\log(2N)\big)$ and $O(MN)$, respectively, where $N$ is the total number of spatial points and $M \ll N$. The method is also naturally parallelizable. We establish its consistency and accuracy through a rigorous error analysis that quantifies the local interplay between solution regularity and the spatially varying order $\alpha(\mathbf{x})$. To the best of our knowledge, this is the first work to quantify how these two factors jointly determine convergence rates, with numerical experiments demonstrating this dependence. Extensive numerical experiments confirm the accuracy and efficiency of the proposed method. Applications to anomalous diffusion in composite materials further demonstrate its ability to capture the effects of material heterogeneity.

We present an $\infty$-categorical refinement of classical $KK$-theory and $E$-theory for graded, $G$-equivariant for a locally compact second countable group $G$, Real $C^*$-algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$-theory spectra, and explicitly analyse the induced endomorphisms on $K_0$-groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$- and $E$-categories are not a Dwyer-Kan localization of graded $C^*$-algebras.

Let $q$ be a prime power. An $(r, i)$-regular fat $\mathbb F_q$-linear set has exactly $r$ points of weight $i\ge2$, called fat points. All of its remaining points have weight one, and at least one such point occurs. We solve two open problems posed by three of the authors of this paper. First, we derive divisibility constraints on the point weights using the scalar stabilizer of the defining subspace. Consequently, a regular fat linear set of rank $2i$ in $\operatorname {PG}(1,q^n)$ has at most two or exactly $q + 1$ fat points. In the latter case, the fat points form an $\mathbb F_q$-subline. Second, we construct higher-dimensional regular fat linear sets whose fat points do not form an $\mathbb F_q$-subgeometry. In particular, for every $k\geq 3$ we obtain examples of rank $2k$ whose fat locus consists of an $\mathbb F_q$-subgeometry $\operatorname {PG}(k-2,q)$ in $\operatorname {PG}(k-1,q^4)$ together with one external point. This construction is then generalized further. Finally, we apply our projective-line results to $q$-matroid representability and present an infinite family of non-representable rank-two $q$-matroids arising from partial spreads.

A class of models which brings to the fore the inner logic of the calculus of probability instead of assigning probabilities to sentences is introduced and investigated.

For the counts $(X_1,\dots,X_n)\sim\mathrm{Mult}(N;1/n,\dots,1/n)$ of $N$ observations in $n$ ordered bins we study the variance of the discrete total variation $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$. An exact decomposition splits $\mathrm{Var}(d)$ into per-edge, adjacent-pair and disjoint-pair terms and yields, for fixed $n$ as $N\to\infty$, $\mathrm{Var}(d)/N=\alpha(n)-\beta(n)/N+\gamma(n)/N^2+o(N^{-2})$. For every $n\ge 2$, $N\alpha(n)$ is the classical normal-theory variance (Hoel, Kamat) of the sum of absolute successive differences of $n$ independent $N(0,N/n)$ variables; the correction $\beta(n)$ is obtained in closed form for $n\ge 3$ (the case $n=2$ is treated separately). Two disjoint edge differences $D_1=X_2-X_1$ and $D_3=X_4-X_3$ are exactly uncorrelated and asymptotically independent, yet $\mathrm{Cov}(|D_1|,|D_3|)=-1/(\pi n)+O(1/N)$: a correlation of relative order $1/N$, invisible in the Gaussian limit and absent under Poissonized sampling, whose constant is proportional to the exact joint fourth cumulant $\kappa_{2,2}(D_1,D_3)=-4N/n^2$ and is proved here on the lattice. The adjacent pair contributes the lattice-Edgeworth constant $a_*=-2\sqrt{3}/(9\pi)$, of which one quarter is a smooth Edgeworth term and three quarters a lattice-corner term; the disjoint skeleton $h(m)$ is closed through a Bessel generating function; and $\gamma(n)$ ($n\ge 5$) is expressed through a single triangular-lattice constant whose closed form we conjecture. Together with the mean of Part 1, the variance supplies the two moments used by moment-matched approximations to the null law of $d$. This is Part 2 of a series; the cyclic statistic $d^{c}$ will be treated in Part 5.

Ideal Poisson–Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson–Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson–Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson–Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson–Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg–Poisson–Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.

The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to $\C^N$, with prescribed finite nerves, or are unions of Fatou–Bieberbach domains whose nonempty finite intersections are again Fatou–Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some $\C^N$, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in $\C^5$. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.

Let $G$ be a semisimple Lie group that is not isogenous to the isometry group of real or complex hyperbolic $n$-space. We show that the rational rank is a profinite invariant for irreducible lattices in $G$, unless $G$ has type $D_{2n}$ which forms a genuine exception. In particular, profinitely isomorphic lattices in $G$ of type $D_{2n}$ can have differing virtual cohomological dimension. As another corollary, uniformity is a profinite property for irreducible lattices in $G$ outside type $D_4$. Assuming CSP, the existence of triality symmetries gives uniform and non-uniform profinitely isomorphic lattices in type $D_4$.

Spatial sudden death of entanglement refers to the disappearance of quantum correlations between subsystems after a certain distance called the entanglement length. Specifically, given a density matrix $\rho_{ABC}$ on a tripartite system $ABC$ with $B$ spatially separating $A$ and $C$, the entanglement length is the distance $\ell(A,C)$ after which the reduced density matrix $\rho_{AC}$, given by tracing out the $B$ subsystem, is separable. We show in the case of gapped ground states in any dimension that the entanglement length is finite on finite subsystems. Restricting to one-dimensional systems, we prove that normal matrix product states have a finite entanglement length independent of $A$ and $C$, and that a finite pure state with such uniform entanglement length has a matrix product state representation with bond dimension independent of $A$ and $C$. Consequently, in one dimension, entanglement length distinguishes matrix product states from all other pure states.

In many practical applications of Bayesian inference, it is difficult or undesirable to compute the gradient of the negative log-posterior, motivating the study of derivative-free samplers, also called zeroth-order samplers. The aim of this paper is to generalize and rigorously analyze a multiscale derivative-free sampler recently proposed in the literature by Pavliotis, Stuart and Vaes (2022). In the discrete-time setting, the multiscale sampler we study bears a strong resemblance to other recently proposed methods, such as the so-called zeroth-order perturbed stochastic gradient descent algorithm (Jin et al., 2018) and the zeroth-order Langevin Monte Carlo algorithm by Roy et al. (2022). We present error estimates with respect to all the parameters of the problem, both in the continuous-time and discrete-time settings. The error estimates are in three different metrics: strong convergence, convergence of ergodic averages, and uniform-in-time weak convergence.

We study whether an unnormalized graph Laplace operator can determine both the geometry of a manifold and its sampling density. On a fixed compact smooth manifold without boundary, we use a known positive, continuous, strictly decreasing radial kernel at a fixed positive bandwidth. For positive sampling densities, the operator built from geodesic distances determines the Riemannian metric and identifies the density up to equality almost everywhere. When the metric is induced by a Euclidean embedding, the operator built from ambient distances also determines the embedding up to rotations, reflections and translations, and the density. On compact parameter classes with positive continuous densities, minimizing the distance between empirical and popula- tion operators gives estimates that converge to the true parameters under independent, identically distributed sampling. This recovery result assumes that the empirical operator can be evaluated at arbitrary points of the manifold.

We present and analyze a spatially and temporally adaptive high-order method for the one-dimensional Richards equation describing vertical water transport in unsaturated soil, with an emphasis on resolving sharp wetting fronts while satisfying a fully discrete water mass balance. The spatial discretization uses an arbitrary-order local discontinuous Galerkin spectral-element formulation with collocated Legendre-Gauss-Lobatto quadrature, and the time integration is based on a high-order Rosenbrock-Wanner method. We focus primarily on the mixed formulation of the Richards equation, which advances water content as the conserved variable while evaluating the nonlinear diffusion operator in terms of pressure head; the resulting fully discrete scheme satisfies the water mass balance in exact arithmetic, independent of spatial resolution and time step size, including under spatial and temporal adaptivity. In contrast, the pressure-head formulation advances pressure head through a nonlinear capacity function and satisfies the corresponding conservation identity only at the semi-discrete level. Numerical experiments examine convergence through spatial and temporal refinement studies, with mass conservation assessed under spatial and temporal adaptivity using the solver flux output method (SFOM). The SFOM results confirm machine-precision mass conservation for the mixed formulation, whereas the pressure-head formulation exhibits mass balance errors above roundoff. Moreover, in the temporal refinement studies, the mixed formulation yields more accurate solutions than the pressure-head formulation for a prescribed time step size. The combination of high-order accuracy, spatial and temporal adaptivity, and fully discrete mass conservation makes the proposed mixed-form discontinuous spectral-element method a promising approach for simulating unsaturated flow in next-generation land models.

By virtue of their relation to Fulton–MacPherson compactifications, the cohomological field theory of Hilbert schemes of points on a surface, in extremal curve classes, is universally and effectively determined by the characteristic numbers of the surface and by the corresponding theory of Hilbert schemes of the affine plane. As a consequence, we compute the extremal genus 1 Gromov–Witten invariants of Hilbert schemes of all surfaces.

We study the boundary corrector $\theta_\epsilon$ in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet–Masmoudi and Prange, on smooth uniformly convex domains $\theta_\epsilon$ converges to the solution $\theta^*$ of a homogenized problem with boundary data given by the boundary layer tail $d(n)$, which is determined by a half-space problem for each normal $n$. In the lifted form of Gérard-Varet and Masmoudi, $d(n)$ is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator $N_n$ of this half-space problem. Our main result is that $N_n$ satisfies a nonlinear equation on the period cell alone, so that $d(n)$ and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along $n$. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute $d(n)$ at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for $N_n$ that captures the leading term of $d(n)$ for small contrast, for the truncated problem. Finally, we approximate $\theta_\epsilon$ at finite $\epsilon$ by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other $\theta^*$, and test it on the disk.

This paper investigates a shift-sum decoding algorithm that employs dual codewords. Existing dual codeword shift-sum decoding schemes mainly rely on minimum-weight dual codewords. However, the number of such codewords is often limited, which restricts the amount of reliability information available to the decoder. To address this issue, we extend the component counting mechanism from minimum-weight dual codewords to dual codewords whose Hamming weights lie within a prescribed upper bound. We further introduce a normalized component counting rule to suppress the overcounting bias caused by the much larger populations of higher-weight classes. Based on this framework, we develop a hard-decision iterative error reduction decoder and a soft information-aided iterative error reduction decoder. Numerical results show that the proposed method consistently outperforms existing minimum-weight shift-sum baselines especially for short codes where additional dual codeword information is particularly valuable.

Recovering piecewise-smooth functions with jump discontinuities from noisy, indirect, and incomplete data is a central task in computational inverse problems. Bayesian neural networks (BNNs) with heavy-tailed weight priors have emerged as effective edge-preserving priors for such problems. However, posterior inference for these models remains computationally challenging, and existing approaches settle for point estimates obtained by directly optimizing heavy-tailed log-densities. We propose GSBL-BNNs, which model the neural network parameters hierarchically using sparsity-promoting priors from generalized sparse Bayesian learning (GSBL). The hierarchical formulation recovers several popular marginal priors, including Laplace and Student's $t$ (including Cauchy). A key modeling ingredient, novel among edge-preserving BNNs and native to the hierarchical formulation, couples each neuron's bias to its weights through a shared hyperparameter, so that the induced jump discontinuities are free to occur anywhere in the domain while the conditionally Gaussian structure underlying efficient inference is preserved. Furthermore, we develop a block-coordinate descent (BCD) algorithm for maximum a posteriori (MAP) estimation. This BCD approach alternates between standard network training with iteratively reweighted $\ell^2$-regularization and exact, closed-form hyperparameter updates for gamma and inverse gamma hyperpriors. Building on the BCD algorithm, we perform approximate Bayesian inference and uncertainty quantification (UQ) using deep ensembles. Computational experiments—including signal denoising, deblurring, and inverting a two-dimensional hyperbolic conservation law— show that GSBL-BNNs deliver accurate, edge-preserving reconstructions with built-in UQ. Compared with Gaussian- and Cauchy-BNNs, they yield more accurate predictive means and avoid overconfident uncertainty estimates.

We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains $\Omega\subsetneq\mathbb S^n$, $n\ge2$. Set $p_n^*=2n/(2n-3)$. For the Lane–Emden equation $-\Delta_g u=u^p$, $1<p\le p_n^*$, we prove that the set of positive solution pairs $(p,u)$ has a connected component containing the near-linear solution curve and projecting onto $(1,p_n^*]$. Every pair on this component satisfies $\nabla_g^2(u^{-(p-1)/3})>0$ in $\Omega$. For the torsion equation $-\Delta_g u=1$, we establish $\nabla_g^2\sqrt{u}<0$ for its unique Dirichlet solution in all dimensions $n\ge2$, without an additional diameter or circumradius assumption. In both settings, the solutions specified above have a unique nondegenerate maximum and strictly geodesically convex positive superlevel sets. For the non-power reaction $f(s)=s^p/(1+\lambda s^\delta)$, with $1<p\le p_n^*$, $\lambda>0$, and $0<\delta\le(p-1)/3$, we also prove the existence of at least one positive solution $u$ for which the transform $\left(\int_u^\infty ds/f(s)\right)^{1/3}$ has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.

Rotation symmetric Boolean functions are Boolean functions invariant under cyclic permutation of their variables and can be represented by binary necklaces. We express local sensitivity as a weighted bichromatic degree in the necklace quotient and derive an exact formula for the multiplicities between neighboring necklaces. For an odd prime number of variables, we obtain explicit sensitivity formulas and show that multiplicity corrections occur only on an asymptotically negligible family of necklaces. For monomial rotation symmetric Boolean functions, we prove that every odd-degree function has maximum sensitivity, determine the sensitivity of every quadratic function, and determine the sensitivity of the consecutive quartic family. We also determine exactly the sensitivity of cyclic run-threshold functions for arbitrary length and threshold.

We provide several results showing that sums of “small” Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that $C+C=[0,2]$. 2) For any $n\ge 2$ there exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that a Minkowski sum of $n$ copies of $C$ is an interval $[0,n]$, while a sum of any smaller number of copies of $C$ is still a Cantor set. 3) For any $n\ge 2$ there exists a collection of Cantor sets $C_1, C_2, \ldots, C_n\subset [0,1]$ of zero lower box-counting dimension such that a Minkowski sum of any $n-1$ sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all $n$ sets is an interval $[0,n]$. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff $h$-measure for an arbitrary given gauge function $h$.

We develop an intrinsic rigidity theory for bar-and-joint frameworks whose vertices are constrained to a Riemannian manifold and whose edges are given by minimizing geodesic arcs. Such frameworks are called rigid if they cannot be deformed beyond the ambient isometries, while preserving the geodesic edge lengths. For admissible placements, for which adjacent vertices are joined by unique minimizing geodesics, we define a coordinate-free rigidity operator as the differential of the squared geodesic distance of the edges. By the first-variation formula, its edge rows are expressed in terms of the Riemannian logarithm, providing an intrinsic analogue of the classical Euclidean rigidity matrix. Trivial motions are characterized by orbits of the diagonal action of the isometry group, leading to notions of infinitesimal and local rigidity. We further develop a covariant second-order rigidity theory using the Riemannian Hessians of the edge constraints and characterize second-order rigidity through self-stresses and associated quadratic forms. Finally, we show that infinitesimal and second-order rigidity can be used as sufficient conditions for the local rigidity of an intrinsic framework. Our overarching goal is to provide a unified differential-geometric formulation of intrinsic framework rigidity on smooth manifolds.

We introduce a new criterion which guarantees that a conservative local diffeomorphism is both non-uniformly expanding (in particular has all its Lyapunov exponents positive) and stably ergodic. The criterion involves averaging expansion rates over preimages and allows us to construct explicit examples in a variety of homotopy classes, including those represented by linear endomorphisms with contracting directions.

We study the asymptotic behavior of extreme events in multivariate stationary sequences, with particular emphasis on stochastic processes generated by dynamical systems. We develop a general framework for describing the dependence structure of extremes through two complementary objects: the tail process and the spectral measure. We first extend the enriched point-process convergence of the univariate theory to the multivariate setting, proving that the decorated point process of exceedances converges to a marked Poisson cluster process whose marks retain the componentwise and temporal structure of extreme clusters. We then construct a spectral measure for stationary sequences, which assigns weight to distinct extremal profiles and characterizes the asymptotic dependence between components, and establish a de Haan-Resnick type representation. We also relate the spectral measure to the tail process, showing that its normalized profile distribution is recovered from the angular component of the tail process. Finally, we recover the classical univariate and multivariate exceedance processes as special cases. We illustrate the theory through applications to dynamical systems, using examples in which different dynamical mechanisms give rise to distinct extremal clustering patterns and cross-dependence structures. These applications demonstrate the ability of the proposed framework to capture genuinely multivariate features of extreme events and to extract information about the underlying dynamics that is not accessible through classical exceedance processes.

We give a construction of a body of constant width from a unit Reuleaux polyhedron that preserves all its symmetries. As opposed to the Meissner surgery, our construction modifies both edges of each dual pair and requires no choice between them. For the Reuleaux tetrahedron it recovers a previously known body of constant width with six elliptic edges and tetrahedral symmetry.

Real data are often arbitrary, indirect observations of an unknown underlying process, and those observations can distort its geometry. Existing geometric methods equip data with the induced geometry of the ambient space, but this entangles the true geometry of the process with the distortion of the observation. To address this, we introduce Intrinsic Geometry Matching (IGM): a scalable approach to learn the intrinsic geometry of the underlying process. Specifically, we learn the carré du champ operator of the process, allowing us to recover its diffusion geometry. Our method uses the fact that data often come with natural perturbations, like steps in a time series, to recover the intrinsic carré du champ while transforming naturally with changes in observation: it is observation-equivariant. Furthermore, since this equivariance relates the geometry of different spaces, we can train a model using additional sources of data, even when that data lies in another space. We evaluate IGM on molecular dynamics, image, and video data and find better performance and scaling than existing neural and graph-based geometric tools.

We characterize the randomized component-oracle complexity of finite-sum nonconvex-concave optimization with a bounded dual domain. For $\Phi(x)=\max_{y\in Y}n^{-1}\sum_{i=1}^n f_i(x,y)$, the goal is to find an $\varepsilon$-stationary point of the Moreau envelope $\Phi_{1/(2L)}$ with probability at least $2/3$ and a fixed query budget. Let $A=L\Delta/\varepsilon^2$, $B=LD_Y/\varepsilon$, and $\bar B=\max\{1,B\}$, where $\Delta$ bounds the initial primal gap and $D_Y$ bounds the dual diameter. For $A$ above a universal constant, the worst-case complexity is \[ \widetilde\Theta\!\left(n+A\min\left\{ \sqrt n\,\bar B+\bar B^4,\ n^{3/4}\bar B\right\}\right). \] The bounds agree within constants for $B\le n^{1/4}$ and within one logarithm for larger $B$. Upper bounds require only mean-squared averaged smoothness and concavity of the average in the dual variable; lower bounds already hold under individual smoothness. The lower constructions cover adaptive component selection, arbitrary query norms, private randomization, and outputs that were never queried. A cone construction gives the small-diameter bound, while a serial branch construction and balanced replication give the remaining two regimes. On the upper side, a persistent estimator with primal averaging complements accelerated proximal dual steps equipped with an explicit primal certificate. The characterization takes the worst case over admissible known dual domains and finite dimensions.

We derive a Fredholm-Pfaffian formula for the one-time multi-point transition probabilities of half-space polynuclear growth with any nonnegative boundary nucleation rate and general deterministic upper semi-continuous initial profiles. We prove the formula by showing that it satisfies an integral Kolmogorov equation with a unique bounded solution. We also discuss the formal 1:2:3 scaling limit of the kernel and its relation to the half-space KPZ fixed point.