Papers

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

A general formulation of metriplectic dynamics is presented, where the metriplectic four-bracket is constructed by multiplying two skew-symmetric brackets. The new formulation is then used to introduce irreversibility in a generalized two-dimensional (2D) quasigeostrophic (QG) upper-ocean model involving advected quantities, with the thermal QG model as a special case. By construction, the resulting dynamics ensure the conservation of internal energy and the generation of entropy, in accordance with the first and second laws of thermodynamics. Our metriplectic dynamics formulation allows for a flexible specification of irreversibility, ranging from a type that results in nearly material conservation of potential vorticity to the representation of realistic forcing and dissipation in 2D QG ocean modeling with advected quantities.

Let $G$ be a group. We prove that matrix stability for either $G$-algebras or $G$-graded algebras guarantees Morita invariance. As a consequence, bivariant algebraic K-theory (either $G$-equivariant or $G$-graded) is Morita invariant. In particular, we show that if $G$ is a finite group acting freely on a finite simplicial set $X$, then $\ell^X\rtimes G$ and $\ell^{X/G}$ are kk-equivalent. Here, $\ell^Y$ denotes the $\ell$-algebra of piecewise polynomial functions on $Y$ with coefficients in the ground ring $\ell$.

The social compass model has been recently proposed as a model for depolarization in populations where individuals have multiple, possibly correlated, opinions. Previous work has focused on the steady state of this model, but has not addressed the dynamics leading to depolarization. We show that the macroscopic dynamics of the social compass model can be described using the Ott-Antonsen Ansatz and that, for initially clustered opinions, the resulting equations reduce to a finite-dimensional system of ordinary differential equations. We study the linear stability of the polarized state and find a dispersion relation for the growth rate of perturbations from this state. We find that the critical coupling for depolarization depends only on the first inverse moment of the conviction distribution, whereas the rate of depolarization depends on higher moments. Consequently, conviction distributions with the same critical coupling can exhibit vastly different depolarization timescales. We also demonstrate how our analysis can be extended to study depolarization in the presence of community structure.

Solving stochastic eigenvalue problems has long been essential for informed decision-making, advancing scientific knowledge, and ensuring the reliability of engineering designs and applications. This paper underscores the need to continue enhancing existing numerical methods for solving the stochastic eigenproblem in order to improve convergence rates, computational efficiency, and robustness. Specifically, we propose a novel spectral-chaos method for solving the stochastic (linear) eigenvalue problem, employing Halley's method as the root-finding algorithm to leverage its cubic convergence properties. Our method achieves maximal convergence in solving stochastic eigenvalue problems since its rate cannot be further improved using a higher-order Householder method due to the quadratic nature of the resulting system of equations. Additionally, due to the complexity of the resulting system of equations, a tensorial approach was developed to tackle the challenges associated with the dimensional multiplicity of the stochastic eigenvalue problem, without which the solution would have been intractable. The method is derived rigorously, with a detailed error analysis that highlights the benefit of using our approach when the eigenvector components are nearly known, the computational cost of the method is also rigorously presented, and an illustrative example is provided to demonstrate the implementation of the method. Subsequently, a case study is demoed to analyze the results and validate the advantages of using Halley's method over Newton's method and Monte Carlo simulations.

Classical Morse-Conley theory provides powerful tools for relating dynamical and topological invariants of smooth manifolds. In this paper, we extend this perspective to pseudomanifolds with spherical-cone singularities. By introducing and investigating singular Morse-Smale flows on pseudomanifolds with isolated singularities whose links are homeomorphic to finite disjoint unions of spheres. We establish formulas for the Conley indices of spherical-cone singularities in terms of their local dynamics, prove the existence of global Lyapunov functions, and investigate the structure of the associated Lyapunov graphs. These results yield alternative formulas for the Euler-Poincaré characteristic expressed in terms of Conley-theoretic invariants. To relate the singular and smooth settings, we introduce a global morsification procedure that associates a smooth manifold $\widetilde{X}$ to a singular pseudomanifold $X$. This construction allows us to compare the topology of $X$ and $\widetilde{X}$ and, in particular, to derive formulas relating their Euler-Poincaré characteristics. Finally, we study the intersection homology of pseudomanifolds with spherical-cone singularities. We establish connections between intersection homology, singular homology, and the Morse homology of the morsification, thereby providing a dynamical approach to the computation of intersection homology.

We study safety verification for linear time-invariant systems with bounded inputs in continuous time. The standard approach reduces to a reachability analysis in two steps: first discretize time and then apply a forward analysis in the discretized system. Existing algorithms use either a fixed time step or an adaptive time step that changes based on the approximation error compared to the underlying continuous system. In this paper, we present an efficient reachability algorithm that adapts the time step based on a given safety property. Essentially, our algorithm makes the largest possible time step such that it can still prove safety. For this approach to be scalable in practice, we discuss several optimizations such as avoiding the repeated expensive calculation of the matrix exponential during discretization and a careful balance how we tame the approximation error stemming from the states and the inputs. This allows our algorithm to yield a moderate approximation error even when using a large time step, thus requiring much fewer steps than prior algorithms. We demonstrate the effectiveness and scalability on the large-scale SLICOT benchmark suite, where our algorithm consistently outperforms other state-of-the-art approaches.

Completing partial Latin squares is NP-complete. Ryser characterized precisely when an $r\times s$ Latin rectangle can be completed to an $n\times n$ Latin square. We extend this theorem to a broader setting. Let $M$ be an $n\times n$ array whose top-left $r\times s$ subarray is filled with symbols from $\{1,2,\dots,k\}$, where $1\leq k\leq n^2$. We determine necessary and sufficient conditions under which the remaining cells of $M$ can be filled so that each symbol $\ell\in\{1,2,\dots,k\}$ appears exactly $\rho_\ell$ times in total, while the numbers of occurrences of each symbol in any two rows and in any two columns differ by at most one. Equivalently, our result characterizes when a partial edge-coloring of $K_{r,s}$ can be extended to an edge-coloring of $K_{n,n}$ in which each color class is spanning and almost regular, with prescribed color-class sizes. In this sense, our theorem generalizes Baranyai's construction of almost-regular colorings of complete uniform multipartite hypergraphs, restricted to the bipartite case. Our theorem unifies and generalizes several classical results. When $s=k=\rho_1=\cdots=\rho_k=n$, it reduces to Hall's theorem, and when $k=\rho_1=\cdots=\rho_k=n$, it yields Ryser's completion theorem for Latin rectangles. Additional special cases recover results of Goldwasser, Hilton, Hoffman, and Özkan, as well as a theorem of Bahmanian. Thus, our work may be viewed as a common generalization of these completion theorems and Baranyai's theorem.

The category of Harish-Chandra bimodules for quantum groups was first appeared in the works about topological quantum field theory of surfaces. In this paper, we study this category when the quantum parameter q is an odd order root of unity. We relate the category to the category of affine Soergel bimodules and to non-commutative Springer resolution.

In \S7 of Generalized Analytic Functions \cite{vekua}, the reduction of a first-order elliptic system to canonical form proceeds through a factor of the characteristic equation, which Vekua selects in real form~(7.13) and then restates, without derivation, in complex (Beltrami) form~(7.14). We supply that conversion. With the standard Wirtinger convention used below, the complex form of~(7.13) is the negative of the coefficient printed in~(7.14) (p.~126, 1962 Pergamon edition), and we confirm the correct sign against Vekua's own factorization~(7.12) and his canonical coefficient~(7.17). A related sign defect appears in the second-order Beltrami coefficient~(7.23) (p.~127): a coordinate solving (7.23) as printed reduces the equation to the canonical form~(7.26) only in the special symmetric case $a=c$. In both instances the error is confined to the displayed coefficient and leaves the surrounding reduction, carried out independently of it, intact; we record the corrected coefficient in each case.

For fixed $t_0 \in [0,\sqrt{3}]$ we give asymptotically sharp lower bounds on the quantity $L_K \text{vol}_{d-1}(K \cap H)$, where $H$ is a hyperplane at distance $t_0 L_K$ from the origin, $K$ is any symmetric isotropic convex body in $\mathbb{R}^d$, and $L_K$ stands for the isotropic constant of $K$.

This work proposes a framework for sampling from the Gibbs distribution of a given potential using hybrid stochastic dynamics. In this framework, two distinct sampling dynamics are run in different regions of the state space. The two dynamics are coupled across the interface through natural transmission conditions that preserve the target distribution. Using a specially constructed regularization scheme, we establish an exponential rate of convergence for the hybrid dynamics to equilibrium. We also analyze the metastability properties of the hybrid dynamics in a radially symmetric landscape, showing that the hybrid scheme can improve the mean exit time. This advantage is further confirmed by the numerical experiments.

We show that very small distances in a planar integral point set are essentially one-dimensional. Let P be a non-collinear set of n points in the plane, all of whose pairwise distances are integers. We prove that, for a sufficiently small $c>0$, at most one pair of points can determine a distance below $n^{c\log\log{n}}$ unless all such short pairs are supported on a single line. We also give a construction showing that the line-supported alternative is necessary: there are arbitrarily large non-collinear integral point sets with one off-line point and with the first two distinct distances 2 and 4. We conjecture that in the general case (no three points are collinear) even the smallest distance should be large, at least $n^{c\log\log{n}}$, however we can prove a linear lower bound only.

Given a closed, real, non-singular 1-form on a compact manifold $\Omega$, global properties of the associated differential complex are studied. We completely characterize global solvability in the first and last levels of the complex. Furthermore, in the particular case where the $1$-form is rational, we prove global solvability for every degree and give a complete description of the cohomology spaces. Finally, a complete characterization for global hypoellipticity is obtained, building on the work of A. Meziani (Comm. PDE., 2002). In all cases, it is shown that the conditions depend exclusively on the arithmetic nature of the form's periods.

We prove in this paper some analogues of the well-known group-theoretical Grün's lemma, stating that in a perfect group the first and the second centre coincide, and Baer's theorem, stating that if the quotient by the $n$th centre of a group is finite, then so is the $(n+1)$th term of the lower central series, in the scope of infinite skew left braces. These results represent significant improvements over previous work. The trifactorised group associated with a skew left brace will be crucial for our proofs.

We show that the polar degrees of the coordinate-wise inverse of a linear subspace $L \subseteq \mathbb{P}^n$ are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid $\mathrm{M}(L)$. Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of $\mathrm{M}(L)$. As a corollary, we settle two open conjectures regarding matroid discriminants.

Scott and Sokal asked for a structural description of the finite graphs \(G\) for which inverse powers \(T_G^{-\beta}\) of the spanning-tree polynomial are completely monotone. We prove the following bounded-treewidth criterion: if \(G\) is a finite connected simple graph with \(\operatorname{tw}(G)\le k\), then \(T_G^{-\beta}\) is completely monotone for every \(\beta>(k-1)/2\). Consequently, every partial \(3\)-tree is covered throughout the first Scott–Sokal open interval \(1<\beta<3/2\), including finite Apollonian networks, \(K_5-e\), and the four-spoke wheel \(W_4\). The proof combines the real Riesz/Wishart integral for determinants, star–mesh elimination of simplicial vertices, and a Gaussian Laplace kernel for the degree-\(d\) star. General bounded-treewidth graphs follow by chordal completion and monotone deletion of completion edges.

The prediction of the extremely large values of a response variable $Y$ in terms of a vector of covariates $X=(X_i)_{i=1}^d$ is a fundamental problem arising in many scientific and engineering domains. The scarcity of data in the extremes makes the optimal solution of this problem of particular importance. The optimal predictors of such events can be explicitly characterized in just a few cases and it is of fundamental practical and theoretical interest to develop optimal estimators over large classes of models and predictors. In this work, the focus is on the case where $(Y,X)$ have a multivariate regularly varying distribution and one seeks an optimal predictor expressed as a positive homogeneous function $h(X)$ of the covariates. The asymptotic prediction precision in this setting coincides with the tail-dependence coefficient $\lambda(Y,h(X))$ and it can be expressed as an integral functional of the associated angular measure of $(Y,X)$. Thus, finding asymptotically optimal homogeneous predictors amounts to solving a variational problem. We obtain a general solution to this problem, which is expressed in terms of a non-extreme conditional quantile of a tilted distribution derived from the angular measure. This leads to a general inference methodology for the optimal predictors in the peaks-over-threshold framework form extreme value theory. We establish the universal consistency for these estimators over large classes of angular measures. A general-purpose implementation of the resulting inference procedure is shown to work remarkably well against optimal oracle estimators, as well as in the challenging problem of extreme solar flare prediction.

We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux $b$, we construct an open region in the three-dimensional parameter space $(b,\mu,\lambda)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane $\lambda=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2\pi\sigma^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

Computing the structure of the Hochschild (co)homology and the cyclic homology of an algebra can be hard work, but Etingof and Eu showed that it can be done surprisingly easily for preprojective algebras of ADE Dynkin type, at least if one only wants to know the graded vector space structure of each Hochschild cohomology group. Their method is based on exploiting strong structural features of such a preprojective algebra via a graded Euler characteristic that can computed using the algebra's graded Cartan matrix. In this paper, we present a generalization of the method used by Etingof and Eu to higher preprojective algebras. We also apply our generalization to the higher preprojective algebras of the 2-representation finite algebras that arise as tensor products of representation finite hereditary algebras of type A. For this, it turns out to be enough to know the graded vector space structure of the center and the zeroth Hochschild homology to be able to deduce the graded vector space structure of the Hochschild (co)homology and the cyclic homology in all other degrees.

We investigate ideas from the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001) in the setting of graph isomorphism. This provides us a playground of finite combinatorial objects on which to explore these techniques. We seek to separate non-isomorphic pairs of graphs using vector spaces of polynomials that are set-wise invariant under permutations (so-called separating modules). We characterize the power of this method for distinguishing graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ are equivalent in power to the counts of $O(k)$-vertex subgraphs. - We show that separating modules of symmetric algebraic circuit size $n^{\Theta(k)}$ are equivalent to $\Theta(k)$-dimensional Weisfeiler-Leman. This generalizes and strengthens the result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025). - When considering only the representation-theoretic multiplicities of separating modules, we show that two graphs are separated by multiplicities if and only if their automorphism groups have different multiplicity of cycle types (cycle index). The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect support size (from the study of WL) to complexity measures on $S_n$ (Dafni, Filmus, Lifshitz, Lindzey, & Vinyals, ITCS '21); as well as connections between invariant polynomials, the Graph Reconstruction Conjectures, and Forman's "invariants of finite type" (Adv. Math., 2004).

We review the representations of so(1,D), the algebra of isometries of D dimensional de Sitter space. We cover the representations in all D, including mixed symmetry representations and fermionic representations, and connect them to the various types of fields that can propagate on de Sitter space. The presentation is from a physics point of view, favoring concrete constructions over abstract considerations.

In this work, we present a general Graph Neural Network (GNN) framework for learning algebraic properties of finite groups from their Cayley graph representations. The framework provides a unified computational pipeline consisting of a common graph construction procedure, feature representation, training methodology, and GNN architecture, with only the target labeling function varying across classification tasks. To demonstrate the generality of the proposed approach, we consider three representative algebraic properties: abelianity, nilpotency, and solvability. Experiments were conducted on a benchmark of 176 finite groups drawn from several classical families, with all groups included in each classification task. To address class imbalance, class-weighted cross-entropy loss was employed where appropriate during training. Furthermore, the family PSL(2,q) was reserved exclusively for testing, enabling evaluation of the framework's ability to generalize to previously unseen group families. The best-performing models achieved test balanced accuracies of 1.000, 0.856, and 0.875 for abelianity, nilpotency, and solvability, respectively. Although the same computational framework was employed across all tasks, different GNN architectures proved optimal for different algebraic properties, suggesting that the representational complexity required to learn a property depends on its underlying algebraic structure. These results demonstrate that GNNs can effectively learn multiple algebraic properties directly from Cayley graph representations while exhibiting strong generalization to unseen group families. More broadly, the proposed framework establishes a computational methodology for studying algebraic properties of finite groups using graph-based machine learning.

The continued fraction corresponding to the Askey-Wilson polynomials for $q^N=1$ is explicitly expressed as a ratio of two terminating $q$-hypergeometric series.

This article explores the comparative strengths of Higher-Order Unconstrained Binary Optimization (HUBO) and Quadratic Unconstrained Binary Optimization (QUBO) models in the context of probability optimization using tensor sampling techniques. HUBO can represent interactions beyond pairwise terms without introducing auxiliary variables, while QUBO benefits from a simpler algebraic structure and a wide range of established solvers. We combine a theoretical analysis of formulation expressiveness and reduction overhead with empirical experiments on synthetic polynomial objectives, RSA factorization, and Max-Cut instances. The results show that native HUBO formulations are often preferable when the original problem contains high-order interactions, because reducing such problems to QUBO increases the effective dimension seen by tensor samplers. At the same time, QUBO remains competitive for naturally quadratic problems. These findings provide practical guidance for selecting an optimization framework that balances model fidelity, dimension growth, and solver performance.

We study mixing rates for locally Hamiltonian flows on compact surfaces with asymmetric logarithmic singularities. For a full measure set of such flows, we show that the decay of correlations of smooth observables cannot be uniformly faster than a power of $\log t$. In particular, there exist sequences of times and observables for which correlations admit lower bounds of order $(\log t)^{-2-\nu}$ for any $\nu>0$. We further show that for a typical Arnol'd flow on $\mathbb T^2$, the self-correlation of every box in the minimal component is bounded below by $(\log t)^{-1}$ along an unbounded sequence of times. Motivated by questions in spectral theory, we also construct examples of such flows for which the self-correlation of a box fails to be square-integrable. These results complement previous upper bounds for correlations in both settings, which are also of polynomial order in $\log t$, and show that logarithmic decay rates are essentially sharp along sequences of times.

Given a triangulated category with a t-structure, we introduce a method for inducing t-structures on its semiorthogonal components, based on the construction of an associated perverse t-structure on the ambient category. As applications, we construct bounded t-structures in many new examples, including: almost all known phantom and quasiphantom categories; the semiorthogonal complement of the structure sheaf on a Fano variety; the residual component of an Enriques surface; the categorical resolution of a nodal cubic curve appearing in an early counterexample to the Jordan-Hölder property for semiorthogonal decompositions; and Brill-Noether modifications of the derived category of a curve.

For a set of $m$ subsets of a universe set of size $n$, we construct a Boolean polynomial $\mathrm{ICC}_{m,n}(X)$ such that the Union-Closed Conjecture is true for this $m$ and $n$ if and only if $\mathrm{ICC}_{m,n}(X)$ is the zero Boolean polynomial. We use an equivalent formulation, called the Intersection-Closed Conjecture.

Finding all modes of a multimodal black-box function is a fundamental challenge in optimization, Bayesian inference, and scientific computing. Existing approaches – basin-hopping, CMA-ES, multistart gradient descent – operate sequentially and cannot exploit the massive parallelism of modern GPU hardware. We introduce \chisao{} (Convergence-Halt-Invert-Stick-And-Oscillate), a GPU-native population optimizer that runs an entire sample batch simultaneously and exploits a deliberate convergence-anticonvergence oscillation cycle to escape local traps while freezing confirmed modes. The structural move is asymmetric: samples that reach true peaks are frozen (“stuck”) and preserved, while the rest keep exploring via momentum-based anti-convergence and stochastically smoothed gradients. Adaptive reseeding via two complementary strategies (Repulse Monkey and Golden Rooster) maintains population diversity throughout. On all 42 functions of the Simon Fraser University optimization benchmark suite across dimensions $d \in \{2, 4, 8, 16, 32, 64\}$, \chisao{} achieves 100% mode recovery where all CPU baselines collapse at $d \geq 8$ on the hardest multimodal functions, at up to \textbf{$34\times$} speedup over basin-hopping on functions where all methods succeed (Michalewicz $d=64$) and up to \textbf{$39\times$} on unimodal functions (Rotated Hyper-Ellipsoid $d=64$, pure GPU dividend). All benchmarks evaluate the objective by value alone – gradients come from finite differences – so the reported speedups are a derivative-free worst case. Under substantial likelihood noise ($\sigma_{\mathrm{noise}}$ up to 1.0), mode detection remains 100% reliable. The algorithm is available as a standalone open-source Python package on PyPI.

For piecewise deterministic samplers such as Randomized Hamiltonian Monte Carlo (RHMC), Bouncy Particle Sampler (BPS) or Zig-Zag Process (ZZP), long-time exponential convergence rates have been established in previous works using Harris or $L^2$ hypocoercivity approaches. In particular, in the $L^2$ framework, a so-called diffusive-to-ballistic speedup was known for log-concave targets, according to which the convergence rates of these samplers, with suitable parameters, are quadratically improved with respect to the standard overdamped Langevin diffusion process. A recent work by Jianfeng Lu showed that this speedup also holds for the kinetic Langevin diffusion process when the convergence is stated in terms of relative entropy, raising the question whether this also holds for piecewise deterministic samplers. The present work provides a positive and a negative answer to this: first, we show that the speedup holds in entropy for RHMC; second, we show that for BPS or ZZS, even for a standard Gaussian target, a similar result cannot hold, and even that exponential convergence (at any rate) in entropy fails.