Many generative tasks in chemistry and science involve distributions invariant to group symmetries (e.g., permutation and rotation). A common strategy enforces invariance and equivariance through architectural constraints such as equivariant denoisers and invariant priors. In this paper, we challenge this tradition through the alternative canonicalization perspective: first map each sample to an orbit representative with a canonical pose or order, train an unconstrained (non-equivariant) diffusion or flow model on the canonical slice, and finally recover the invariant distribution by sampling a random symmetry transform at generation time. Building on a formal quotient-space perspective, our work provides a comprehensive theory of canonical diffusion by proving: (i) the correctness, universality and superior expressivity of canonical generative models over invariant targets; (ii) canonicalization accelerates training by removing diffusion score complexity induced by group mixtures and reducing conditional variance in flow matching. We then show that aligned priors and optimal transport act complementarily with canonicalization and further improves training efficiency. We instantiate the framework for molecular graph generation under $S_n \times SE(3)$ symmetries. By leveraging geometric spectra-based canonicalization and mild positional encodings, canonical diffusion significantly outperforms equivariant baselines in 3D molecule generation tasks, with similar or even less computation. Moreover, with a novel architecture Canon, CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.

This paper studies a flat degeneration P_n of the classical coinvariant algebra R_n, a bigraded Artinian Gorenstein algebra that arises from the coordinate ring of the Segre embedding of the n-fold self-product of the projective line. The Frobenius character of P_n is computed by a natural bigraded refinement of the classical Lusztig--Stanley formula for the character of the coinvariant algebra. Young invariants in P_n get related to coordinate rings of general Segre embeddings of products of projective spaces; their bigraded Hilbert polynomials get expressed in terms of major-descent generating functions of words in multisets. Relations to the diagonal coinvariant algebra, cohomological interpretations including quantum cohomology, and Garsia-Stanton-style bases are also explored.

Given a finitely generated group $Γ$ and $g\inΓ$, we prove sufficient conditions in terms of various growth/decay functions for freeness of the action of $g$ on the Furstenberg boundary of $Γ$. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of $Γ$.

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

We propose ALiA, a novel adaptive variant of the alternating direction method of multipliers (ADMM). Specifically, ALiA is a variant of function-linearized proximal ADMM (FLiP ADMM), which generalizes the classical ADMM by leveraging the differentiable structure of the objective function, making it highly versatile. Notably, ALiA features an adaptive stepsize selection scheme that eliminates the need for backtracking linesearch. Motivated by recent advances in adaptive gradient and proximal methods, we establish point convergence of ALiA for convex and differentiable objectives. Furthermore, by introducing negligible computational overhead, we develop an alternative stepsize selection scheme for ALiA that improves the convergence speed both theoretically and empirically. Extensive numerical experiments on practical datasets confirm the accelerated performance of ALiA compared to standard FLiP ADMM. Additionally, we demonstrate that ALiA either outperforms or matches the practical performance of existing adaptive methods across problem classes where it is applicable.

A random geometric graph (RGG) with kernel $K$ is constructed by first sampling latent points $x_1,\ldots,x_n$ independently and uniformly from the $d$-dimensional unit sphere, then connecting each pair $(i,j)$ with probability $K(\langle x_i,x_j\rangle)$. We study the sharp detection threshold, namely the highest dimension at which an RGG can be distinguished from its Erdős--Rényi counterpart with the same edge density. For dense graphs, we show that for smooth kernels the critical scaling is $d = n^{3/4}$, substantially lower than the threshold $d = n^3$ known for the hard RGG with step-function kernels \cite{bubeck2016testing}. We further extend our results to kernels whose signal-to-noise ratio scales with $n$, and formulate a unifying conjecture that the critical dimension is determined by $n^3 \mathop{\rm tr}^2(κ^3) = 1$, where $κ$ is the standardized kernel operator on the sphere. Departing from the prevailing approach of bounding the Kullback-Leibler divergence by successively exposing latent points, which breaks down in the sublinear regime of $d=o(n)$, our key technical contribution is a careful analysis of the posterior distribution of the latent points given the observed graph, in particular, the overlap between two independent posterior samples. As a by-product, we establish that $d=\sqrt{n}$ is the critical dimension for non-trivial estimation of the latent vectors up to a global rotation.

We study the hyperbolic defocusing sinh-Gordon model with parameter $β^2>0$ and its associated Gibbs dynamics on the two-dimensional torus. We establish global well-posedness of the model for a certain range of parameters $β^2>0$ with the corresponding Gibbs measure initial data and prove invariance of the Gibbs measure under the flow, thereby resolving a question posed by Oh, Robert, and Wang (2019). Our physical space approach hinges on developing a novel $L^\infty$-based well-posedness theory for wave equations with exponential-type nonlinearities, going beyond the classical $L^2$-based framework. This refinement allows us to fully leverage structural properties of Gaussian multiplicative chaos. As a by-product of our method, we also obtain an improved well-posedness theory for the hyperbolic Liouville model.

This paper undertakes the study of the topology of T-manifolds of arbitrary codimension obtained by combinatorial patchworking with real phase structure as described by Brugallé, López de Medrano and Rau (2024). We prove new bounds on the number of connected components of T-curves and T-surfaces. For sufficiently high codimension, this improves the results of Brugallé, López de Medrano and Rau (2024). In addition, we present a new description of patchworking à la Viro for T-manifold of codimension 2. We use this method to construct a family of maximal real algebraic curves in $\mathbb RP^3$.

The full lattices in a finite dimensional commutative ${\mathbb Q}$-algebra form a commutative semigroup. In the case of an algebraic number field the top part of a certain quotient semigroup is the class group. For a separable algebra some basic results, especially the Jordan-Zassenhaus theorem, are known for this quotient semigroup. This paper considers also algebras which are not separable. It studies the commutative semigroup of full lattices in such an algebra and also the quotient semigroup. This leads in this commutative, but not separable situation to a certain extension of the Jordan-Zassenhaus theorem. One application concerns $GL_n({\mathbb Z})$-conjugacy classes of regular integer $n\times n$ matrices.

Given data $\{({\boldsymbol x}_i,y_i): i\le n\}$, with ${\boldsymbol x}_i$ standard $d$-dimensional Gaussian feature vectors, and $y_i\in{\mathbb R}$ response variables, we study the general problem of learning a model parametrized by ${\boldsymbol θ}\in{\mathbb R}^d$, by minimizing a loss function that depends on ${\boldsymbol θ}$ via the one-dimensional projections ${\boldsymbol θ}^{\sf T}{\boldsymbol x}_i$. While previous work mostly dealt with convex losses, our approach assumes general (non-convex) losses hence covering classical, yet poorly understood examples such as the perceptron and non-convex robust regression. We use the Kac-Rice formula to control the asymptotics of the expected number of local minima of the empirical risk, under the proportional asymptotics $n,d\to\infty$, $n/d\toα>1$. Specifically, we prove a finite dimensional variational formula for the exponential growth rate of the expected number of local minima. Further we provide sufficient conditions under which the exponential growth rate vanishes and all empirical risk minimizers have the same asymptotic properties (in fact, we expect the minimizer to be unique in these circumstances). We refer to this phenomenon as `rate trivialization.' If the population risk has a unique minimizer, our sufficient condition for rate trivialization is typically verified when the samples/parameters ratio $α$ is larger than a suitable constant $α_{\star}$. Previous general results of this type required $n\ge Cd \log d$. We illustrate our results in the case of non-convex robust regression. Based on heuristic arguments and numerical simulations, we present a conjecture for the exact location of the trivialization phase transition $α_{\text{tr}}$.

We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation and a hybrid mixed first-order system variant (FOSPG). We establish optimal a priori error estimates for each variant, which are verified numerically. We conclude by applying the method to American option pricing, free boundary problems in porous media, advection-diffusion with a semipermeable boundary, and the enforcement of discrete maximum principles.

We consider the mean-field noisy Kuramoto-Daido model, which is a McKean-Vlasov equation on the circle with bimodal interaction $W(θ)=\cosθ+m\cos2θ$ for $m\ge 0$ and interaction strength $K$, generalizing the celebrated noisy Kuramoto model corresponding to $m=0$. Our first contribution is to characterize the phase transition threshold $K_{c}$ by comparing it to the linear stability threshold $K_\# = \min (1, m^{-1})$ of the uniform distribution. When $m \leq 1/2,$ $K_{c}=1$, coinciding with that of the Kuramoto model. On the other hand, for $m \geq 2$, we show $K_c= m^{-1}$. We also classify the regimes in which the phase transition is continuous or discontinuous. Our second contribution is to analyze the linear stability of a global minimizer $q$ (the ``ordered phase'') of the mean-field free energy in the supercritical regime $K>1$. This stationary solution of the Kuramoto-Daido equation is unique up to translation invariance and distinct from the uniform distribution (the ``disordered phase''). Our approach extends the Dirichlet form method of Bertini et al. from the unimodal to bimodal setting. In particular, for $m \leq 1.590 \times 10^{-4}$ and $K>1$, we show an explicit lower bound on the spectral gap of the linearized McKean-Vlasov operator at $q$. To our knowledge, this is the first rigorous stability analysis for this class of models with bimodal interactions.

We consider the general problem of learning a predictor that satisfies multiple objectives of interest simultaneously, a broad framework that captures a range of specific learning goals including calibration, regret, and multiaccuracy. We work in an online setting where the data distribution can change arbitrarily over time. Existing approaches to this problem aim to minimize the set of objectives over the entire time horizon in a worst-case sense, and in practice they do not necessarily adapt to distribution shifts. Earlier work has aimed to alleviate this problem by incorporating additional objectives that target local guarantees over contiguous subintervals. Empirical evaluation of these proposals is, however, scarce. In this article, we consider an alternative procedure that achieves local adaptivity by replacing one part of the multi-objective learning method with an adaptive online algorithm. Empirical evaluations on datasets from energy forecasting and algorithmic fairness show that our proposed method improves upon existing approaches and achieves unbiased predictions over subgroups, while remaining robust under distribution shift.

Max-min bilinear optimization models, where one agent maximizes and an adversary minimizes a common bilinear objective, serve as canonical saddle-point formulations in optimization theory. They capture, among others, two-player zero-sum games, robust and distributionally robust optimization, and adversarial machine learning. This study focuses on the subclass whose variables lie in the completely positive (CP) cone, capturing a broad family of mixed-binary quadratic max-min problems through the modelling power of completely positive programming. We show that such problems admit an equivalent single-stage linear reformulation over the COP-CP cone, defined as the Cartesian product of the copositive (COP) and CP cones. Because testing membership in COP cones is co-NP-complete, the resulting COP-CP program inherits NP-hardness. To address this challenge, we develop a hierarchy of semidefinite relaxations based on moment and sum-of-squares representations of the COP and CP cones, and flat truncation conditions are applied to certify the tightness. We show that the tightness of the hierarchy is guaranteed under mild conditions. The framework extends existing CP/COP approaches for distributionally robust optimization and polynomial games. We apply the framework to the cyclic Colonel Blotto game, an extension of Borel's classic allocation contest. Across multiple instances, the semidefinite relaxation meets the flat-truncation conditions and solves the exact mixed-strategy equilibrium.

We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation $σ_2/σ_1=1$ in $\mathbb{R}^n$. The proof is based on the observation that after rewriting the quotient operator as the $σ_2$ operator, acting on a new function, one can refer to the recent result of Shankar and Yuan on Liouville's theorem for $σ_2$ equation.

We study the existence of almost complex structures on even-dimensional sphere bundles over complex projective spaces. For bundles $ξ_{n,q}$ with fibre $S^{2q}$ over $\mathbb{C} P^n$, we establish a necessary condition: if $q \ge a(n)$ for an explicit function, then the total space $E_{n,q}$ does not admit an almost complex structure. As an application, we analyse a concrete family associated with the canonical line bundle and obtain non-existence criteria in terms of $p$-adic valuations; for $p=2$ this yields a simple numerical bound. The proofs rely on Chern class computations and divisibility properties of characteristic classes. The results leave open the question of existence in the range $4 \le q < a(n)$.

This paper focuses on infinite-horizon optimal control problems for dissipative systems and the relations to their finite-horizon formulations. We show that, for a large class of problems, dissipativity of the state equation, when a coercive storage function exists, implies that infinite-horizon optimal controls can be obtained as limits of the corresponding finite-horizon ones. This property is referred to as pattern preservation, or pattern-preserving property. Our analysis establishes a formal link between dissipativity theory and the variational convergence framework in optimal control, thus providing a concrete and numerically tractable condition for verifying pattern preservation. Numerical examples illustrate the effectiveness and limitations of the proposed sufficient conditions.

In machine unlearning, $(\varepsilon,δ)-$unlearning is a popular framework that provides formal guarantees on the effectiveness of the removal of a subset of training data, the forget set, from a trained model. For strongly convex objectives, existing first-order methods achieve $(\varepsilon,δ)-$unlearning, but they only use the forget set to calibrate injected noise, never as a direct optimization signal. In contrast, efficient empirical heuristics often exploit the forget samples (e.g., via gradient ascent) but come with no formal unlearning guarantees. We bridge this gap by presenting the Variance-Reduced Unlearning (VRU) algorithm. To the best of our knowledge, VRU is the first first-order algorithm that directly includes forget set gradients in its update rule, while provably satisfying ($(\varepsilon,δ)-$unlearning. We establish the convergence of VRU and show that incorporating the forget set yields strictly improved rates, i.e. a better dependence on the achieved error compared to existing first-order $(\varepsilon,δ)-$unlearning methods. Moreover, we prove that, in a low-error regime, VRU asymptotically outperforms any first-order method that ignores the forget set.Experiments corroborate our theory, showing consistent gains over both state-of-the-art certified unlearning methods and over empirical baselines that explicitly leverage the forget set.

Let $D$ be a multidigraph. We study the simplicial complex $\mathrm{Dlf}(D)$, whose vertices are the directed edges of $D$ and whose faces correspond to directed linear forests, that is, vertex-disjoint unions of directed paths. We also consider the related directed tree complex $\mathrm{DT}(D)$. Our main approach is to associate with $D$ a simple graph encoding the local incompatibilities among the edges of $D$. Under mild acyclicity assumptions, we show that $\mathrm{Dlf}(D)$ and $\mathrm{DT}(D)$ can be realized as the independence complexes of respective graphs. This correspondence allows us to apply structural results from the theory of independence complexes to obtain graph-theoretic criteria guaranteeing vertex decomposability, shellability, and sequential Cohen-Macaulayness of these complexes. In particular, we describe explicit forbidden induced directed subgraphs that obstruct vertex decomposability, and we identify classes of multidigraphs-including certain acyclic multidigraphs and multidigraphs whose underlying graphs are forests or cycles-for which $\mathrm{Dlf}(D)$ and $\mathrm{DT}(D)$ are vertex decomposable. We also provide examples showing that these properties do not hold in general.

Let $U$ be a maximal unipotent subgroup in a symplectic group over a finite field of sufficiently large characteristic $p$. According to the Kirillov's orbit method, the coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. Almost all important classes of orbits and characters studied to the moment can be uniformly described as the orbits and characters associated with so-called orthogonal rook placements. In the paper, we construct a semi-direct decomposition for the corresponding irreducible characters in the spirit of the Mackey little group method. As a corollary, we present an explicit formula for the character corresponding to an orbit of maximal possible dimension.

The Robinson-Schensted-Knuth (RSK) algorithm maps an integer matrix to a pair of semi-standard Young tableaux (SSYTs) whose underlying shape has the same integer partition. We study the set of matrices associated with a given partition $λ$ vis-a-vis the number of inversions of the matrix. In the case where the integer matrix is a permutation matrix, the resulting tableaux are standard Young tableaux or SYTs. Han (EJC, 2005) combinatorially studied the set of permutations that map to SYTs of shape $λ$ under the RSK algorithm and counted the permutations with the minimum number of inversions in that set, as well as formulated the minimal number of inversions. Han's work can be extended to a case where the matrix is a general integer matrix and the tableaux are semi-standard Young tableaux. We have conjectured a formula for the minimal number of inversions in the set of matrices with a fixed shape $λ$. We further provide a conjecture for the characterisation of the minimal generalised matrices.

We study the approximation of $L_p$-functions, $p\in (0,\infty]$, on cylindrical space-time domains $Ω_T:=[0,T]\times Ω$, $0<T<\infty$, $Ω\subset \R^d$ Lipschitz, $d\in \mathbb{N}$, with respect to continuous anisotropic space-time finite elements on prismatic meshes. In particular, we propose a suitable refinement technique which creates (locally refined) prismatic meshes with sufficient smoothness and the desired anisotropy, and prove complexity estimates. Furthermore, we define a (quasi-)interpolation operator on this type of meshes and use it to characterize the corresponding approximation classes by showing direct and inverse estimates in terms of anisotropic Besov norms.

Both the antiferromagnetic Ising model and the hard-core model could be said to be tractable on line graphs of bounded degree. For example, Glauber dynamics is rapidly mixing in both cases. In the case of the hard-core model, we know that tractability extends further, to claw-free graphs and somewhat beyond. In contrast, it is shown here that the corresponding extensions are not possible in the case of the antiferromagnetic Ising model.

Residual-based a~posteriori error estimators are derived for the modified Morley FEM, proposed by Wang, Xu, Hu [J. Comput. Math, 24(2), 2006], for the singularly perturbed biharmonic equation and the nonlinear von Kármán equations. The error estimators are proven to be reliable and efficient. Moreover, an adaptive algorithm driven by these error estimators is investigated in numerical experiments.

We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group $\mathrm{SO}_3$ over a local field. This generalizes the $6j$ symbol whose theory was developed by Racah, Wigner, and Regge. We give several formulas for this invariant, including in terms of hypergeometric-type integrals and functions, and show that it admits a symmetry by the the $23040$-element Weyl group of $\mathrm{Spin}_{12}$. We then interpret these results in terms of relative Langlands duality, where the dual story comes from the action of $\mathrm{Spin}_{12}$ on a $16$-dimensional cone of spinors.

In this paper we prove that a recent condition of Lyons--Martínez--Navarro--Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups, which when combined with their results gives a new characterization of finite groups with a normal $\ell$-complement for a prime $\ell$. This leads us to study the distribution of characters in unipotent blocks of disconnected reductive groups and show that this is well-behaved under a generalization of $d$-Harish-Chandra theory. We go on to study the blockwise Galois--McKay (also known as the Alperin--McKay--Navarro) conjecture for the blocks of almost (quasi-)simple groups above unipotent blocks.

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we establish a numerical inequality relating the rank, the minimal support of a representing divisor and the modular dimension of the family. When the modular map is dominant, we obtain a sharp classification: equality occurs only for multiples of odd theta characteristics or of the canonical section. As applications, we derive geometric consequences for plane curves, obtaining results on intersections with very general quartic curves, in the spirit of the work of Chen-Riedl-Yeong, and with quintic curves.

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra.

Let $M^{n}$ be an $n$-dimensional complete spacelike linear Weingarten submanifold immersed in a locally symmetric semi-Riemannian space $\mathbb{L}_{q}^{n+p}$ of index $q$, with parallel normalized mean curvature vector field and flat normal bundle. Assuming that $M^{n}$ satisfies suitable curvature constraints, we investigate rigidity results for such submanifolds. By combining a Simons-type formula for spacelike submanifolds with analytic techniques involving the Cheng-Yau modified operator $\mathcal{L}$, we establish sharp inequalities relating the traceless second fundamental form and the gradient of the mean curvature. As applications, we obtain several characterization results showing that $M^{n}$ must be either totally umbilical or isoparametric. More precisely, we derive rigidity results under three distinct frameworks: via the Omori-Yau maximum principle, via the $\mathcal{L}$-parabolicity of the underlying manifold, and under an integrability condition on the gradient of the mean curvature. These results generalize and unify known classification theorems for spacelike submanifolds satisfying linear Weingarten relations in semi-Riemannian ambient spaces.

Reinforcement learning with verifiable rewards (RLVR) has been a main driver of recent breakthroughs in large reasoning models. Yet it remains a mystery how rewards based solely on final outcomes can help overcome the long-horizon barrier to extended reasoning. To understand this, we develop a theory of the training dynamics of RL for transformers on compositional reasoning tasks. Our theory characterizes how the effectiveness of RLVR is governed by the smoothness of the difficulty spectrum. When data contains abrupt discontinuities in difficulty, learning undergoes grokking-type phase transitions, producing prolonged plateaus before progress recurs. In contrast, a smooth difficulty spectrum leads to a relay effect: persistent gradient signals on easier problems elevate the model's capabilities to the point where harder ones become tractable, resulting in steady and continuous improvement. Our theory explains how RLVR can improve performance at the edge of competence, and suggests that appropriately designed data mixtures can yield scalable gains. As a technical contribution, our analysis develops and adapts tools from Fourier analysis on finite groups to our setting. We validate the predicted mechanisms empirically via synthetic experiments.