Predictions from statistical physics postulate that recovery of the communities in the Stochastic Block Model (SBM) with a fixed number $K$ of communities is possible in polynomial time above, and only above, the Kesten-Stigum (KS) threshold. This conjecture has given rise to a rich literature, proving that non-trivial community recovery is indeed possible in SBM above the KS threshold. Failure of low-degree polynomials (LDP) below the KS threshold was also proven, as long as $K\ll \sqrt{n}$, where $n$ is the number of nodes in the observed graph. When $K\geq \sqrt{n}$, Chin et al.(2025) recently proved that, in a \emph{sparse regime}, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough led them to postulate a new threshold for the many-communities regime $K\geq \sqrt{n}$. In this work, we provide evidence supporting their conjecture:\\ 1- We prove that, for \emph{any graph density}, LDP fail to recover communities below the threshold postulated by Chin et al.(2025) ;\\ 2- We prove that community recovery is possible in polynomial time above the postulated threshold, not only in the \emph{sparse regime} considered in Chin et al.~(2025), but also in \emph{moderately sparse regimes}, by counting occurrences of some specific motifs inspired by the LDP analysis.\\ In particular, counting self-avoiding paths of length $\log(n)$, which is closely related to spectral algorithms based on the Non-Backtracking operator, is optimal only in the sparse regime. More complex motifs based on the blow-up of a cycle must be considered in denser regimes.

We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of masses must lie on the base of an isosceles triangle, and the other pair on its symmetric axis with one mass positioned inside the triangle formed by the other three. Using a rigorous computer-assisted analytical approach, we prove that for any non-negative mass ratio, the number of such configurations is either zero, one, or two, thereby providing a complete classification of this family. Furthermore, we show that the unique configuration corresponding to a specific mass ratio is a fold-type bifurcation point within the reduced subspace. We also give a clear and complete bifurcation picture for both symmetric and asymmetric cases of this concave type across the entire planar 4-body configuration space.

In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree introduced in arXiv:1309.0049, replacing a positive symmetric bilinear form by a Hermitian form. Various examples are presented to show the robustness of the machinery.

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.

In this paper, we consider a contextual stochastic optimization problem in which unknown parameters follow distributions that depend on contextual covariates and decisions. The problem is motivated by transportation infrastructure decisions such as facility location or network design. In such high-stakes settings, decisions must often be communicated, justified, and reconsidered under alternative stakeholder requirements. To this end, we propose a framework for computing relative counterfactual explanations. These explanations identify the smallest changes in the covariates required for a solution to satisfy prescribed constraints while limiting the performance loss to a controlled level. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge, this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term in the objective. Beyond improving tractability, this regularization yields explanations with desirable structural properties: it produces sparser counterfactuals, induces smoother transitions in the underlying choice distributions, and keeps the counterfactual behavior close to realistic demand patterns. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse, plausible, and interpretable explanations. We further validate the framework on a real-world case study of electric vehicle charging station planning in Montreal, where the explanations reveal the minimal capacity investments and environmental conditions required to justify including a candidate location in the charging network.

We propose a fully mixed virtual element method for the numerical approximation of the coupling between linear poroelasticity equations with strong symmetry of total poroelastic stress (using the Hellinger--Reissner principle) and stress-altered solute diffusion (where diffusive flux depends on the poroelastic stress and nonlinearly on the concentration gradient). Because of the nonlinear coupling, the function spaces associated with the nonlinear diffusion sub-problem are of Banach type. To handle this structure, the solvability of both the continuous and discrete problems is established through a decoupled fixed-point strategy. The linear poroelasticity component is analysed using the theory for perturbed saddle-point problems, whereas the nonlinear diffusion problem, relies on the classical Minty--Browder theorem for monotone global operators. The existence of solutions for the fully coupled system is rigorously proven via Schauder's fixed-point theorem. Additionally, we establish rigorous a priori error estimates for the discrete scheme, successfully handling the strongly cross-coupled nonlinearities. These findings are supported by computational evidence, demonstrating that the formulation asymptotically recovers optimal convergence rates in practice. As a key contribution, both the numerical scheme and its underlying analysis prove to be robust with respect to the poromechanical parameters. Finally, several numerical examples are presented to illustrate the properties and applicability of the proposed scheme in the study of solute transport in the context of brain multiphysics.

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

A family of compact n-manifolds is locally combinatorially defined (LCD) if it can be specified by a finite number of local triangulations. We show that LCD is equivalent to the existence of a compact branched n-manifold W, such that the family is precisely those manifolds that immerse into W. In subsequent papers, the equivalence will be used to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD.

Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23]. A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$ in running time $n^{O(\operatorname{poly}(1/\varepsilon) \cdot \operatorname{exp}(m) )}$: showing a PTAS exists for constant $m$.

Data assimilation (DA) estimates a dynamical system's state from noisy observations. Recent generative models like the ensemble score filter (EnSF) improve DA in high-dimensional nonlinear settings but are computationally expensive. We introduce the ensemble flow filter (EnFF), a training-free, flow matching (FM)-based framework that accelerates sampling and offers flexibility in flow design. EnFF uses Monte Carlo estimators for the marginal flow field, localized guidance for observation assimilation, and utilizes a novel flow path that exploits the Bayesian DA formulation. It generalizes classical filters such as the bootstrap particle filter and ensemble Kalman filter. Experiments on high-dimensional benchmarks demonstrate EnFF's improved cost-accuracy tradeoffs and scalability, highlighting FM's potential for efficient, scalable DA. Code is available at https://github.com/Utah-Math-Data-Science/Data-Assimilation-Flow-Matching.

We analyse the classic two-level additive Schwarz domain-decomposition GMRES preconditioner for finite-element discretisations of the Helmholtz equation with large wavenumber $k$, where both the fine and coarse spaces consist of piecewise polynomials with polynomial degree increasing like $\log k$. We exhibit choices of these fine and coarse spaces such that -- up to factors of $\log k$ -- both are pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES iterations is proved to be bounded independently of $k$. These are the first $k$-explicit convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free and does not consist of problem-adapted basis functions.

In this paper, we investigate the existence of nontrivial least-energy solutions for the super-Liouville equation with positive coefficient functions on the two-dimensional sphere. Firstly, we derive a global Pohozaev-type identity by analyzing the behavior of solutions under conformal transformations, which generalizes the classical Kazdan-Warner obstruction for the two-dimensional Nirenberg problem. Secondly, by exploiting conformal symmetry, we establish a pointwise estimate that bounds the norm of the spinor component by the scalar component, and show that the $H^1 \times H^{1/2}$ energy of the spinor part remains uniformly bounded. As a byproduct of our analysis, parallel techniques are applied to the Dirac-Einstein equations on the 3-sphere, demonstrating that nontrivial solutions are uniformly bounded away from the trivial solution in the $H^1 \times H^{1/2}$ norm. Moreover, the compactness of the solution space is also analyzed from two perspectives: in the low-energy regime, and modulo the action of the Möbius group. Finally, by introducing a new natural constraint $\mathcal{A}$ and employing variational methods, we obtain a supersymmetric generalization of the Moser-Trudinger-Onofri inequality and establish the existence of least-energy solutions for even coefficient functions. In particular, these solutions are shown to be nontrivial provided that a certain spectral parameter associated with the coefficients satisfies $λ_1(h_2, h_1) < 1$. Concurrently, we provide a complete classification of nontrivial least-energy solutions in the case of positive constant coefficients.

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. While we show that ISS in general does not imply the existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded, we prove that indeed quadratic ISS Lyapunov functions always exist for $p$-admissible input operators with $p<2$, provided the semigroup is similar to a contraction on a Hilbert space. The constructions are semi-explicit and rely on classical results on analytic semigroups and similarity to contractive ones. In the case of self-adjoint generators, they coincide with the canonical Lyapunov function being the norm squared.

In this paper, we prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,\rm fin}$ of perfectoid algebras. More precisely, if $V$ is an integral perfectoid valuation ring and $A$ is an integral perfectoid $V$-algebra, then the tilting functor induces a weak equivalence \[ \mathrm{dMGL}^{a,\rm fin}(A) \simeq \mathrm{dMGL}^{a,\rm fin}(A^\flat). \] This invariant is a finite syntomic, derived, and non-$\mathbb{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed characteristic bases. To prove the result, we first establish the corresponding finite non-unital statement and isolate a form of excisive approximation for pointed $\infty$-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with the framework of Heuts. We also define approximation functors along natural transformations and apply them to the comparison between periodic algebraic cobordism and homotopy $K$-theory, obtaining Bott periodicity and Gabber rigidity.

We introduce the Cycle Walk, a new Markov chain Monte Carlo method for sampling distributions on balanced graph partitions, motivated by applications in political redistricting. The method operates on spanning forests and combines two types of updates: local "cycle" moves within districts and global moves that exchange population between adjacent districts while preserving balance constraints. This construction enables efficient Metropolis--Hastings correction while allowing proposals at multiple spatial scales. We show that the Cycle Walk naturally interpolates between existing approaches based on local updates and a class of global update methods derived from recombination (RECOM). Through a range of numerical experiments on synthetic graphs and real-world precinct data, we demonstrate that the Cycle Walk exhibits improved empirical convergence diagnostics for distributions that place weaker weight on spanning-tree counts, a regime that is challenging for existing methods. In particular, the algorithm remains effective when incorporating alternative compactness measures that more closely reflect policy-relevant criteria. These results suggest that the Cycle Walk provides a flexible and computationally efficient framework for sampling from a broader class of redistricting distributions than previously accessible with MCMC techniques.

We introduce a purely Lie algebraic formalization of the Feigin--Tipunin's geometric construction of logarithmic CFTs/VOAs. After reformulating the geometric representation theory of FT construction under this new setting, within this framework, we uniformly construct the (multiplet) principal W-algebras at positive integer level associated with any simple Lie algebra $\mathfrak{g}$ and Lie superalgebra $\mathfrak{osp}(1|2r)$, thereby establishing Weyl-type character formulas and simplicity theorems that extend the first author's previous results.

Let $K$ be a number field, let $S$ be a finite set of primes of $K$ containing all archimedean primes, and let $G_{K,S}$ denote the Galois group of the maximal extension of $K$ unramified outside $S$. In this paper, we study the second partial Euler--Poincaré characteristic $χ_{2}(G_{K,S},M)$ for a finite $G_{K,S}$-module $M$, without imposing the condition that the order of $M$ is an $S$-unit. By adjoining a further finite set of primes of $K$, which can be chosen to be disjoint from any prescribed set of primes of density zero, we obtain an explicit formula for the corresponding second partial Euler--Poincaré characteristic. As an application, we investigate the presentation of the Galois group $G_{K,S}$. Furthermore, for any number field, we construct counterexamples to the dimension conjecture for Galois deformation rings.

We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov-Witten theory of the projective line and is dictated by $\mathbb{Z}_2$-equivariant topological recursion.

We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form $n$ congruent copies of the original object, generalising a known result by Raphael Robinson.

The maximum entropy principle, as applied to quantum systems, is a fundamental prescript positing that for a quantum system for which we only have partial knowledge, the maximum entropy state consistent with the partial knowledge is a valuable choice as the system's state. An intriguing result is that in case the only prior knowledge is of a fixed energy, the maximum entropy state turns out to be the thermal state, a ubiquitous state in several arenas, especially in statistical mechanics. We extend the consequences of this principle from static quantum states to dynamic quantum processes. We establish that a quantum channel attains maximal output entropy under a fixed energy constraint if and only if it is an absolutely thermalizing channel, where the fixed output is the thermal state corresponding to that energy. Our results have potential implications for understanding the informational and thermodynamic utility of quantum channels under physical constraints. As an application, we examine the consequences for private randomness distillation from fixed energy constrained quantum processes.

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it turns out, the patterns corresponding to separable permutations play an important role in the computational features of the statement. We prove that the avoidance of any separable permutation is equivalent to the existence of an infinite homogeneous set in standard models, while this property fails for any other pattern. For this, we develop a novel argument for relativized diagonal non-computation.

In this work the standard kinetic theory assumption of instantaneous collisions is lifted. As a continuation of of a previous paper by Kanzler, Schmeiser, and Tora [KRM, 2024], a model for higher order non-instantaneous alignment collisions is presented and studied in the asymptotic regime of short collision duration. A first order accurate approximative model is derived as a correction to the instantaneous limit. Rigorous results on its well-posedness and on the instantaneous limit are proven. The approximative model is a system of two equations. An equally accurate scalar approximation is suggested.

We prove that the $τ$-tilting graph of any quasi-silted algebra is connected and has the reachable-in-face property. Our approach utilizes $τ$-reduction and wall and chamber structures. In particular, we observe a sufficient condition on the wall and chamber structure under which the connectivity of $τ$-tilting graphs is preserved under taking quotients of algebras. As an immediate consequence, the connectivity of $τ$-tilting graphs is also established for several new classes of algebras.

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.

We find identities involving differential operators in the generic artinian reduction of the Stanley-Reisner ring of a simplicial sphere in any positive characteristic. These identities generalize the characteristic 2 identities used by Papadakis and Petrotou to give a proof of the algebraic g-conjecture. We show that these identities are a shadow of an identity on the degree map, and we use them to prove the anisotropy of certain forms on the generic artinian reduction of the Stanley--Reisner ring and to prove weak Lefschetz results.

We study a Navier-Stokes/Cahn-Hilliard system modeling the evolution of a compressible binary mixture of viscous fluids undergoing phase separation. The novelty of this work is a free energy potential including the physically relevant Flory-Huggins (logarithmic) entropy, as opposed to previous studies in the literature, which only consider regular potentials with polynomial growth. Our main result establishes the existence of global-in-time weak solutions in three-dimensional bounded domains for arbitrarily large initial data. The core contribution is the derivation of new estimates for the chemical potential and the Flory-Huggins entropy arising from a density-dependent Cahn-Hilliard equation under minimal assumptions: non-negative $γ$-integrable density with $γ>\frac32$. In addition, we prove that the phase variable, which represents the difference of the mass concentrations, takes value within the physical interval $(-1,1)$ almost everywhere on the set where the density is positive.

Statistical literature provides pointwise limiting distributions of the nonparametric maximum likelihood estimator (NPMLE) in monotone binary regression for the two extremal cases: If the feature-label relation is strictly monotone and sufficiently smooth, it converges at a cube-root-$n$ rate with scaled Chernoff-type limiting distribution, and it converges at the parametric $\sqrt{n}$-rate if the underlying relation is flat. In this article, we provide the complete picture of the distributional metamorphosis of the NPMLE, revealing a new limiting distribution which provides a significantly better distributional approximation for small samples in case of a weak feature-label relationship. It is shown to continuously interpolate between the two extremal cases. The innovative way to determine this distribution is to generate it as a limit of the NPMLE in the newly introduced weak-feature-impact triangular array for a particular parameter-sample-size constellation. Moreover, the phase transition is likewise observed for the suitably rescaled $L^{1}$-error in this weak-feature-impact scenario. As a by-product, its limiting distribution for flat regression functions is obtained, which was unknown before. The proof develops a completely new strategy, notably not based on the switch relation. A novel type of local minimax lower bounds accompanies these results.

The stochastic cross-dock door design problem entails determining the number of doors and their nominal capacities under uncertainty. The inbound flow of commodities from origin nodes is assigned to the entry doors consolidated in the platform, and the outbound flow is assigned to the exit doors to be delivered to the destination nodes. This problem combines three high computational difficulties, namely, NP-hard quadratic combinatorics, uncertainty in the main parameters, and ambiguity in their probability distribution. Distributionally robust optimization is considered to deal with these uncertainties. A two-stage mixed binary quadratic model is presented, where the first stage decisions are related to the design of the platform and the second stage ones are related to the assignment of the commodity flow to the doors in the members of the ambiguity set. The goal is to minimize the highest total cost in the ambiguity set, subject to the constraint system for each of those members. In addition to the risk-neutral version, a risk-averse formulation is presented. Given the difficulty of this problem, a min-max matheuristic scheme based on a scenario cluster decomposition is proposed for obtaining lower and upper bounds. A computational study is conducted to compare the solutions provided by the straightforward use of the state-of-the-art solvers CPLEX and Gurobi, as well as to validate the proposed matheuristic scheme.