Papers

We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu–Wood and Sawin–Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu–Wood and Sawin–Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.

This paper studies the three-dimensional Muskat problem for two immiscible fluids in a porous medium subject to gravity and surface tension. We consider the setting in which one fluid forms a bounded bubble surrounded by the other. For any value of the physical parameters, we prove the global-in-time nonlinear asymptotic stability of a translating spherical bubble for small initial perturbations in the critical Lipschitz class. This enlarges the class of critical spaces considered in previous two-dimensional bubble results. The proof combines spectral analysis of the linearized operator on $\mathbb{S}^2$ with multilinear singular integral estimates that exploit a crucial null structure in the nonlinear terms.

We prove that every bounded stable solution of the Allen–Cahn equation in $\mathbb R^3$ is one-dimensional. As a consequence, this shows the validity of De Giorgi's conjecture for monotone solutions in dimension four.

A $d$-set mapping is a function acting on a domain $X$ equipped with a partition into $d$ disjoint subsets. While standard functions represent $1$-set mappings, generalizations to arbitrary $d$-partite structures appear naturally across discrete mathematics. In this paper, we develop an analytic combinatorial framework to quantify the functional graphs of these mappings. By leveraging generating functions and singularity analysis, we derive exact asymptotic expansions for macroscopic graph properties as the cardinality of $X$ tends to infinity, including the expected number of connected components, cyclic nodes, and tail lengths. We demonstrate the efficacy of this framework by recovering the classical bipartite mapping results of Hansen and Jaworski, and successfully generalize these mechanisms to arbitrary $d$-set mappings, providing the foundational architecture to establish their probabilistic limit laws.

A standard question in computational real algebraic geometry is to compute all real solutions to a system of polynomial equations with real coefficients. One classical and promising approach is to track along a connected component of a real curve defined by a Newton homotopy, which is dependent upon the selected start point. As the start point varies, different subsets of real solutions may be obtained. This yields a partition of the space of start points into cells, and it is important to understand the structure of this partition in order to develop efficient algorithms based on Newton homotopies. The structure of the boundary of such cells and the number of cells in the corresponding partition are investigated for bivariate systems. Several examples are included to demonstrate the results.

We import the theory of $R$-links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of Lagrangian and symplectic sliceness, and more generally establishes one approach to what we call the Lagrangian slice-ribbon conjecture. Our main theorem asserts that a Legendrian knot with Thurston-Bennequin invariant $-1$ is Lagrangrian slice (resp. regularly Lagrangian slice) if and only if it supports a tight transverse (resp. Legendrian) $R$-link derivative.

We address the inverse problem of reconstructing the initial temperature distribution in a one-dimensional non-homogeneous heat equation with Dirichlet boundary conditions from a finite number of pointwise-in-time measurements at a single spatial location. We propose an explicit approximation scheme that incorporates the source term and utilizes a refined sequence of measurement times. Unlike previous approaches that often rely on exponentially growing observation times, our chosen time sequence allows for efficient recovery within a fixed time horizon. We establish algebraic convergence rates under suitable regularity assumptions on the initial data and the forcing, and we validate the theoretical sharpness with numerical experiments for varying numbers of measurements.

We introduce GridSFM, a framework that combines a pretrained foundation model across grid topologies with physics-informed fine-tuning for solving AC Optimal Power Flow (AC-OPF) at scale. It is a $15$ million parameter physics-inspired graph neural network pretrained across $54$ topologies of $500$ to $4{,}000$ buses. Our model attains a $2.45\%$ zero-shot generation-cost error on a $10{,}000$ bus case held-out operating conditions with no degradation as system size grows. Building on this, we pair the pretrained backbone with a physics-informed fine-tuning design based on Newton's method for power flow. With only $100$ solved instances, GridSFM adapts to unseen grids up to $10{,}000$ buses. We show it out performs single topology, dedicated neural network models that are trained more data, both in terms of cost and solver iterations when deployed as warm starting points. In designing this foundation model, we overcome the fact that the feasible set for AC-OPF can be disconnected. This is an obstruction that prevents any continuous neural network from approximating the solution map. To do so, we lift the problem and relax its constraints with logarithmically penalized slacks. We prove that the resulting elastic feasible set is contractible, that the AC-OPF minimizers remain minimizers of the elastic problem above an explicit penalty threshold, and that projecting an approximate solution back onto the AC-OPF feasible set is well posed. We release all models, data, and code so that the community can build on a shared starting point for AC-OPF.

We develop the thinning–sprinkling technique, a general method for proving robustness of graph properties under random vertex sampling. Using it, we show that random induced subgraphs of tough graphs, high-degree connected vertex-transitive graphs, and nearly regular sublinear expanders retain strong connectivity or expansion properties with very high probability. We also prove that every $k$-connected graph with $k=\omega(\log n)$ contains a spanning bipartite subgraph that is $\Omega(k)$-connected. Using these robustness results, we further develop a general framework for constructing almost Hamilton cycles from randomly sampled highly connected subgraphs. As a consequence, we show that tough graphs, connected vertex-transitive graphs and nearly regular expanders contain a cycle of length at least $(1-o(1))n$ whenever the toughness or degree is polylogarithmically large. This gives asymptotic solutions of longstanding conjectures of Chvátal and Lovász on Hamiltonicity of tough and vertex-transitive graphs.

Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to establish code families with constant encoding rate and linear distance, simultaneously achieving optimal asymptotic scaling in both parameters. In particular, we characterize explicit logical representatives of the full gauged code space, and provide a fundamental treatment of non-Abelian code distance in the sense of Knill–Laflamme based on combining expansion with cleaning to establish protection against arbitrary errors. We further construct an almost-good family with entire code spaces exhibiting long-range magic. Moreover, we use linearly addressable cup product to directly generate linearly many encoded magic states from gauging measurements of Clifford operators. These results extend good qLDPC codes beyond the Pauli stabilizer setting, lay the foundation for pursuing the no low-energy trivial magic conjecture and shed light on non-Abelian quantum phases and orders beyond geometric locality.

This note record a finitely generated extension of Caprace's example of a non-inner-amenable group is also not properly proximal, which answers the original question by Boutonnet, Ioana and Peterson. The extension is an explicit ICC subgroup of $\operatorname{GL}_4(\mathbb{F}_2(t))$, which has no property (T). Failure of proper proximality follows from the work of Ishan-Peterson-Ruth and the passage of proper proximality to co-amenable subgroups. Non-inner-amenability follows from a property-(T) subgroup and an elementary centralizer computation.

Let $(M, g)$ be an $n$ $(\ge 3)$ dimensional, non-collapsed compact Riemannian manifold and $\operatorname{Ric}^-$ be the negative part of the Ricci curvature and $\beta \in (\frac{2n}{n+2}, 2)$. We prove a relative volume comparison theorem when $|\operatorname{Ric}^-|^\beta$ is in the Kato class (cf. Definition 1.1), which results from a new integral Laplace comparison theorem in the spirit of \cite{PW} for a suitable conformal metric. This partly addresses an expectation in \cite{TZZZZ}, where the same result was proven when $|\operatorname{Ric}^-|^2$ is in the class.

Let $(X_i,d_i,\mathfrak m_i)$ be compact $\mathrm{RCD}(K,N)$ spaces converging to a space $X$ of rectifiable dimension $m$. Their first Betti numbers can drop by at most $N-m$. When equality holds, the maximal abelian covers have a non-collapsed limit $Y$ carrying a free isometric $\mathbb{R}^{N-m}$-action. We turn this limiting symmetry into fibrations of the approximating spaces. More precisely, after passing to a subsequence, there is an open full-measure set $G\subset X$, containing every regular point, on which $X$ is a topological orbifold and the $X_i$ admit local Seifert fibrations with $(N-m)$-torus fibres. Each finite local group acts on the fibre by translations. If the $X_i$ have no boundary, then $X\setminus G$ has Hausdorff codimension at least two. If $Y$ has no bubbling and $X$ is a smooth closed Riemannian orbifold, the local fibrations may be chosen as restrictions of a single global Seifert fibration. An affine replacement on the smooth manifold cover shows, in addition, that a finite cover of $X_i$ is homeomorphic to a product with $\mathbb{T}^{N-m}$; here we use the classification of affine torus bundles at the Betti number equality of Peng, Wang and Wang. When the base is a closed Riemannian manifold, the maps are torus bundles, confirming a conjecture of Zamora and Zhu for possibly singular RCD total spaces. The new ingredients are a pointwise linearization of the collapsing action at regular orbits, an invariant harmonic transverse coordinate compatible with finite isotropy, and an exactly equivariant orbit coordinate built from an elementary rounding-and-doubling argument in the collapsing deck group.

A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in the sharp integer form $\gpr(G)\le 2\lfloor |V(G)|/4\rfloor$ for every finite simple cubic bipartite graph $G$. The proof combines a directed contraction along a perfect matching, switching arguments based on dominator trees, a four-symbol boundary calculus for two-edge cuts, and the Gallai–Edmonds decomposition. Equality is attained by $K_{3,3}$ when $|V(G)|\equiv2\pmod4$ and by the cube $Q_3$ when $|V(G)|\equiv0\pmod4$.

We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree $\Delta\ge2$. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most $\Delta$ other terms, the Gibbs state is a mixture of product Pauli eigenstates whenever \[ \beta \le z_\Delta := \operatorname{arctanh}\left[\max_{0\le x \le 1}x\left(\frac{1-x}{1+x}\right)^{\Delta-1} \right]. \] For every $\beta>z_\Delta$, a finite commuting Hamiltonian with maximum overlap degree at most $\Delta$ has an entangled Gibbs state. At any fixed $\beta<z_\Delta$ strictly below the threshold, a classical polynomial-time algorithm produces samples from a distribution over product Pauli eigenstates approximating the Gibbs state in trace distance.

For a compact connected Hamiltonian $T$-space (with $T$ a torus), the components of the set of (relative) regular values of the momentum map are open convex polytopes, whose closures (called the chambers) form a polyhedral decomposition of the momentum polytope. In a previous paper we showed that there is a canonical affine stratification of the momentum polytope with the property that, while varying through a stratum, the fibers of the momentum map do not change as $T$-spaces. In this paper we show that the strata of this stratification are the relative interiors of the faces of the chambers. In doing so, we also give a proof of the fact that these faces form a polyhedral complex, which (it seems) was missing from the literature. We further extend the above to Hamiltonian $T$-spaces whose momentum map is proper as map into a convex set and, in the compact Kähler case, we point out a description of the above strata in terms of the $T_\mathbb{C}$-orbit closure polytopes.

High-dimensional extreme eigenvalue problems often arise from molecular vibrational models, electronic structure calculations, and quantum mechanics. Directly solving these problems suffers from the curse of dimensionality. Instead of tackling the problem in high-dimensional ambient space, we reformulate the problem through low-rank tensor formats, which can significantly reduce the computational cost and storage. Specifically, we consider the Rayleigh–Ritz problem on bounded-rank tensors in the tensor train format, which enables a more flexible choice of rank parameters. Moreover, we consider a smooth parametrization for bounded-rank tensors by introducing slack variables, leading to a smooth manifold structure. The original Rayleigh–Ritz problem is therefore transferred to an optimization problem on the manifold. We develop the Riemannian geometry and propose optimization methods for solving extreme eigenvalue problems. In practice, the Kronecker-product structure in Hamiltonian and PDE operators is employed to simplify the computation. Numerical experiments on the harmonic oscillator, the Laplace operator, the Schrödinger equation, the layered cluster problem, and the Bose–Einstein model demonstrate that the proposed method achieves accuracy comparable to full-space eigensolvers with reduced computation time and avoids storing full vectors. The results also show numerical rank reduction during iteration and robustness when the rank parameter is over-estimated. In addition, the fixed-point iterations illustrate that the proposed methods are able to serve as an inner eigensolver for solving nonlinear eigenvalue problems.

We determine the Doob-Martin boundary of Hoffman's leaf-grafting graph on finite unlabelled non-plane rooted trees. Its full and minimal boundaries coincide and are parametrized by deterministic recursive paintboxes, identified when their finite sampling laws agree. Every central measure is a unique mixture of the corresponding extremal laws, and its limiting boundary point generates the completed tail field. We also show that the boundary is homeomorphic to the space of unordered root masses marked by child boundary classes. For the recursive Ewens family, we obtain the unique extremal decomposition from independent Poisson-Dirichlet splits, including the uniform recursive-tree and rooted-tree Plancherel cases.

Extra-large multiple-input multiple-output (XL-MIMO) systems {serving users across near-field and far-field regions} experience spherical wavefront propagation that provides enhanced spatial resolution over traditional far-field systems. In this work, we propose a novel weighted rate (WR) ordering-based successive interference cancellation (SIC) scheme that exploits the spatial degrees of freedom inherent in near-field propagation. We also develop an iterative detection and decoding (IDD) framework that integrates the proposed WR ordering with low-density parity-check (LDPC) coding and channel estimation. We then analyze the information rates and the impact of near-field channel characteristics. Numerical results show that the proposed WR-SIC and the IDD scheme outperform existing approaches.

The main purpose of this paper is to establish the local quadratic convergence of Newton's method for nonlinear optimal control problems. Under suitable smoothness assumptions, the cost functional is first shown to be twice Gâteaux differentiable with respect to the control. An explicit operator representation of its second derivative is derived, and its continuity in different control spaces is examined. The second-order coercivity condition in $L^2(0,T;\maathbb{R}^m)$ is then shown to be equivalent to the existence of a strongly regular solution to an associated matrix Riccati differential equation, thereby expressing an infinite-dimensional quadratic-form condition in terms of the solvability of a matrix differential equation. On this basis, local quadratic convergence of Newton's method in $L^\infty(0,T;\maathbb{R}^m)$ is established using linear-quadratic optimal control theory. Under suitable structural assumptions, local quadratic convergence in $L^2(0,T;\maathbb{R}^m)$ is also established. Explicit estimates of the corresponding convergence neighborhoods are derived, and the theoretical results are illustrated by numerical examples.

Let $G$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ assigns to each vertex and each edge a label from $\{1,\ldots,k\}$. The weight of a vertex is the sum of its label and the labels of its incident edges. A total labeling is vertex irregular if all vertex weights are distinct. The total vertex irregularity strength $\text{tvs}(G)$ is the smallest $k$ for which $G$ has a vertex irregular total $k$-labeling. For an $r$-regular graph $G$ on $n$ vertices, a counting argument gives $\text{tvs}(G)\ge\lceil(n+r)/(r+1)\rceil$. The restriction of a conjecture of Nurdin, Baskoro, Salman, and Gaos to regular graphs asserts that this bound is attained. We prove this assertion for cubic and $4$-regular graphs. We also show that, for every fixed $r\ge2$, a recent theorem on prescribed degree frequencies implies the assertion for all sufficiently large $r$-regular graphs.

We study the vanishing-noise limit of the rate function for the Donsker-Varadhan large deviation principle for one-dimensional diffusion processes on a circle. As is well known, the rate function can be represented either as the Legendre transform of the principal eigenvalue of the perturbed infinitesimal generator or by a variational formula. We analyze the asymptotic behavior of the principal eigenvalue as the parameter in front of the noise goes to zero and compare the Legendre transform of the limit with the expression obtained as the limit of the variational representation. We prove, in particular, that the resulting expressions do not always coincide, leading to continuity and discontinuity phenomena in infinite-dimensional functional spaces. Moreover, we use the previous analysis to pass to the limit in the LDP, using the notion of {\Gamma}-convergence.

This paper investigates the Schrödinger-Poisson system involving electromagnetic fields. Our approach is mainly based on the variational method. By developing the global compactness result, we establish the existence of higher-energy bound state and ground state solutions under suitable assumptions. In addition, under symmetry conditions, the existence and multiplicity of intertwining solutions are obtained for both finite and infinite symmetry group.

Statistical model transformations based on randomly stopped sums, maxima and minima are widely used to extend statistical models. We characterize the complete set of stopping models for which randomly stopped sum and extreme model transformations function as statistically stable (idempotent) model extensions. Stability requires the underlying stopping model to be closed under pgf composition. We prove that any finite-dimensional, connected stopping model closed under pgf composition is necessarily a family of random variables whose pgfs commute. Using the corresponding Koenigs function, we establish that these models form a statistical manifold admitting a global, one-dimensional parametrization $\theta = \Pr(N=1) \in (0, \theta_*]$, where the probability mass at $i$ is a polynomial in $\theta$ of degree at most $i$. Finally, we establish a duality between stopping models closed and containing the identity variable (the ones yielding stable extensions) and the set of probability distributions supported on the positive integers. These findings disprove the long standing conjecture that statistical stability occurs only under geometric stopping.

We prove that for every isotropic log-concave distribution $P$ on $\mathbb{R}^d$ and every even $m\ge2$, the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares, where $C>0$ is a universal constant. This improves on the Poincaré-dependent bounds (Kothari and Steinhardt, 2017), recovering the optimal moment bounds for log-concave distributions. As an immediate corollary, we obtain computationally efficient algorithms with dimension-free error guarantees for a wide range of statistical estimation problems. Our proof proceeds by using stochastic localization to decompose $P$ as an average of strongly log-concave measures, whose centered moments admit the subgaussian certificates (Diakonikolas et al., 2025). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from the variance inequality for quadratic forms (Letwin, 2026) suffices to control this averaging at every even degree.

This paper establishes that transcendental entire functions do not have wandering Fatou components whose forward orbit is bounded. This settles a long-standing question in transcendental dynamics. A stronger statement is proved using the same method: the orbit of every point in a wandering domain of a transcendental meromorphic function is unbounded. In particular, transcendental meromorphic functions do not admit orbitally bounded wandering domains. The proofs rely mostly on uniform estimates for hyperbolic area, following the approach of Ye [arXiv:2609.23834v1].

We study the largest common subtree of two independent unlabeled uniform attachment trees (also known as random recursive trees). Our main result shows that, when the two trees have $n$ vertices each, their largest common subtree has at least $n^{0.83}$ vertices with high probability. This is obtained by starting with the common subtree induced by the Ulam–Harris labels in the two trees and improving using local optimization steps. We also give some upper bounds and bounds for general random tree growth models. We leave as an intriguing open question to understand the magnitude of the size of the largest common subtree.

We study Fibonacci polyominoes, a class of column-convex polyominoes whose lower boundary is a staircase path made of unit horizontal and vertical steps. We derive generating functions that enumerate these polyominoes by area and semiperimeter. The resulting formulas give refinements of the classical Fibonacci enumeration and lead to explicit expressions involving Catalan numbers. We give a bijection with labeled Dyck paths that provides combinatorial proofs of these formulas and translates natural statistics on Fibonacci polyominoes into peaks, returns, and related statistics on Dyck paths. This yields refinements by descents and left-to-right minima, including distributions governed by Narayana numbers. We also study consecutive columns of equal height, leading in the diagonal case to a refinement of the Catalan enumeration governed by Motzkin numbers.

Let $R$ be a commutative ring and $\Phi$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $\Phi$-reduced and $\Phi$-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with $\Phi$ are determined by $\mathcal{L}$. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When $\Phi$ is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module-theoretic framework for studying generalized torsion and completion with respect to families of ideals.

We establish global $W^{2,p}$ solvability for the Poisson equation with mixed Dirichlet–Neumann boundary conditions in two classes of domains in all dimensions $n\geq 2$. For $C^{1,\alpha}$ domains with a Reifenbeg flat interface, we obtain the optimal range $1<p<4/3$, provided that $\alpha>1-1/p$. For Lipschitz polyhedra with facewise boundary decompositions, we obtain solvability for $p$ close to $1$. In both settings, we establish endpoint $W^{2,1}$ solvability for data in an adapted atomic Hardy space. Applications of the methods to Lamé systems are also discussed.