Papers

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random directed Erdős–Rényi bipartite graph $\vec G(n,\lceil\alpha n\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\vec G(n,\lceil\alpha n\rceil,v)$ as $n\to\infty$. We prove this conjecture. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto $H$, for each finite abelian $p$-group $H$. However, in the case of $p$-Sylow subgroups of sandpile groups of random directed bipartite graphs, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this by restricting to a high-probability subset of graphs on which the surjective moments are well-behaved, and discarding a rare exceptional set of graphs whose contribution to the distribution vanishes but whose contribution to the surjective moments often diverges. Computing the conditional surjective moments on the good set and applying Wood's universality theorem yields the desired convergence in distribution.

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

We give the structure theorem of Galois self-covers $f: \mathbf{P}^n \to \mathbf{P}^n$. As an application, we show that the essential dimension of every such nontrivial cover attains its maximum possible value $n$. As another application, we prove that the pair $(\mathbf{P}^n, R_f/(q-1))$ is log Calabi-Yau as conjectured by Gongyo, where $R_f$ is the ramification divisor and we write $f^*\mathcal{O}(1) = \mathcal{O}(q)$.

We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of $N$ distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the $N\times N$ matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general $\Gamma$-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele–Shaw problem with surface tension (and contact angle conditions at triple junctions).

Finiteness of the right little finitistic dimension of a finite dimensional algebra $\La$ is known to be equivalent to existence of a (possibly infinite dimensional) tilting right $\La$-module $T_f$ whose tilting class is $\{ T_f \}^{\perp_\infty} = (\mathcal P ^{< \infty})^{\perp}$, \cite{AT}. We use this equivalence to interpret the recent surprising results of Cummings \cite{C} concerning the asymmetry of left and right finitistic dimensions of the triangular matrix algebras $\tilde{A}$ built from arbitrary basic finite dimensional algebras $A$. In particular, we determine the structure of the tilting right $\tilde{A}$-module $T_f$ in the case when the algebra $A$ has finite right global dimension.

This paper studies the complexity of a dual dynamic programming (DDP) method for solving a class of convex optimization problems with linear coupling constraints. Existing complexity results based on DDP depend on the dimensions of the state vectors and, in particular, grow exponentially with dimension. The goal of this paper is to establish a complexity bound that is independent of the dimension. Our approach first studies an unconstrained strongly convex problem and develops a flexible framework for DDP, called FDDP, for solving the associated dynamic programming equations, and establishes an iteration-complexity bound for it that is independent of the dimension. A dimension-independent complexity bound for the original linearly constrained problem is then obtained by applying FDDP to a corresponding unconstrained strongly convex problem obtained via smoothing and penalization. Inspired by the literature on bundle methods, FDDP updates lower approximations of cost-to-go functions through a generic procedure that includes both the classical multi-cut DDP and a new two-cut DDP variant as special cases. The two-cut variant maintains only two affine cuts per stage at each iteration, making it more memory-efficient while retaining the same theoretical guarantees. Finally, numerical experiments illustrate the practical behavior of multi-cut and two-cut DDP, including their dependence on problem parameters and their performance relative to a direct quadratic constraint reformulation.

We compare two existence principles for maximal ideals, a classical one for vector lattices with a strong unit and a second, newly introduced one for vector lattices with a strong truncation. Although the latter strictly generalizes the former, we show that the two statements are equivalent over ZF set theory.

Nivat's conjecture asserts that every two-dimensional configuration $\eta : \mathbb{Z}^2 \to \mathcal{A}$ whose rectangular pattern complexity satisfies $P_{\eta}(k,n) \leq kn$ for some $k,n \in \mathbb{N}$ is periodic. A theorem of Cyr and Kra \cite{CyrKra16} establishes the conjecture in the short-rectangle case $P_{\eta}(k,n) \leq kn$, with $k \leq 3$. Using the algebraic framework of Kari-Szabados \cite{KariSzabados20} and recent advances on periodic decompositions and one-sided nonexpansive directions \cite{Colle23,Colle22}, we extend the Cyr-Kra result to the case $P_{\eta}(4,n) \leq 4n$: every configuration satisfying this complexity bound is periodic. The key new ingredient is an intermediate structural theorem of independent interest: for any non-periodic configuration with low convex pattern complexity and integer-valued alphabet $\mathcal{A}$ contained in $\mathbb{Z}_+$, there exist a configuration $\vartheta$ in the orbit closure of $\eta$, a $\mathbb{Z}$-minimal periodic decomposition $\vartheta = \vartheta_1+\cdots+\vartheta_m$, a prime $p \in \mathbb{N}$ with $\mathcal{A} \subset [[p]]$, and pairs of disjoint half-planes $U_i,V_i \subset \mathbb{Z}^2$ such that the reductions modulo $p$ of the components $\vartheta_i$ are fully periodic on $U_i$ and on $V_i$ simultaneously, for each $1 \leq i \leq m$.

Submodular set functions, characterized by the diminishing-returns property, provide a unifying combinatorial framework for many subset-selection problems in decision and control. Although exact maximization is NP-hard in general, the structural properties of submodular functions enable simple greedy algorithms that achieve constant-factor approximation guarantees for monotone objectives, with randomized greedy-based variants extending such guarantees to the non-monotone case. This survey reviews the theory, algorithms, and applications of submodular optimization with a focus on systems and control. We cover the structural properties of submodular functions, including curvature and the submodularity ratio, the constraint families that arise in practice (matroids, knapsack, and $p$-systems), and the main approximation algorithms for monotone and non-monotone submodular maximization, with up-to-date approximation ratios and hardness results. We then survey applications across sensor scheduling, multi-agent coordination, robust submodular optimization, leader-follower systems, distributed submodular optimization, game theory, system theory, resource allocation, social networks, and informative path planning. The survey emphasizes practically implementable greedy-based algorithms and instance-dependent refinements via curvature and the submodularity ratio. We close with observations on canonical control-theoretic objectives: certain functionals are submodular (the log-determinant and rank of the controllability Gramian, and the log-determinant of the Kalman filter information matrix), whereas closely related objectives fail to be sub- or supermodular (the steady-state Kalman filter error covariance, and the average control energy obtained from the inverse Gramian). We also highlight the cross-cutting open directions that follow.

Let $\mathcal{S}_{ex}^{\ast}$ denote the class of normalized analytic functions $f$ in the unit disk satisfying \[ \frac{zf'(z)}{f(z)} \prec e^{\alpha z},\qquad 0<\alpha\le 1. \] We obtain sharp bounds for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$. In particular, the bound for $|\Gamma_3|$ has a four-branch structure with transition values \[ \alpha_\star \approx 0.542712,\qquad \alpha_b \approx 0.679103,\qquad \alpha_c \approx 0.691095. \] We also establish sharp upper and lower bounds for the successive coefficient difference \[ |\Gamma_2|-|\Gamma_1|, \] for the inverse logarithmic Hankel determinant \[ H_{2,1}\bigl(F_{f^{-1}}/2\bigr), \] and for the third-order Hermitian–Toeplitz determinant $T_{3,1}(f)$, whose sharp lower bound changes at \[ \alpha_H=\frac{-1+\sqrt{241}}{15}\approx0.9683. \] Furthermore, we completely solve the sharp generalized Fekete–Szegő problem for the functional \[ |a_3-\lambda a_2^2|-\mu|a_2|. \] All estimates are sharp, and the corresponding extremal functions are explicitly constructed by using Carathéodory function representations.

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

Let $X$ be a smooth projective curve over a field $k$ with an action of a finite group $G$. A well-known result of Chevalley and Weil describes the $k[G]$-module structure of cohomologies of $X$ in the case when the characteristic of $k$ does not divide $\# G$. In case when $G$ has a cyclic $p$-Sylow subgroup, it is known that the $G$-structures of the module of holomorphic differentials and of de Rham cohomology of $X$ are completely determined by the ramification data of the cover $X \to X/G$. In this article we extend this result to the crystalline cohomology of $X$. Also, we provide an explicit description of the structure of the crystalline cohomology when $G = \mathbb Z/p^n$. The main used tool is the theory of Yakovlev diagrams - algebraic objects that classify the $G$-modules over Witt vectors.

We show that the partitions that always appear with nonnegative $e$-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

An element of a group is called generalized torsion if a finite product of its conjugates equals the identity. We characterize the generalized torsion subset of upper triangular matrix groups over commutative rings and determine when these groups satisfy positive generalized identities in terms of torsion and exponent properties of the unit group of the underlying ring. Moreover, as an application of the developed theory, we construct triangular matrix groups satisfying positive generalized identities of prescribed degrees and obtain lower bounds for the minimum degrees of such identities. We also provide a counterexample to a question of Sherman concerning groups generated by elements of infinite order.

We study the frog model with death and drift on $\mathbb{D}_{m,d}$, the free product of $d+1$ copies of the complete graph of order $m$. Active and inactive particles are located at the vertices of $\mathbb{D}_{m,d}$. Each active particle performs a $\alpha$-biased random walk towards the root of $\mathbb{D}_{m,d}$, dying after a random lifetime with a geometric distribution of parameter $1-p$. Each inactive particle remains dormant until an active particle visits its location. We present conditions on the parameters $\alpha$ and $p$ for the process to die out almost surely and to survive with positive probability. Our proofs are based on comparisons of the model with simple and multi-type branching processes.

We analyze the complexity of Bidirectional Stochastic Dual Dynamic Programming (BSDDP) algorithm applied to multistage stochastic optimization problems with strongly convex cost functions. Under standard regularity assumptions and without discount factor, we establish an explicit complexity bound on the expected number of iterations needed to obtain an epsilon-optimal policy. Our bound depends on the strong convexity constants and Lipschitz constants of the stagewise cost functions but is independent of the dimensionality of the state space. This improves upon the complexity of SDDP for convex problems from (Lan, 2020), particularly when the number of stages is moderate and the state dimension is large. The analysis further reveals that the complexity decreases as the strong convexity of the cost functions increases, highlighting the algorithm efficiency in structured problem instances.

This is a review paper about the use of transfer operators to study the statistical properties of hyperbolic billiards. The main focus is on equilibrium states, sequential billiards, and decay of correlation for billiard flows. In addition to reviewing the literature, we try to flesh out the main ideas, highlight the relations between them, and provide the larger context in which the problems discussed here are situated.

In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice with side length 5 or larger, and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

We $\textit{unconditionally}$ prove a $\textit{generic}$ categorical local Langlands conjecture for a large class of quasi-split reductive $p$-adic groups $G$, including all quasi-split classical groups and some non-classical groups. More precisely, we construct a natural fully faithful functor from the stable $\infty$-category of $\textit{generic}$ Bernstein blocks on the automorphic side to the stable $\infty$-category of ind-coherent sheaves on the moduli stack of (arithmetic) $L$-parameters, generalizing earlier work of the first author with Ben-Zvi, Chen and Nadler [BZCHN24] for $\mathrm{GL}_n$. Moreover, for an $\textit{arbitrary}$ quasi-split reductive $p$-adic group $G$, we formulate a classical local Langlands framework under which a classical correspondence can be lifted to an $\infty$-categorical correspondence. Our result builds upon the phenomenal recent work of Zhu [Zhu25] on the unipotent block, as well as structure results such as [Sol22] on the automorphic side and [DHKM25] on the spectral side. In particular, our work establishes [HM26,Conjecture 8.2.1], which implies that the conditional proof of [HM26] for the Fargues-Scholze categorical local Langlands equivalence [FS24] (conditional on the conjectured compatibility of the Fargues-Scholze construction with spectral Eisenstein series) applies as well to a large class of quasi-split reductive $p$-adic groups $G$.

The category of noncommutative quadratic quadric hypersurfaces, ${\tt Quad}\text{-}{\tt QHS}$, consists of pairs $(A, f)$, where $A$ is a quadratic algebra and $f \in A$ is a nonzero degree $2$ element. We associate to such $(A, f)$ a pair $(\bar{A}^!, f^!)$, and show that this association makes ${\tt Quad}\text{-}{\tt QHS}$ into a category with duality. We construct a faithful functor from the category of graded modules over $\bar{A}^!$ to the homotopy category of curved DG modules over a canonical curved DG algebra $(A \otimes \bar{A}^!, d, f \otimes f^!)$. If $A$ satisfies the left strong rank condition and $f \in A$ is not a right zero divisor, we show that the restriction of our functor to a natural full subcategory of the category of graded modules over $\bar{A}^!$ is valued in a stable category of noncommutative matrix factorizations of $f$. When $A$ is Koszul of finite global dimension and $f \in A$ is normal and regular, we prove that the even Clifford algebra, $\bar{A}^![(f^!)^{-1}]_0$, is isomorphic to a canonical PBW-deformation of a Zhang twist of the $2$-Veronese subalgebra of the Koszul dual $A^!$. Finally, we study several classes of Artin-Schelter regular algebras to illustrate our results.

For every $n\geq3$, we determine the union of the spectra of row-stochastic matrices supported on a directed $n$-cycle with self-loops and strictly positive cycle edges. These matrices describe progression through cyclic stages with geometric waiting times, and also arise by uniformization of unidirectional continuous-time reaction cycles. For the latter, the classification determines the minimum transition-rate bound required to realize a prescribed nonreal eigenvalue and provides rates attaining that bound. The angles $m=\mathrm{Arg}\lambda$ and $M=\mathrm{Arg}(\lambda-1)$ reduce the nonreal eigenvalue problem to determining the range of a strictly convex logarithmic sum over angles with prescribed sum. Jensen's inequality gives its minimum, simplex vertices give its finite maximum, and connectedness makes these bounds sufficient for realization. The resulting branch graphs are strictly decreasing and their horizontal sections are ordered. This ordering gives the complete boundary, which alternates between uniform-cycle segments and one-loop algebraic arcs. We state a direct membership criterion and construct a realizing matrix for every spectral point; every nonreal point admits a realization with at most two distinct self-loop weights. The real spectral set is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. Allowing deleted cycle edges adds only $0$ in odd dimension and adds nothing in even dimension. We also apply the classification to a branching network with matched holding probabilities on parallel paths. The proofs are independent of Karpelevich's theorem.

We use the exactly-solvable t Hooft model of 1+1D large-N_c QCD as a rigorous laboratory for the breakdown of analyticity of a causal response function, the meson two-point function. A PT-symmetric deformation i gamma(x-1/2) of the light-cone meson operator, the analogue of an imaginary chemical potential, drives the lowest two mesons to an exceptional point (EP) at gamma_c. Recasting the resolvent as a Jacobi continued fraction yields gamma_c in closed form: 2 pi g^2 N_c at the two-pole level, converging to 7.966 g^2 N_c by depth five – an analytic, not numerical, threshold. The square-root exponent nu=1/2 is fixed by the 2x2 Jordan form and confirmed by finite-size scaling to N=1999. The breakdown has an unambiguous time-domain signature: the propagator norm is bounded for gamma < gamma_c, grows linearly at gamma_c (the Jordan secular law), and exponentially beyond – observable, since the deformed operator is a non-Hermitian Wannier-Stark ladder, in photonic and topolectrical analogues. The threshold is locked to confinement, gamma_c propto g^2 N_c, and recurs as a uniform EP cascade; a second, non-reciprocal deformation yields an exactly-exponential non-Hermitian skin effect. This is the first analytically-controlled instance of exceptional-point analyticity breakdown in a confining gauge theory.

Asymptotic analysis is one of the classical tools for bridging models across scales. Behind many such derivations lies a common symbolic structure: an ansatz, a substitution, an order-by-order matching procedure, and the extraction of effective equations or interface conditions. This article revisits that structure through two representative bridges: the kinetic-to-fluid limit, illustrated by radiative transfer with interface layers and by neural-network approximations of Boltzmann equations, and the quantum-to-classical limit, illustrated by the Frozen Gaussian Approximation and its Dirac extension. We then explain why such derivations are natural candidates for AI-assisted formalization: their recurring symbolic structures can be organized, verified, and reused. In this sense, a carefully structured expository paper may serve not only as a review, but also as a mathematical seed for future AI-assisted environments.

We study local eigenvalue spacing for finite truncations of a two-sided Jacobi matrix with two movable endpoints. In particular, we show that a suitable analog of clock spacing follows from a pointwise reflectionlessness condition. We obtain this as a consequence of a new scaling limit for Christoffel–Darboux kernels with a movable starting point. Without reflectionlessness, we obtain a new class of limit kernels, which combine distinct contributions from $\pm\infty$. We also show that clock spacing in the two-sided setting is a fragile phenomenon, which can be destroyed by the change of a single Jacobi coefficient; in particular, it is not merely a consequence of absolutely continuous spectrum.

The number of perfect matchings in a user-behavior bipartite graph is a natural measure of anonymity: more matchings mean greater uncertainty for an attacker. A fundamental question is which graph structure maximizes this count for a fixed infrastructure cost, represented by the number of edges. We prove that the answer is $k$-anonymity by partitions. Using Brègman's Theorem, we show that partitioning users into equal-sized groups and making each group a complete bipartite component achieves the theoretical upper bound on perfect matchings. For edge counts where an exact partition is impossible, we construct a family of graphs that asymptotically attains this bound as the group size grows. We further prove that this optimality is robust: after an attacker de-anonymizes a user by the most damaging choice, the resulting graph is still a partition graph and remains optimal. Together, these results provide a combinatorial justification for the widespread use of $k$-anonymity by partitions in anonymity system design.

Many numerical methods require solution of a sequence of linear systems with the same matrix and similar right-hand sides. Krylov subspace methods are a common tool for solving such linear systems, and a carefully chosen initial guess for the solution can reduce the total number of iterations, and thereby the total computational cost, required for convergence to a specified numerical tolerance. This paper introduces the WING algorithm, a modification of Fischer's second algorithm, which lowers the cost of forming an acceptably close initial guess by skipping orthogonalization and solving the possibly singular normal equations with a pseudoinverse. We demonstrate the efficacy of the new algorithm, particularly for solving linear systems with coarse relative tolerances, with numerical benchmarks based on fluid-structure interaction, mantle convection, and earthquake models.

Our aim in this paper is to study principal eigenvalues, maximum principles and estimates for Lane-Endem related systems in nondivergence form.

Current-status data arise when an event time is observed only through an indicator of whether it occurred before an examination time. This paper studies a nonparametric neural-network sieve maximum likelihood estimator of the conditional cumulative distribution function of the event time. Under Hölder smoothness assumptions, we establish an explicit convergence rate by combining approximation theory for rectified linear unit neural networks with empirical-process arguments. This result provides theoretical support for neural-network estimation and subsequent inference under current-status observation.

We study finite-sum composite optimization under two departures from classical stochastic gradient descent theory that are central in practice: incremental gradient access and heavy-tailed gradient noise. Specifically, we consider fixed cyclic passes over component gradients and assume that, at the optimum, component gradients have a bounded $q$-th centralized moment for some $q\in(1,2]$. This setting is much closer to modern ML training practice than the assumptions used in classical SGD theory, yet its theoretical understanding remains limited. We propose a Dual Averaging Power-Prox method for incremental gradients and establish, to the best of our knowledge, the first convergence analysis in this regime. We further show that our method achieves a better asymptotic convergence rate than the corresponding SGD method with i.i.d. (with-replacement) sampling.

We prove that the von Neumann algebra $\cL(G)$ associated with any hyperbolic group $G$ satisfies the following amenable absorption property: for any infinite maximal amenable subgroup $H \leqslant G$ and any amenable von Neumann subalgebra $\mathcal{Q} \subset \cL(G)$ with diffuse intersection with $\cL(H)$, one must have $\mathcal{Q} \subset \cL(H)$. This strengthens a result of Boutonnet and Carderi \cite{BC2}. We also establish similar amenable absorption results for the broader class of acylindrically hyperbolic groups, including relatively hyperbolic groups, mapping class groups, and limit groups.