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 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 prove a generic categorical (arithmetic) 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 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 [BZCHN24] for $\mathrm{GL}_n$. Moreover, for an 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. Furthermore, combined with the recent work of Hansen-Mann [HM26] and assuming the expected compatibility of Fargues-Scholze construction with spectral Eisenstein series, our results give the full Fargues--Scholze categorical local Langlands equivalence [FS24], without the genericity condition, for a large class of quasi-split reductive $p$-adic groups $G$.

For every $n$, we determine the complete eigenvalue region of the $n$-cycle stochastic family. For $n\ge 2$, write $A_n(α)$ for the matrix indexed by $\mathbb Z/n\mathbb Z$ with $$ (A_n(α))_{j,j}=α_j,\qquad (A_n(α))_{j,j+1}=1-α_j,\qquad 0\le α_j<1, $$ and all other entries zero, and set $C_n=\{A_n(α):α\in[0,1)^n\}$. Writing $Σ_n$ for the corresponding spectral union, the trivial cases are $Σ_1=\{1\}$ and $Σ_2=[-1,1]$. For $n\ge 3$, we give an explicit description of $Σ_n$ in angular coordinates $m=\mathrm{Arg}(λ)$ and $M=\mathrm{Arg}(λ-1)$. Under the map $$ Λ(m,M)=\frac{\sin M}{\sin(M-m)}e^{im}, $$ the upper half of $Σ_n$ is the image of a finite union of $K=\lfloor(n-1)/2\rfloor$ vertical angular sectors. Its exposed boundary is an alternating chain of Jensen chords, arising from the Jensen-equality lines $M=ϕ_k$, and algebraic one-loop arcs joining the relevant roots of unity to $0$; the lower boundary is obtained by complex conjugation. The real spectral part is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. The proof is independent of Karpelevich's theorem and reduces the two-monomial characteristic equation to sharp argument bounds on a simplex, obtained by Jensen, majorization, and finite visibility arguments.

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 $\pm\infty$.

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.

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 \emph{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.

We prove that the billiard flow in any finite polygon has at least one periodic orbit. The proof by contradiction is based on a fundamental result on the dynamics of the billiard flow by Galperin, Krüger and Troubetzkoy, on the geometry of a one-parameter scaling of the natural Riemannian metric on the unit tangent bundle, and on the topology of the skeleton or cut-locus of the scaled metrics.

Railway crew scheduling consists of assigning a set of tasks to crew members in the form of feasible duties so as to minimise costs, and is one of the key planning problems faced by railway operators. While column generation is the dominant solution method, literature on exact branch-and-price approaches is scarce. Moreover, few publicly available benchmark instances exist. We present a state-of-the-art branch-and-price algorithm and conduct a systematic computational study of its components, including branching rules, primal heuristics, and reduced cost fixing techniques. To this end, we introduce a novel library of 75 benchmark instances, ranging in size from 450 to 3,016 tasks. The instances are generated by mimicking the full railway planning process, validated against data from Netherlands Railways, and publicly available with best known solutions and a solution checker. We report average optimality gaps well below 1% across all instance sizes, driven primarily by the strong performance of primal heuristics, whereas branching and reduced cost fixing contribute little to closing the gap. Our findings provide support for the popularity of column generation-based primal heuristics, and show that improving the dual bound remains the main challenge for exact methods in railway crew scheduling.

We prove the realizability of genus-$0$ branch data of the form $(2^r,a \mathop{|} 3^s,b \mathop{|} 3^t,c)$ and $(2^r,a \mathop{|} 3^s \mathop{|} 3^t,b,c)$, where $a$, $b$, $c$ are not divisible by $2$, $3$, $3$ respectively. The proof uses an explicit combinatorial description of coverings of the sphere branched over $3$ points via dessins d'enfants. As a corollary, we establish realizability for a broader class of branch data with more critical values.

Quantum optimal transport (QOT) is a rapidly developing field. Among the many formulations of this adaptation of classical optimal transport (OT) to spaces of density matrices, we numerically study a family of distances based on a dynamical formulation inspired by the Benamou-Brenier OT formulation. We introduce an interior-point regularized method to compute geodesics between positive semidefinite matrices and visualize the results in terms of integral kernels and densities, inspired by quantum chemistry applications. We show that dynamical QOT may provide a good approximation to certain problems in quantum chemistry with appropriate parameter tuning. We also study the numerical properties of the distances at hand, and the convergence of the objects when the size of the matrices increases.

In a symmetric Bernoulli process, all binary strings, or ``words'' of the same length have the same long term frequency. However, between two such words, one may have a ``frequency advantage'' in the sense that in any long enough segment of the Bernoulli process, the probability that the word occurs more times than the other word is greater than the probability the other way around. To characterize the frequency advantage in the long run, the asymptotics of the difference between the two probabilities as the length of the segment of the Bernoulli process tends to infinity is derived.

Let $(X,d,μ)$ be a complete separable metric measure space satisfying a doubling condition and a $(1,1)$-Poincaré inequality. We develop a rigorous framework unifying two lines of analysis: the isoperimetric-profile approach of Coulhon-Grigor'yan-Levin \cite{CGL2003} and the grand/small Lebesgue-Sobolev scale introduced by Fiorenza-Formica-Gogatishvili \cite{FFG2018}. An explicit profile-to-scale transform $\PhiX$, defined via an inverse integral of $\IX$, converts geometric data into grand Lebesgue parameters. Sharp, up to universal constants, embeddings $W^{1,1}(X) \hookrightarrow \mathcal{G}_X$ with explicit constants (Theorem \ref{thmmain}). A converse: controlled grand embeddings imply explicit lower bounds on $\IX$ (Theorem \ref{thmconverse}). Concrete examples in genuinely non-Euclidean settings: the Heisenberg group $\mathbb{H}^1$, a model manifold with logarithmic volume growth, and Gaussian measure on $\R^n$ treated as a locally doubling space. All arguments are carried out on general metric measure spaces without reference to charts or a smooth structure; the gradient is the upper gradient in the sense of Heinonen-Koskela, and perimeter is the outer Minkowski content.

Let $\cX$ be a (projective, geometrically irreducible, non-singular) algebraic curve of genus $\ge 2$ and positive $p$-rank $γ(\cX)$, defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p>0$. Contrary to what occurs for the genera, no function $h(γ)$ exists such that $|\aut(\cX)|\le h(γ)$ whenever $γ=γ(\cX)$. Thus, to have a bound on $|\aut(\cX)|$ only depending on $γ(\cX)$, some restrictions on $\cX$ and $\aut(\cX)$ are needed. In this context, the following theorem is proven. Let $Γ$ be a subgroup of $\aut(\cX)$. Assume the existence of a point $P\in \cX$ such that if $S_P$ is the Sylow $p$-subgroup of $Γ_P$ fixing $P$, then the quotient curve $\cX/S_P$ is rational. Then %$γ(\cX)\ge 2$ and the following $p$-rank analog of the Riemann-Hurwitz bound \begin{equation*} %\label{eq18122025} |Γ|<900 \left(\frac{p}{p-1}\right)^4 γ(\cX)^4 \end{equation*} holds, unless a subgroup of index $\le 2$ of $Γ$ fixes $P$. This bound is sharp apart from the constant.

The mean first passage times are among the most critical characteristics of a Markov chain. In this paper, we focus on the scenario in which one or more states of a higher order ergodic Markov chain are modified to be absorbing. We prove that the resulting chain has to be absorbing. For a higher order absorbing Markov chain, we prove that the equation its fundamental tensor satisfies must be nonsingular and provide a MATLAB function {\tt fund} for solving the equation. Besides, we connect each horizontal slice of the mean first passage time tensor with a fundamental tensor obtained when one state of a higher order ergodic Markov chain is modified to be absorbing, which also leads to a tensor series representation for selected mean first passage times.

We prove a duality theorem for quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, with respect to a smooth and Kiehl partially-proper morphism. This includes an identification of the dualizing object with volume forms. The functional analysis underlying our theory does not use condensed mathematics, but rather Ind-Banach spaces, following Ben-Bassat--Kelly--Kremnizer. Nevertheless, our overall strategy is inspired by that of Clausen--Scholze in the complex-analytic setting.

We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. Using this adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $Q$-, the $S$-, the cobordism model, and the squares construction.

\emph{Magic braids} are used in leatherworking to make intricate straps and bands. They appear to be impossible to make. To determine which braids can be made with this leatherworking technique, we explore their relation to several braid groups.

Liu and Wan introduced T-adic exponential sums as a way to interpolate all character sums with character having p-power conductor. In this paper, we generalize their T-adic theory to partial T-adic exponential sums. We prove that the associated L-functions are T-adic meromorphic, and as a consequence, give a p-adic proof of rationality for all partial L-functions of characters with p-power conductor. We also give Newton-over-Hodge estimates.

We consider Schrödinger operators with dynamically defined potentials that are anti $α$-Hölder and generated by hyperbolic dynamics. We prove an anti Lipschitz estimate on functions related to the orbit of $\vec{e}_1$ under an associated projectivized cocycle. After, we apply the estimate on subshifts of finite type and maps with one expanding direction on $\mathbb{T}^d$ where $d\geq 1$. In particular, the application leads to uniformly positive Lyapunov exponents with sufficiently large coupling constants.

We study the Laplace equation with Dirichlet and Neumann boundary conditions posed on multi-arcs, i.e., collections of open arcs meeting at junction points. We begin by introducing a scale of Sobolev spaces constructed using the Sobolev spaces on open arcs as main building block and extend the definition of trace operators. We reformulate the boundary value problems using boundary integral formulations. We then establish a well-posed integral formulation for the Dirichlet problem, which can be discretized using standard numerical methods. We further investigate the singular behavior of the solution densities at branch points through numerical experiments and observe that these singularities are comparable to the corner singularities arising in polygonal domains. For the Neumann problem, we show that the associated hypersingular operator is not necessarily invertible on classical Sobolev spaces and provide numerical evidence that solutions may develop jump discontinuities at branch points.

To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce automorphic measures with complex eigenvalues for rational maps on the Riemann sphere. In particular, these measures extend the classical notions of quasi-invariant and conformal measures by allowing the respective Radon--Nikodym derivative to be complex-valued and proportional to a multiplicative cocycle. \[ j_{(s,t)}(R) = |R'|^{s} \left(\frac{|R'|}{R'}\right)^{t}, \] which plays the role of a generalized automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between geometric and dynamical properties of rational maps. We show that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. Specifically, for a weakly dissipative rational map admitting a $( -1, 1)$-unimodular measure, there exists an integer $q \ge 1$ such that the map is $q$-unstable. This result generalizes earlier instability criteria involving pseudoconformal measures and connects the presence of such measures to the failure of structural stability. Furthermore, we establish ergodic and combinatorial conditions ensuring the existence of unimodular measures or vector fields -- most notably, through bounded recurrence, bounded velocity of arguments, and relations with the Milnor--Thurston kneading theory. These criteria provide a unified framework linking automorphic measures, Lyapunov spectra, and the geometric deformation spaces of rational maps.

We prove several conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The conjectures considered here concern determinants and permanents of special matrices, such as Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries are defined by powers of differences. As tools we use row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and $LU$-factorizations. We also obtain closed-form formulas for several related integer sequences for which no such formulas were conjectured.

Expanding upon the work of arXiv:2502.08799, we provide a non-explosion criterion for branched rough differential equations (bRDEs) where we continue to allow for the drift and the rough coefficient to be unbounded in the branched setting. Moreover, by providing a characterization of ``pure area'' branched rough paths -- branched rough paths over the zero path, we provide two different constructions of bRDEs which explode in finite time, and thus demonstrating the sharpness of the criterion. Finally, by realizing a trade-off between the growth of the coefficient of the bRDE and the decay of its higher-order derivatives, we provide a new non-explosion principle for bRDEs which allows for the coefficient to grow even faster than what is provided in arXiv:2502.08799.

We study the stacky approach to cohomology in the super setting. We introduce the classic transmutation stacks (Betti, de Rham and Dolbeault) due to Simpson, which are geometric realizations of locally constant sheaves, D-modules and Higgs bundles respectively and we give new proofs for results due to Penkov on $D$-Modules and the isomorphism between de Rham cohomology and super de Rham cohomology. To do this, we will develop the theory of derived categories on superstacks establishing, amongst others, base change and recollement theorems. The goal of this paper is to demonstrate the usage of ideas and methods coming from derived algebraic geometry in the supergeometric setting as derived and super have geometric similarities, which lead to the same considerations when adapting classical notions to their respective settings.

We identify a previously unrecognised structure in the finite-temperature geometry of Blahut--Arimoto (BA) rate-distortion optimisation. The starting point is an exact partition identity. For every source density (p) and every inverse temperature $β>0$, the BA partition function $Z(x)=\int q^*(y)e^{-β|x-y|^2}dy$ satisfies $$ Z(x)=\left(\fracπβ\right)^{d/2}p(x). $$ This identity, obtained from the BA fixed-point equation, implies that the BA effective score $g_β=-\nabla\log Z$ coincides exactly with the classical Fisher score $s=-\nabla\log p$ for all temperatures. Moreover, if $v=-\nabla\log q^*$ denotes the translation mode generated by the quadratic-distortion symmetry, then its BA projection satisfies $\mathcal P v=-s$. These observations lead to the central identity $$ J(p)=\mathcal R(v):=\langle v,\mathcal G v\rangle_{L^2(q^*)}, $$ where $\mathcal G$ is the BA relaxation kernel. Thus Fisher information is exactly the Rayleigh quotient of the translation mode and is therefore a temperature-invariant spectral quantity in the BA framework. This yields a geometric interpretation of the Fisher information inequality: the inequality $$ J(X+Y)^{-1}\ge J(X)^{-1}+J(Y)^{-1} $$ becomes the parallel-combination law of a Rayleigh quotient under convolution. The entropy power inequality then follows through the standard heat-flow argument. The contribution is not a new proof of the entropy power inequality, but the identification of a hidden geometric structure: Fisher information as the spectral charge of the translation mode in BA rate-distortion geometry, with the entropy power inequality emerging as a consequence of this temperature-invariant fact.