Papers

We prove that any two rank-one systems are sub-$L^1$ orbit equivalent, fully clarifying quantitative orbit equivalence for a generic class of transformations. As a corollary, we establish unconditional optimality of Belinskaya's theorem.

A well-known phenomenon in statistical analyses of populations of phylogenetic trees in the Billera-Holmes-Vogtmann space is that the topology of the Fréchet mean tree can contain multifurcations (i.e., internal nodes with more than two children), which raises the practical question of whether this reflects a population-level branching structure (hard polytomy) or merely sampling variability in the data (soft polytomy). This is an instance of the more general phenomenon of "stickiness" in non-Euclidean statistics, whereby the sample Fréchet mean in certain non-positively curved stratified spaces becomes permanently trapped in a lower-dimensional stratum. In this work, we identify a particular multidimensional random walk embedded within the Fréchet mean process, and we show that the time at which stickiness occurs is determined by the largest last-passage time above zero of the coordinates of this random walk. Using this representation, we develop a fully nonparametric procedure for estimating the probability that trifurcations in a sample Fréchet mean tree will bifurcate at some future time if more observations are collected. Lastly, we apply our methodology to a problem in phylogenetics where we consider whether an observed trifurcation in the species tree of primates, glires, and tree shrews is genuinely trifurcated at the population level.

Let $C\subset \PP^2$ be a reduced irreducible complex plane curve with complement $U$. In this paper we prove that $\chi(U)\geq \nu(C)$, where $\nu(C)$ is the freeness defect of $C$ defined in \cite{Dim24}. When $C$ is a rational curve with unibranched singularities, this inequality proves the conjecture of Dimca and Sticlaru, that a rational unibranched plane curve is either free or nearly free.

We develop a unified spectral and complex analytic theory of the Lame equation through its canonical Heun realization. Starting with ellipsoidal separation, we derive essential elements, including the Lame-preserving anharmonic parameter action and finite accessory spectral polynomials. We construct the global Heun monodromy representation on the four-punctured sphere, connected to the (2,2,2,2) pillowcase orbifold and the SL(2,C) character surface. The accessory parameter is framed within a Riemann-Hilbert map that links spectral values to monodromy conditions. Utilizing Seifert van Kampen theory reveals the fundamental group of the punctured sphere, and the Nielsen Schreier rank formula applies to monodromy coverings. The vanishing of higher homotopy groups for unbranched covers and the emergence of a Riemann-Hurwitz genus formula post-compactification are noted. Colored Schreier graphs are employed for a visual calculus of the covering data, connecting elliptic geometry and topology.

Gödel-Löb provability logic $\GL$ is a propositional modal system that on one hand enjoys completeness with respect to conversely well-founded Kripke frames and on the other hand captures all modal principles about $\PA$-provability that are provable in $\PA$ itself. In the present paper we carry out an initial investigation into the question of what the infinitary counterpart of $\GL$ is. We develop a non-well-founded deep inference proof system $\dgla$ for the modal language with at most countably infinite conjunctions and disjunctions. We show that the calculus is sound and complete for well-founded transitive Kripke frames. Using Kripke-Platek set theory we develop an interpretation of the infinitary modal language in terms of infinitary provability over admissible sets. Then we show that a natural Hilber-style variant of infinitary $\GL$ is sound for this interpretation. We leave open, however, the question if $\dgla$ proves any additional theorems in comparison with the Hilbert-style calculus. Nevertheless, under certain conditions we do show that the infinitary provability logic arising from certain admissible sets lies between the set of theorems of the Hilbert-style calculus and the non-well-founded deep inference system.

We show that a sequence of simplicial polytopes approximating a smooth convex body has primitive Betti numbers diverging from the upper bound given by the $g$-theorem. This complements a result of Adiprasito-Nevo-Samper, who proved the corresponding divergence from the lower bound. The proof uses a new lower bound for the size of the shadow of a set of monomials.

We study the stochastic Landau–Lifshitz–Baryakhtar equation with multiplicative noise and homogeneous Neumann boundary conditions on smooth bounded domains in dimensions $d\le 3$. Relying on stochastic maximal regularity results, we establish global well-posedness in a variational strong, intermediate, and weak setting, as well as for initial data in the scaling-critical Besov spaces $B^{d/q-1}_{q,p}$ for every $q\in[2,\infty)$ and $p\in(2,\infty)$.

We establish an optimal leafwise width estimate for compact, spin, foliated bands with positive leafwise scalar curvature and infinite vertical $\widehat{A}$-cowaist. This result addresses a special case of the foliated version of Gromov's band width conjecture and generalizes the scalar and mean curvature comparison theorem for spin bands due to Cecchini and Zeidler. Our approach, which also applies to spin foliations, relies on a local boundary value problem for deformed sub-Dirac operators on the Connes fibration. A key technical ingredient is a Lichnerowicz-type formula in the adiabatic limit for almost isometric foliations, which captures the sharp coefficient $\operatorname{rank} F/(\operatorname{rank} F - 1)$ associated with the integrable subbundle $F$.

A ghost-point immersed boundary method is coupled with auxiliary differential equations to impose a locally-reacting impedance condition on a wall that does not conform to a Cartesian grid, in the framework of the linearized Euler equations. The ghost-point reconstruction follows the normal-stencil approach: an elliptical least squares cloud of fluid nodes is used to build a high-order polynomial along the local wall normal, which is then extrapolated onto the ghost points by one-dimensional La grange interpolation. The impedance, modeled as a massspringdamper oscillator, is advanced in time through two auxiliary state variables per ghost point, coupled to the reconstruction at every RungeKutta stage. The method is validated against exact and semi-analytical references on a flat immersed wall, a cylindrical wall reflecting an acoustic pulse, and a cylinder scattering a harmonic monopole. Convergence orders ranging from about three on the curved wall to about four and a half on the flat wall are measured, and a cross-check against an exact harmonic reference shows that a persistent offset observed against a semi-analytical reference is a property of that ref erence rather than of the coupled solver. A spectral stability analysis of the linearized semi-discrete operator further shows that the normal-stencil reconstruction reduces, without eliminating, a known instability that occurs when the immersed wall becomes tangent to the grid.

A Ševera bicomplex is a bicomplex $(M,d_{0},d_{1})$ whose differentials $d_{i}$ are Grothendieck differential operators of order $\leq i$ with respect to a given $A$-module structure on $M$, where $A$ is a graded commutative algebra. One also requires that $(M,d_{0})$ admits a contracting homotopy $K$ that is a differential operator of the same order as $d_{0}$. Under suitable assumptions, the differential $d_{2}$ of the second page of the spectral sequence associated with $(M,d_{0},d_{1})$ is a Batalin-Vilkovisky operator on $E_{2}$, so that when $E_{2}$ is a free rank 1 $A$-module, the graded commutative algebra structure of $A$ is enhanced to a Gerstenhaber algebra structure, and the choice of a $d_{2}$-closed basis element for $E_{2}$ further enhances this to a Batalin-Vilkovisky algebra structure. The prototypical example of this construction is Ševera's description of the Batalin–Vilkovisky algebra structure on the algebra of smooth functions of an odd symplectic manifold. Throughout the whole article we look at Batalin-Vilkovisky algebras through the lenses of Cartan calculus. Indications of a generalization to derived Cartan calculus are briefly discussed in the concluding section.

In the planted clique problem, one observes either an Erdős–Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = \Theta(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = \lambda\sqrt{n}$ planted in a random location with $\lambda > 1 / \sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande–Montanari (2015) that succeeds when $\lambda > 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $\lambda$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{\omega + o(1)}$ for $\omega$ the constant of square matrix multiplication and succeeds when $\lambda > 0.3320$; under the folklore conjecture that $\omega = 2$, this runs in the nearly-linear time of the algorithm of Deshpande–Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon–Krivelevich–Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $\lambda$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $\lambda$.

We provide a clearer presentation of the Chair44 monotile, a new strongly aperiodic monotile in three-dimensional Euclidean space $E^3$.

We construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo constructs a map that is symmetric with respect to a representation of the Lie group formed by the $(m - 1)$-torus; we replace this group with $\mathrm{SO}(3)$. In our examples, the osculating determinant is constant.

We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with parametrized loads. Under standard resolution, oversampling, and discrete-stability conditions, we establish, to our knowledge, the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization. The estimate is two-sided relative to the fine-grid solution and includes a computable bound for the corrector-localization error. In addition, we design a novel family-aware adaptive algorithm using additive Schwarz corrections to construct a shared regional enrichment space for Helmholtz problems with multiple loads. Energy-weighted proper orthogonal decomposition compresses this space. We establish stability and quasi-optimality of the enriched formulation, with bounds independent of the enrichment space. Numerical experiments demonstrate the effectiveness of ALOD for multiple loads and large wavenumbers.

We prove new bounds for Tverberg's theorem with tolerance. We show that $N = rt+\Theta_{d,r}(t^{1/2-1/(2d)})$, where $N$ is the smallest number such that any set of $N$ points in $\mathbb{R}^d$ has a partition into $r$ parts such that the convex hulls of the parts intersect even if we remove any $t$ of the points. We extend Tverberg's theorem with tolerance to families of hyperplanes in $\mathbb{R}^d$, and show that for any set of $rt + O_{d,r}(t^{1/2-1/(2d)}\sqrt{\log (t+1)})$ hyperplanes in $\mathbb{R}^d$ there exists a partition of them into $r$ parts such that the regression hulls of the parts intersect even if any $t$ hyperplanes are removed. Our bounds follow from establishing a connection between Tverberg-type results and discrepancy theory.

The purpose of this paper is to prove that every finite simple group of type $\mathsf B$ or $\mathsf C$ satisfies the inductive blockwise Alperin weight condition at every prime $\ell$ dividing its order. For ${\rm PSp}_{2n}(q)$, where $q$ is odd and $n\geq3$, a permutation lattice argument based on Conlon's induction theorem removes the unitriangularity assumption from earlier results on the inductive condition at odd nondefining primes. At $\ell=2$ in odd defining characteristic, errors and omissions in the underlying classification of radical $2$-subgroups affect an earlier parametrisation of weights for arbitrary blocks. We verify the required weight parametrisation for principal blocks and use Jordan reduction to establish the inductive condition for all blocks. At odd $\ell$ in even defining characteristic and rank at least four, we prove the inductive condition using generic weights and Jordan reduction, without assuming unitriangularity. For $\Omega_{2n+1}(q)$, where $q$ is odd and $n\geq3$, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At $\ell=2$, we prove the stabiliser and extension property for Brauer characters of $\operatorname{Spin}_{2n+1}(q)$ that was assumed in earlier work on the inductive condition. We also give proofs of the inductive condition for the remaining sporadic groups $J_4$, $Fi'_{24}$, the Baby Monster and the Monster, without making a priority claim. Together with the previously established cases, these results imply that the blockwise Alperin weight conjecture holds at $\ell$ for every finite group each of whose nonabelian simple sections of order divisible by $\ell$ is of type $\mathsf B$, of type $\mathsf C$, or sporadic.

Let $K=\mathbb{F}_q(\theta)$, and let $\mathcal{Z}$ be the $K$-algebra generated by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$. We prove that $\mathcal{Z}$ is a polynomial algebra and construct an explicit polynomial basis of $\mathcal{Z}$ over $K$. We also determine the transcendence degree of $K[\zeta_A(\mathfrak{s}): \operatorname{wt}(\mathfrak{s}) \leqslant w]$ over $K$ for every integer $w \geqslant 1$, generalizing a result of Ngo Dac–Nguyen Chu–Pham. The proof uses Chang's grading theorem, the linear basis theorem proved independently by Chang–Chen–Mishiba and Im–Kim–Le–Ngo Dac–Pham, and the carry relations of Im–Kim–Ngo Dac. The key step is to show that suitable derivations obtained from deconcatenation and linear functionals descend through the carry relations to $\mathcal{Z}$.

A vanishing sequence $V_n$ of divergence-free matrix fields is one that is bounded in $L^1$ and is carried by open sets $A_n$ of vanishing volume. Recently established, Bouchitté's vanishing mass conjecture says that the directions such a sequence can carry are rigidly constrained: their limiting distribution must be a superposition of microstructures whose barycenters are singular matrices. We prove the converse in the non-symmetric setting: every such superposition is attained by a vanishing sequence. In fact, we are able to construct divergence-free fields $V_n$ $supported$ on $A_n$, whose relative boundary is a smooth compact manifold.

We prove that every smooth immersed disk in Euclidean three-space with nonzero constant mean curvature, regular up to the boundary and mapping its boundary diffeomorphically onto a circle, is an embedded spherical cap.

We develop and analyze angular maps as a numerical tool for visualizing and detecting rotational dynamics in nonlinear, nonautonomous dynamical systems in discrete and continuous time. An angular map assigns to each initial point the angular spectrum of the variational equation along the corresponding trajectory. This spectrum describes the long-time average rotation of subspaces transported by the linearized dynamics, measured by maximal principal angles. Building on the theory of angular spectra, we develop efficient numerical algorithms based on forward and backward subspace iteration. We prove that these algorithms asymptotically provide angular spectral values for subspaces that are dominant in either forward or backward time. Applications to Hénon maps, a planar flow, and the Lorenz system illustrate how angular maps reveal rotational features across phase space. For continuous-time systems, we prove that suitably rescaled angular spectra of exact time-step maps converge to the continuous angular spectrum in the Hausdorff metric as the step size tends to zero. For autonomous systems, we also justify a simplified algorithm that uses successive trajectory points to approximate the angular range associated with the flow direction.

Let $\Gamma\subset\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$ be an irreducible, non-uniform lattice and define the space $X = \operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})/\Gamma$. Let $U$ be the standard horospherical subgroup in $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$. We show that for every $\delta>0$ and $x\in X$ with dense $U$-orbit, the $U$-orbit evaluated at almost squares, $\{(u_{(n^{2-\delta},m^{2-\delta})}~:~n,m\in\mathbb{N}\}\cdot x$, is dense in $X$. The main idea of the proof is to shadow large periodic $U$-orbits and reduce the statement to a density statement within the periodic $U$-orbit, i.e. a statement for the $2$-torus $\mathbb{T}^2$. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic $U$-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in $\operatorname{SL}_2(\mathbb{R})$.

The result of Greenbaum, Pták, and Strakoš that for a given set of eigenvalues, any convergence curve is possible [SIMAX 1996] and the subsequent parameterization of such matrix-right-hand side pairs $(A,\mathbf{b})$ of Arioli, Pták, and Strakoš [BIT 1998] demonstrated that the behavior of the \gmres could not be completely characterized by the eigenvalues of $A$ alone. In this paper, we consider how to use this theory to understand the admissible and attainable \gmres behavior for matrices with constrained structure, focussing on non-Hermitian (non-symmetric) tridiagonal Toeplitz matrices. We show that Toeptliz structure necessarily constrains the how the theory from these papers can manifest but that a continuum of admissible behaviors is still attainable.

The growing demand for real-time, data-driven decision-making in complex and dynamic systems is placing increasing pressure on Operational Research (OR) methodologies. Sequential decision-making under uncertainty has long been studied in OR through Markov decision processes, dynamic programming, and their approximations, and reinforcement learning (RL) provides simulation-based methods for learning decision policies in such problems. Recent research shows an increasing interest in integrating RL with OR to address dynamic decision-making problems and enhance heuristic and exact methods for combinatorial optimization. The overarching goal across these efforts is to improve solution quality, computational efficiency, and robustness. Given the diversity of integration approaches and application settings, there is a clear need for a systematic and technically detailed review of how RL empowers OR methods. To address this gap, this paper presents a structured review of two methodological roles of RL in OR. The first concerns sequential decision problems in dynamic environments, where states evolve and decisions are repeated over time. The second concerns static combinatorial optimization problems, where inputs are fixed before the solution process begins and RL either constructs solutions or supports heuristic and exact algorithms. Digital twins provide an optional deployment and evaluation layer for training, testing, and validating these methods. We critically synthesize recent advances across these roles and uses, highlighting their advantages, implementation requirements, limitations, and challenges. Finally, we consolidate guidance on when the reviewed approaches may be considered and outline directions for future research on the methodological and practical integration of RL and OR.

In response to a conjecture of Petr and Turek, we prove that every largest intersecting temperate collection of sets $\mathcal{A}\subseteq [2k]^{(k)}\cup [2k]^{(k+1)}\cup [2k]^{(k+2)}$ includes all $k$-sets containing some given $a\in[2k]$, all $(k+1)$-sets, and all $(k+2)$-sets not containing $a$.

We study sticky Brownian motion on $\mathbb{R}$ with a set of sticky points $A$. In contrast to the classical case of a single sticky point, the set $A$ may have a more complicated structure and, in particular, may have accumulation points. The stickiness at the points of $A$ is described by a locally finite purely atomic measure $\nu$. We extend the classical time-change approach for sticky Brownian motion to this setting and use it to construct a weak solution of the corresponding stochastic differential equation with a local-time condition. We then prove uniqueness in law of the solution.

For each sufficiently large $N$, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group $C_N$ of rotations through multiples of $2\pi/N$. For each such $N$, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.

We show a large deviation principle for the weighted spectral measure of the Gaussian beta ensemble in the high temperature regime, when $n\beta_n\to 2c \in [0,\infty)$. The resulting rate function has a novel and interesting structure: it is finite only for discrete measures and moreover, in the very high-temperature regime (when $c=0$), additional terms may arise that penalize each support point beyond the first.

We prove that a divisor graph on an interval of integers is a planar graph whenever the interval $[n,m]$ satisfies $m\leq 7n$, and that 7 is the greatest such real number for which we can guarantee that the graph is planar. We prove these graphs are 3-colorable.

We report the first calculation of the lowest-known energy configuration for $N=100\,000$ classical point charges confined to a disk and interacting via the $1/r$ Coulomb potential - a system size never before achieved for the Thomson problem in a disk, more than doubling the previous record of $N=40\,886$ reported by Amore and Zarate. An adaptive defect-targeting subdomain optimization strategy yields an approximately linear growth of the wall-clock time per cycle with $N$; for $N=100\,000$ a single $24$-core CPU workstation without GPU acceleration required $\approx31$ hours. The energy $E_{\min}=7.80466624157\times10^{9}$ deviates from the standard empirical asymptotic expansion by only $2.525\times10^{-7}$. We demonstrate that this remarkable agreement provides a stringent test of the rigorous next-order asymptotic theory for Riesz interactions, while the residual deviation is consistent with an $O(N^{4/3})$ boundary-layer correction whose absence from the standard fitting basis biases the empirical $N^{3/2}$ coefficient away from its exact crystalline value, thereby tying the deviation to the two-dimensional crystallization conjecture. Bond-orientational order analysis reveals a polycrystalline bulk threaded by radial grain boundaries, establishing a new benchmark for two-dimensional Coulomb systems.

We prove the Brendle isoperimetric inequality for varifolds of finite mass and finite total first variation, allowing an arbitrary singular part. The distributional mean curvature is assumed only integrable. The result applies to integral varifolds and, more generally, to rectifiable varifolds with a positive lower density bound and weight measure concentrated on countably many $C^2$ submanifolds. The proof uses optimal transport and a locality theorem for the tangential Laplacian of an ambient convex function. The isoperimetric constant is sharp in codimensions one and two.