If S and S' are two finite sets of Coxeter generators for a right-angled Coxeter group W, then the Coxeter systems (W,S) and (W,S') are equivalent.

Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more general study of weakly isotropic forms over henselian valued fields and over function fields in one variable.

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a turbulent cascade that regularizes the three-dimensional viscous dynamics, or rely on highly artificial interactions not transparently realized in the true Euler nonlinearity. We also treat the inviscid, unforced case and obtain singularity formation just above the energy level. We conclude with a discussion of the prospects for embedding the behavior of the dyadic model into the full Euler and Navier-Stokes equations.

We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a choice of a divisor and subgroups of $G$ at every point of the divisor. Building on our previous work, we prove that, under mild regularity conditions, the action of a Hecke operator in the deep cusp of $\mathrm{Bun}_G$ in a highly complex ramification mimics an action in a much simpler ramification. This reduces the study to a smaller number of cases which, in particular, involve divisors supported at no more than two points. We demonstrate our methods by computing various examples for $G=\mathrm{PGL}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues.

This article proposes a non-autonomous mathematical model with delay for confrontation between two countries, and examines the stability of its equilibrium state. Our criteria for stability take into account the influence of the factor of hostility between countries. For the autonomous case, the asymptotic stability is studied in a comprehensive way, and the Hopf bifurcations occurring as the delay crosses some critical values are described. For the non-autonomous model, conditions ensuring the global asymptotic stability for both the linear approximation and the nonlinear system are established. The framework of special solutions for delay differential equations is also applied.

We study Walker manifolds, that is, pseudo-Riemannian manifolds $(M^n,g)$ admitting a null parallel distribution $\D$ of rank $r\leq\frac{n}{2}$. We show that $\D$ always integrates to a $G$-Lie foliation $\F_\D$, where $G$ is the simply connected Lie group with Lie algebra equal to the structure algebra $\g_\D$ of $\D$. The transverse holonomy group of $(M,g)$ coincides with the image of the holonomy morphism $h:π_1(M)\to G$. We prove that $\mathrm{Ric}(X,\cdot)=0$ for all $X\inΓ(\D)$, and show that in dimension~$3$ the model group is always $\R$, while in dimension~$4$ with rank~$2$ the structure algebra is always abelian. A local classification distinguishes the abelian, nilpotent, and solvable cases, and a rigidity theorem shows that a minimal nilpotent Walker foliation of dimension~$4$ cannot be deformed into a non-nilpotent solvable one.

Bistable tape spring booms are used on spacecraft for their ability to self-deploy using stored strain energy. However, their uncontrolled deployment can induce mechanical shocks that are variable as a function of material properties and temperature, and may damage sensitive satellite components and disrupt attitude control. Because traditional Finite Element Analysis (FEA) struggles to accurately capture this highly nonlinear behavior, we solve the inverse problem to estimate these loads from dynamic response measurements. Previous data-driven approaches using Vector Fitting required time-consuming retesting for every specific load level due to the boom's load-dependent dynamic behavior. To overcome this limitation, we introduce a parametric data-driven framework where a parametric transfer-function model of a composite tape spring boom is developed using force and velocity measurements. The parametric Adaptive Antoulas-Anderson algorithm (p-AAA) is used to construct a single parametric (multivariate) transfer function capable of capturing the nonlinear response of the boom to load amplitude. To evaluate the proposed framework, the boom is excited at its base at 15 distinct load levels using a single-axis reference input signal. Results demonstrate that the single parametric model outperformed the best discrete non-parametric case, reducing the total relative force estimation error for the reference signal by nearly 38\%. For experimental validation, the boom is subjected to sinusoidal, triangular and square signals. The cross validation results further supported this generalized performance. Collectively, these results show that the proposed parametric model accurately reconstructs input forces from velocity measurements alone, offering a solution for onboard diagnostics in future space missions.

Given a canonically oriented Brieskorn sphere $Y=Σ(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $Σ(3,4,5),$ $Σ(2,5,7)$ and $Σ(2,3,6k+1)$ for $k\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.

We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macrì-Toda for singular threefolds. We also develop relative versions of these constructions, generalizing corresponding results in [BLM+21]. Along the way, we establish Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme. By extending previous techniques, we verify the conjecture for all Fano threefolds with canonical Gorenstein $\mathbb{Q}$-factorial singularities and a series of singular Calabi-Yau threefolds. Furthermore, we construct stability conditions on the relative Kuznetsov components associated with families of singular Fano threefolds, thereby proving a singular analogue of a conjecture of Kuznetsov-Shinder.

We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.

We consider non-Fuchsian monodromy preserving deformations on a Riemann sphere. The associated isomonodromic deformation parameters on this surface comprise the positions of the singularities, together with the Birkhoff (spectral) invariants owing to the presence of irregular singularities. Our first main result is the derivation of the Loewner evolution of these isomonodromic deformation parameters. Using this result, we construct martingale observables for Schramm-Loewner evolution (SLE(4)) processes in the presence of double poles. Geometrically, the expressions contain the pre-Schwarzian and Schwarzian of the Loewner evolution, arising from conformal covariance of the observable. Furthermore, we characterize these SLE(4) observables uniquely in terms of confluent BPZ equations of a CFT with central charge c=1.

The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] It was recently proved by Iriyeh--Shibata \cite{IS2020}, and a shorter proof was later given by Fradelizi--Hubard--Meyer--Roldán-Pensado--Zvavitch \cite{FHMRZ}. Both proofs combine ingenious equipartition arguments of algebraic-topological origin with delicate geometric estimates inspired by Meyer's argument for unconditional bodies. In this paper, we give a new proof of this inequality using a purely geometric approach, based on what we call symmetric admissible shadow systems. This is a natural extension of the new techniques developed in our proof of the three-dimensional non-symmetric Mahler conjecture \cite{CLXX-Mahler}.

In this paper, we consider weighted Dirichlet spaces $\cD_ω$, where $ω$ is a positive superharmonic weight on the unit disc $\DD$. These spaces include the standard weighted Dirichlet spaces $\cD_α$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\cD_ω$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown--Shields conjecture. We develop tools, analogous to those used in the harmonic case, that are needed to treat this problem for superharmonically weighted Dirichlet spaces $\cD_ω$. In particular, we obtain a formula for the Dirichlet integral of outer functions of Carleson--Richter--Sundberg type, estimates for the norm of the reproducing kernel of $\cD_ω$, and several properties on the capacity associated with $\cD_ω$. Using these tools, we provide a description of invariant subspaces when the measure $Δω$ is finite measure or if the $\supp(Δω)\cap \TT$ is countable, where $\TT$ denotes the unit circle. Finally, we prove that a smooth outer function $f \in \cD_α$ such that $\cZ (f) $ is "regular" is cyclic in $\cD_α$ if and only if $c_{α}(\cZ(f))= 0$.

The stationary Kuramoto mean field game models a population of phase oscillators that form synchronized Nash equilibria above a critical interaction strength. We prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave, settling a conjecture of Carmona, Cormier, and Soner. The proof decomposes the second derivative of the self-consistency map into two sign-indefinite moments of the equilibrium--a cubic moment and a gradient moment--and controls their signs through sharp shape estimates for the value function, a pointwise geometric-mean monotonicity that determines the sign of the cubic moment via a cosine-skewness inequality, and a reflection argument combined with a correlation inequality for the gradient moment.

Using techniques from the theory of von Neumann algebras, we propose a framework for addressing questions of controllability of bilinear systems on infinite dimensional Hilbert spaces. In the setup, we assume only that the drift and control terms arising in a bilinear control system are affiliated with a von Neumann algebra of finite type acting on the same Hilbert space. When the control terms satisfy basic norm bound conditions, we prove existence of time-optimal controls. In the more general setting where all operators may be unbounded, we show how the dynamical Lie algebra for the system is still well-defined and may be used to check approximate controllability of the system in question. We discuss how this approach can be applied to classical dynamical systems through the Koopman operator formalism, and investigate potential candidates for the von Neumann algebra which may guide the choice of controls. We illustrate how an affiliation relation naturally arises in both classical and quantum control systems with a few examples.

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,δ)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $δ$. Previous works introduced this concept and study when and how well one can distinguish between such a pair of symmetric distributions by observing $t$ bits. We use a simple hypergeometric smoothing approach and Hahn polynomials to obtain new upper bounds that apply across a wider range of parameters and improve previously available bounds in several regimes. In particular, prior works left open the basic question of whether there exist constants $0<c_1<c_2<1$ and a pair of $(c_1n,0)$-wise indistinguishable distributions such that the $c_2n$-wise marginals have statistical distance $Ω(1)$. One application of our new bounds is to rule this out for all $c_1,c_2$ and to show that the $c_2n$-wise marginals must in fact be exponentially close. Another application in this setting is to show that the $c_2n$-wise marginals must be super-polynomially close even if the $c_1n$-wise marginals are allowed to have statistical distance $δ$ for any $δ\leq\exp\left({-ω(\sqrt{n\log{n}})}\right)$. Our bounds also yield new results in other regimes, for example when $k$ is sublinear or when $t/n$ tends to 1.

Alt $ν$-Tamari lattices constitute a remarkable family of lattices associated with lattice paths that broadly generalize the Dyck and Tamari lattices. To systematically study the structural properties of this family, we introduce a combinatorial model that realizes the canonical join complex of alt $ν$-Tamari lattices. Serving as a universal tool, this model allows us to prove vertex decomposability, establish an explicit shelling order, and reveal the underlying homology of the canonical join complex of alt $ν$-Tamari lattices.

We study the fully generalized Grover's algorithm to find the optimal phase changes for each step of the iteration to maximize gain in probability of observation of the target, and when phase matching is required. We find that classical Grover's algorithm and phase matching remains to be optimal till the target probability gets close 1. However, as the probability of observation approaches 1, the optimal phase changes differ from $π$ and no longer observe phase matching. We provide the optimization statement to find the optimal phase changes given the current amplitude vector and the size of the set. To analyze this formula, we approach it from a numerical and analytical perspective, with the analytical perspective focusing on special cases that simplify the optimization and allow for general statements about its behavior. Finally, we provide an example of a 5 qubit system and show that for the final iteration the optimal phase changes differ from traditional Grover's algorithm and do not observe phase matching, but lead to an increase in the probability of the target.

We study the asymptotic response time tail in the M/G/n multi-server queue with heavy-tailed (regularly varying) job sizes, a setting representative of modern computing workloads. For single-server systems, tail optimization is well understood: under heavy-tailed job sizes, policies such as SRPT that strictly prioritize short jobs are strongly tail optimal, and giving any priority to large jobs is harmful. For multi-server systems, the question has been almost entirely open. This paper gives the first strongly tail-optimal scheduling policies for the M/G/n queue with heavy-tailed job sizes. Our central finding is that the multi-server case is intrinsically different from the single-server case: giving a small amount of ``sympathy'' to large jobs is essential for strong tail optimality. We establish strong (or arbitrarily close to strong) tail optimality across the full stability region, both with and without knowledge of job sizes.

Let $\grad$, $\curl$, and $\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\dv \grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\grad \grad^* + \curl^* \curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial. Furthermore, we determine the Helmholtzian spectrum for certain graph products and characterize Helmholtzian integral graphs, as well as derive bounds for the smallest Helmholtzian eigenvalue. Meanwhile, we pose some open problems for future research.

We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical Lévy processes in Hilbert spaces. The coefficients are assumed to satisfy global Lipschitz conditions, and no moment assumptions are imposed on the driving noise. The principal difficulty arises from the fact that cylindrical Lévy processes exist solely in a generalised sense and typically admit no semimartingale or Lévy-Itô decomposition, which precludes the use of classical existence methods. To overcome these obstacles, we develop a pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times and a fixed-point formulation of the mild solution operator. The resulting approach avoids stochastic calculus techniques relying on semimartingale decompositions and provides a robust and flexible framework for treating multiplicative cylindrical Lévy noise in infinite-dimensional systems.

The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.

In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.

We consider sums of the form $$F_χ(α,β;θ) := \sum_{αp<n\leβp}χ(n)e(nθ),$$ where $χ$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \sqrt p \log \log p \ll \max_{0 \le θ< 1}{\left|F_χ(α,β;θ)\right|} \ll \sqrt{p}\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $χ$, and we consider incomplete mixed character sums.

In statistics permutations typically arise in the context of rank plots for two-dimensional data. Such plots can also be interpreted as discrete copulas. In discrete mathematics, typically in the context of the description of large (non-random) objects, two-dimensional copulas appear as limits of permutations and are then known as permutons if the topology refers to the convergence of pattern frequencies. We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit tests, for two-sample tests, and for tests of symmetry. This includes a suitable variant of the bootstrap for obtaining critical values. Pattern-based procedures are also of interest in a parametric context. We consider two examples, the Farlie-Gumbel-Morgenstern class and a family of delay copulas. We discuss implementation aspects of the resulting procedures and we provide a simulation study that supplements the theoretical results in the nonparametric case.

We provide a Lax-Oleinik-type representation formula for solutions to nonautonomous Hamilton-Jacobi equations posed on networks with a rather general geometry. The networks may possess countably many arcs and allow for the presence of loops. We consider Hamiltonians that are convex and superlinear in the momentum variable, and satisfy a Lipschitz-type condition in the time variable. The representation formula is constructed via an overall Lagrangian that accounts for both the arc-specific dynamics and vertex-based constraints, called flux limiters, which ensure the well-posedness of the problem. We prove that the corresponding action functional admits Lipschitz continuous minimizers without needing to rule out the Zeno phenomenon. Furthermore, we demonstrate that the formula yields the unique solution to the problem even when the flux limiters exceed standard upper bounds.

We develop the theory of p-Lie algebras of finite Morley rank. In particular, we obtain a quite complete characterization in the soluble case

Geometric phases are foundational to isolated quantum systems, yet their thermodynamic role in open systems remains unrevealed Developing a dissipative adiabatic perturbation expansion, we discover a Berry-phase-induced chiral work difference that survives decoherence. This chirality evolves from an interferometric thermodynamic Aharonov-Bohm effect in the unitary regime to a fringe-free signal in the dissipative regime. We illustrate this framework in a two-level system and assess its experimental feasibility. Our findings clarify the role of quantum geometry in the geometric formulation of thermodynamics.

We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\log n)$ bound holds at scale $2γ$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.