Papers

In this paper we use ideas introduced earlier by Moody, Long, Long-Paton, and Bigelow to prove the theorem of the title, that the Burau representation of the classical braid group is faithful for n = 4. An immediate corollary is that the Jones representation of the braid group is also faithful for n = 4.

We study the integrability problem of multisymplectic structures, by identifying them as $G$-structures. Applying the theory of Spencer cohomology, we give conditions on a multisymplectic form for it to admit a chart in which it has constant coefficients. This general study allows for a rough classification of multisymplectic structures of constant linear type, depending on the natural action of the stabilizer group. The theory is illustrated by providing a scheme for proving a Darboux theorem, which is exemplified with several relevant cases. We also build linear types of multisymplectic forms $\varpi_j$ whose flatness strictly requires a condition of order $j$. Finally, the corresponding Lie algebras are computed in the case of field theories.

In this article, we show that the ternary Cantor set is not the Julia set of a rational map.

We determine the exact maximal reflected zero region that forces centered binomial samples of a balanced entire function to have all zeros on the unit circle. In degree $d\ge2$, this region is \[ \Omega_d=\left\{a+ib:\ a^2-\frac{b^2}{d-1}\le\frac d4\right\}. \] The finite theorem is sharp already for a single reflected zero pair, and a phase-preserving canonical-product approximation extends it to balanced entire functions of order at most one. De Bruijn strip contraction and projective Hermite–Kakeya–Obreschkoff theory then give the exact common-zero obstruction, simplicity, strict interlacing of consecutive derivative samples, and a monotone real-pencil root flow. As an application, we prove the derivative-period-polynomial unit-circle theorem for completed $L$-functions of primitive holomorphic newforms, in every derivative order and for arbitrary level and nebentypus. After the standard normalization, every zero of \[ \sum_{j=0}^{k-2}\binom{k-2}{j}\Lambda^{(m)}(f,j+1)z^j \] lies on the unit circle for every weight $k\ge4$, level, nebentypus, and derivative order $m\ge0$. In particular, this proves the full-polynomial unit-circle conjecture of Diamantis and Rolen in its original level-one setting and extends it to arbitrary level and nebentypus. The same source-side theorem also gives simplicity, strict interlacing, and, for each fixed derivative order, conductor-uniform quantitative localization in the weight aspect.

We prove a color-degree criterion for DP-coloring of finite uncrowded hypergraphs of bounded rank. For a $q$-fold cover with edge sizes between $2$ and $r+1$, let $D_\ell$ be the maximum number of forbidden colorings on $(\ell+1)$-edges prescribing a fixed color at a fixed vertex. If $q$ is sufficiently large in terms of $r$, then the cover admits a coloring whenever \[ \sum_{\ell=1}^r\frac{D_\ell}{q^\ell}\leq\log q-20\log\log q. \] For uncrowded $(r+1)$-uniform hypergraphs of maximum degree $\Delta$, this gives a DP-chromatic number at most $(1+O_r(\log\log\Delta/\log\Delta))(r\Delta/\log\Delta)^{1/r}$.

What is the maximum number of points one can place in an $n \times n$ grid such that every Euclidean line contains at most $k$ points? For $k = 2$, this is the notorious no-three-in-line problem of Dudeney. In this paper, we resolve this problem for all other $k$ (and sufficiently large $n$). Namely, for $k \geqslant 3$ and sufficiently large $n$, we show that this maximum is exactly $kn$. To prove this, our key observation is that in the regime $k \geqslant 3$, the problem is dominated in a certain statistical sense by the influence of a small number of "heavy" lines with many grid points. We apply a result of Ehard-Glock-Joos on pseudorandom hypergraph matchings to construct a set of size $kn - o(n)$ with at most $k$ points on each heavy line, and then a crude deletion argument yields a no-$(k+1)$-in-line set of nearly the same size. Finally, we use a randomised switching procedure to complete the construction (building upon ideas of Simkin and Luria). Using similar ideas, we also address the no-four-on-a-circle problem of Erdős and Purdy. Namely, we prove the existence of a set of $2n - o(n)$ points in the $n \times n$ grid such that no four of these points lie on a circle or a line, improving on the previous construction of size $n - o(n)$ due to Dong and Xu.

We investigate elastic binary collisions of relativistic spinning particles in special relativity. The spin of each particle is represented by an antisymmetric second order tensor. Assuming the conservation of total four momentum and total spin tensor, together with the mass shell and spin constraints, we formulate the collision problem in a fully Lorentz covariant setting. We show that the relativistic collision problem admits a simple geometric formulation, reducing the original system of conservation laws to the solution of a quadratic equation on a circle. This reduction yields a complete classification of the postcollisional states together with explicit reconstruction formulas for all postcollisional variables from the conserved quantities. In particular, there are generically only finitely many postcollisional states, with the maximal number equal to eight.

We introduce two new models equivalent to ferromagnetic two-spin systems: a weighted subgraph model and a random cluster type model. Using these new connections, we obtain an efficient sampling algorithm and a new randomised algorithm that efficiently approximates the partition function of ferromagnetic two-spin systems in certain parameter regimes. No efficient sampling algorithms are known before in this regime, and our new estimation algorithm runs in near-quadratic time for bounded degree graphs and in polynomial time for general graphs, improving upon the previous algorithm of Guo, Liu, and Lu (2020).

In this paper, we consider the three-dimensional incompressible rotating Navier–Stokes equations and establish the sharp $L^p$ decay estimates of global solutions. We reveal that the optimal $L^p$ decay rates for $2<p<\infty$ are strictly faster than those obtained in existing results by interpolation between the $L^2$ unitary identity and $L^\infty$ dispersive estimates, although the endpoint cases were known to be sharp. Moreover, the optimality of decay rates is also proved by the lower bound estimate for a specific initial datum. The underlying mechanism lies in the anisotropic degeneracy of the oscillatory integrals arising from the Coriolis force.

We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x; y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree >2 and are ordered by the absolute value of the maximum of the coefficients.

We prove that the number of lozenge tilings of a certain triangular region $\mathcal{T}_n$ is given by the formula \[T_n=\prod_{\substack{1\leq a<b\leq 3n+2\\(a,b)\not=(n+1,2n+2)}}\left|1+\zeta^a+\zeta^b\right|^{1/3},\] where $\zeta=e^{2\pi i/(3n+3)}$. This answers a question of Ciucu and Krattenthaler, both by finding the exact formula and by explaining why $T_n$ has many prime factors. The proof reduces the lozenge tiling enumeration problem to evaluating the determinant of the bipartite adjacency matrix $M_n$ of the dual graph of $\mathcal{T}_n$, and then evaluates this determinant by diagonalising $M_n$.

We exhibit, for any $n \geq 3$, infinitely many pairwise incommensurable complete finite-volume hyperbolic $n$-manifolds, both compact and noncompact, each with nonempty totally geodesic boundary and all sharing the same perimeter-to-volume ratio.

We develop a theory of Hamiltonian group actions on cosymplectic manifolds. These odd-dimensional manifolds combine a codimension-one symplectic foliation with a distinguished Reeb direction, and arise naturally both in stable Hamiltonian geometry and as critical hypersurfaces of $b$-symplectic manifolds. Our approach is based on a compact symplectic thickening process: every cosymplectic manifold $(M,\alpha,\beta)$ gives rise to a symplectic manifold $(M\times \mathbb{S}^1,\ \beta + \mathrm{d}\theta\wedge\alpha)$. We prove that Hamiltonian cosymplectic actions lift canonically to Hamiltonian symplectic actions on this symplectic manifold. This provides a systematic bridge between equivariant symplectic geometry and the cosymplectic setting. Using this bridge, we establish cosymplectic analogues of convexity theorems for torus actions, Delzant theorem for toric actions, ABBV localization, Duistermaat-Heckman formulas, and Kirwan surjectivity. The resulting formulas are not merely formal pullbacks from the symplectic case: the Reeb direction appears explicitly through the factor $\alpha$, and, in the mapping-torus case, the localization and volume formulas are governed by the modular period together with the equivariant geometry of the symplectic fiber. We also explain how these results apply to Hamiltonian geometry on the critical hypersurfaces of $b$-symplectic manifolds.

We introduce the left, right, and full center of an algebra in a monoidal 2-category and prove their Morita invariance. This categorifies Davydov's theory of centers of algebras in monoidal categories, and specializes to give a uniform, structural account of the Drinfeld center of a fusion category, the crossed braided Drinfeld center of a fusion category graded by a finite group, and the center of a central module monoidal category. Along the way, we develop a theory of 2-adjunctions between monoidal 2-categories and a base-change construction for module 2-categories.

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class $\mathcal{C}(\kappa)$ of real random variables with mean $0$, variance $1$, and fourth moment at most $\kappa$, the skewness left free, we compute the worst-case tail probability $V_1(t,\kappa)=\sup_{X\in\mathcal{C}(\kappa)}\mathbb{P}(X\geq t)$ for every threshold $t>0$ and every $\kappa\geq 1$. The answer is a four-regime map: a Cantelli tongue $b(\kappa)\le t\le c(\kappa)$ on which the two-moment bound $1/(1+t^2)$ remains tight and the kurtosis constraint is worthless; a tail regime $t\geq c(\kappa)$ with the closed form $V_1=(\kappa-1)/((t^2-1)^2+\kappa-1)$; a plateau regime, present only for $\kappa\le 3/2$, on which the worst case freezes and the value does not depend on $t$; and a central regime described exactly by an explicit algebraic system, provably admitting no closed form in nested square roots. Beyond $c(\kappa)$ the one-sided and two-sided worst cases coincide: Cantelli's improvement over Chebyshev is annihilated by fourth-moment information. The minimal degree of a sum-of-squares proof of the tight bound is $2$ on the closed tongue and $4$ everywhere else, an exact phase diagram of proof degree. Every closed-form regime carries an explicit dual certificate and an explicit extremal distribution, re-verified on parameter grids by an independent checker in exact arithmetic. The closed forms invert to exact worst-case quantiles, sharpen a median-of-means constant, and give the exact per-direction tail available to degree-4 reasoning under certifiable kurtosis. We found the map through an AI-guided search around the certifying pipeline, LemmaForge, which is validated on classical benchmarks, independently reproduces the symmetric-slice bound of Zelen (1954), and recovers the $2\sqrt{3}-3$ constant of He, Zhang, and Zhang (2010) at $t=0$.

The development of Group Theory in Mathematics as well as its impact in Physics has been spectacular during the twentieth century, and more particularly these last fifty years. If its contribution to Particle Physics deserves a special consideration, its usefulness cannot be neglected in other domains like Statistical Physics for example, and might also be of interest in Theoretical Biology. Some of these aspects will be examined.

We study minimax goodness-of-fit testing for uniformity from $n$ multinomial observations over $N$ categories against $\ell_p$ departures of size $\epsilon_n$. Writing $u_n:=\epsilon_n^2 n\,N^{3/2-2/p}/\sqrt{2}$ for the associated signal-to-noise ratio, we focus on the intermediate regime $N=o(n^2)$ with $u_n\to u^*\in(0,\infty)$, in which the minimax risk converges to a nontrivial constant. In the Poissonized version of the problem this constant equals $2\Phi(-u^*/2)$ \cite{Kipnis2025minimax}, yielding an upper bound on the multinomial minimax risk. Here we prove the matching lower bound. The key step is a conditional central limit theorem for weighted sums under a Poisson mixture prior, conditioned on the total count. Together with the upper bound in \cite{Kipnis2025minimax}, this gives an exact sharp-constant characterization of the multinomial minimax risk in the intermediate regime.

A new transform-based technique that generalizes the unified transform method is developed for bounded doubly connected domains as a novel way to numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian. This work builds on the transform methods for multiply connected circular domains developed by Crowdy (2015, IMA J., 80, 1902-1931) and the methods for simply connected bounded domains developed by Hulse et al. (2025, Proc. A, 481 (2319)). The Szegő kernel of the annulus and a corresponding transformation law is pivotal in the derivation of this new technique. The modified Schwarz problem and a boundary value problem for a point vortex inside of a doubly connected domain are implemented in detail to demonstrate the effectiveness of this new method.

We investigate random freeness of semigroups in the solvable non-virtually nilpotent setting. We focus on a correlated affine model, namely semigroups generated by the two components of a finitely supported random walk $L_n=(L_{n,1},L_{n,2})$ on $\operatorname{Aff}(K)^2$ whose two components share a common linear part. In this model, we show that the long-term behavior of freeness is completely governed by an abelian shadow, namely the projected random walk on the multiplicative group $A\subset K^\times$ generated by the common multipliers. If this walk is transient, then {the semigroup} $\langle L_{n,1},L_{n,2}\rangle_+$ is eventually free almost surely. If it is recurrent, then freeness does not stabilize: the set of non-free times is almost surely infinite, yet has almost sure density zero. Moreover, the obstructions to freeness depends solely on the common linear part and admits an explicit arithmetic description in terms of roots of Littlewood polynomials. The proof combines a local contraction theorem for affine random walks over arbitrary local fields, developed in the appendix and related to the theory of critical affine random walks, with a ping-pong argument played at a time-dependent place.

We generalize Gessel's Formula for the number of Latin rectangles to partial Latin rectangles and non-attacking rook placements in a tridimensional chessboard. We also derive explicit short formulas for the generating series of the numbers of non-attacking rook placements on a chessboard with $2$ or $3$ levels. These series also count partial Latin rectangles with $2$ or $3$ rows. The results are obtained following methods developed by MacMahon and Gessel for counting Latin squares and Latin rectangles, by means of scalar products of multisymmetric functions.

We prove that a discrete effective divisor on the open unit disk $\mathbb{D}$ is the zero divisor of a holomorphic function on $\mathbb{D}$ with integer Taylor coefficients if and only if it is invariant under complex conjugation. The construction uses a one-parameter deformation of the Weierstrass elementary factors in which each modified factor of order $n$ leaves all Taylor coefficients of degree $\leq n$ unchanged while shifting the coefficient of degree $n+1$ by a controlled affine amount. These modified factors act as elementary jet-correction operators: the triangular structure of the coefficient map permits an inductive rounding scheme compatible with canonical-product convergence. As a consequence, every holomorphic function on $\mathbb{D}$ differs from one with Gaussian-integer Taylor coefficients by multiplication by a nowhere-vanishing holomorphic factor.

Over an arbitrary field $\mathbb{F}$, we construct $n+1$ maximal tori $T_1,\dots,T_{n+1}$ in $\operatorname{PGL}_n(\mathbb{F})$ so that the product $T_1\dots T_{n+1}$ is almost the whole $\operatorname{PGL}_n(\mathbb{F})$ and every $g\in T_1\dots T_{n+1}$ can be expressed uniquely as $g=t_1\dots t_{n+1}$ where $t_i\in T_i$. The construction is optimal, as the number of tori with this property attains a general upper bound for connected reductive groups over an algebraically closed field, as well as over finite fields. We also show that $n+2$ suitably chosen maximal tori $T_1,\dots,T_{n+2}$ are enough to cover the whole group, i.e. $\operatorname{PGL}_n(\mathbb{F})=T_1\dots T_{n+2}$, provided $|\mathbb{F}|>n^2$. This is optimal over a finite field and is conjecturally optimal over algebraically closed fields, i.e. the number of such tori is as small as possible.

Cosymplectic manifolds provide a natural geometric framework for codimension-one symplectic foliations and arise throughout geometry and mathematical physics. We develop an equivariant cohomological theory for cosymplectic manifolds, studying the role of symmetry in their topology and their connections to Poisson geometry. Reinterpreting the obstruction theory of Guillemin, Miranda, and Pires, we introduce equivariant obstruction classes for group actions preserving the foliation and characterize the existence of invariant cosymplectic structures. We further show that the vanishing of the first obstruction class is equivalent to equivariant unimodularity of the associated Poisson structure, extending a classical criterion to the equivariant setting via Ginzburg's framework for equivariant Poisson cohomology. For compact cosymplectic manifolds fibering over $\mathbb{S}^1$, we construct equivariant versions of de Rham, foliated, and Poisson cohomologies, establishing formality results in each case via an equivariant Wang sequence. The key input is Kirwan's formality theorem for Hamiltonian actions on the symplectic fiber, which, through the equivariant Wang sequence, yields a complete and computable description of all three equivariant cohomology theories in terms of the monodromy action on the fiber.

We generalize to arbitrary dimensions an example originally introduced by Besicovitch, obtaining for every $d \geq 1$ a purely $d$-unrectifiable set $E \subset \mathbb{R}^{d+1}$ such that $\Theta_{\ast}^{d}(E, x) = \liminf_{r \to 0} \mathcal{H}^{d}(E \cap B_{r}(x))/(2r)^{d} = 1/2$ for $\mathcal{H}^{d}$-almost every point $x \in E$. This establishes the lower bound $1/2$ for the minimal value $\sigma$ such that, if $\Theta_{\ast}^{d}(E, x) > \sigma$ for $\mathcal{H}^{d}$-almost all $x \in E$, then $E$ is $d$-rectifiable. This threshold was conjectured to be exactly $1/2$ by Besicovitch.

Diabetic retinopathy (DR) is a local retinal lesion process and a visible manifestation of systemic microvascular injury. Modern retinal AI can grade images accurately, but often leaves unanswered how local lesion evidence, retinal vascular structure, and systemic disease pathways are connected. This paper introduces Causal-RetiGraph, a compact biomedical informatics framework that links retinal graph phenotypes with NHANES-anchored pathway modelling. The retinal-image fold constructs an interpretable $X1234$ phenotype from vessel maps, lesion evidence, image embeddings, and AutoMorph biomarkers through spatial $X_{12}$ and Jacobian $X_{34}$ branches. The NHANES fold models systemic exposures, covariates, a same-subject retinal mediator family $R^*$, and downstream outcome families. $X1234$ is used for retinal support and pathway prioritisation, while $R^*$ is used for participant-level pathway summaries. On the retinal fold, $X1234$ achieves 0.9055 binary DR accuracy and 0.9711 AUROC, with graded DR QWK of 0.8312. The results show that lesion and biomarker streams improve contextual retinal representation under scarce and imbalanced data. In NHANES, HbA1c, urine albumin, pulse pressure, fasting glucose, and systolic blood pressure are the strongest binary DR anchors. Participant-level pathway analysis identifies glycaemic–renal and glycaemic–haemodynamic pathways as the clearest mediator-style signals. These results suggest that retinal graph phenotypes can help prioritise systemic pathways in DR while preserving the distinction between image-derived support and same-subject mediation.

We define and analyze hockeystick divergences and $f$-divergences for normal positive functionals on general von Neumann algebras, generalizing and unifying previous work in classical probability and finite-dimensional von Neumann algebras. All the main properties of these state distinguishability measures (including in particular monotonicity, convexity, semicontinuity, bounds, state discrimination, data processing inequality) are derived from properties of the Jordan decomposition of selfadjoint normal functionals. This is done by representing the $f$-divergences as integrals over hockeystick divergences, and their significance in quantum hypothesis testing is reviewed. The $f_0$-divergence given by the information function $f_0(t) = t \ln t$ is shown to coincide with Araki's relative entropy, extending results of Frenkel to general von Neumann algebras.

We study quantitative convergence rates for vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the flat torus. The limiting equation contains a spatially dependent diffusion coefficient a(x) >= 0, which is allowed to vanish. Under standard structural assumptions on the Hamiltonian, we first prove a pointwise convergence rate of order O(epsilon |log epsilon|). We then show that, when the error is tested against a smooth probability density, the logarithmic loss can be removed and an averaged O(epsilon) rate holds. The proof is based on the nonlinear adjoint method, weighted Hessian estimates, and entropy estimates for the adjoint density.

This work introduces ADS-PB, an extension of the Adaptive Direct Search (ADS) framework for solving constrained blackbox optimization problems. With ADS, iterates progress without relying on mesh structures or sufficient decrease conditions on the objective function value. Unlike the extreme barrier approach used in ADS, where only unrelaxable constraints are considered, the proposed method also handles quantifiable and relaxable constraints using a Progressive Barrier (PB) mechanism that exploits both constraint and objective function values. A convergence analysis of the proposed framework under mild assumptions is presented. The performance of the proposed method is assessed using sets of analytical and simulation-based constrained test problems and is compared with state-of-the-art blackbox optimization solvers, including the PB approach within the Mesh Adaptive Direct Search (MADS) framework.

We establish the first results on $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators in divergence form with bounded coefficients that depend measurably on all variables. In the case of complex coefficients, we identify the maximal open range of exponents $1<p \leq2$ through the availability of $\mathrm{L}^p$ resolvent bounds. This open range always contains the lower parabolic Sobolev conjugate of $2$ and the result is sharp in spatial dimension $n \geq 2$. For real coefficients, we prove extrapolation to the full range. Our argument relies on novel space-time off-diagonal bounds based on two complementary geometries: parabolic cubes on small scales and regions modeled after the half-order time derivative of a parabolic Bessel potential on large scales.

Workforce scheduling is an NP-hard combinatorial optimization problem requiring simultaneous satisfaction of labor regulations, coverage requirements, employee preferences and operational objectives. Existing CP formulations typically model simplified instances with 6-12 constraints at shift-level granularity and critically lack explicit support for: mandatory break scheduling with midpoint placement control; acuity weighted workload equity; sub-shift temporal granularity enabling demand-driven staffing; inter-week schedule stability; and cross-midnight shift patterns common in 24-hour operations. This paper presents CP-WSP: a declarative CP-SAT framework enforcing 14 hard constraints as mathematically inviolable requirements (zero regulatory violations by construction) while optimizing 15 soft objectives through a unified weighted penalty function – all configurable via a JSON specification with no code changes required. Key contributions include: a shift-window variable decomposition enabling mandatory break scheduling with centrality control; acuity-weighted workload equity; multi-granularity temporal resolution from 30 minutes to 2 hours; inter-week schedule stability; a grid-offset preprocessing technique for cross-midnight shifts; and a reproducible 36-configuration benchmark suite for community comparison. Evaluated on INRC-II benchmarks at both hourly and shift-level granularity and on 36 synthetic configurations.