Papers

An old problem in combinatorial group theory asks, given a group $\Gamma$ and a subset $S \subseteq \Gamma$, when does there exist an ordering $s_1, \cdots, s_k$ of the elements of $S$ such that the partial products $\prod_{i = 1}^j s_i$, $1 \leq j \leq k$, are all distinct. If such an ordering exists, we call $S$ rearrangeable. There have been many conjectures about rearrangeable subsets, the most general being that for every group, every subset not containing the identity element is rearrangeable. We prove an asymptotic version of this: For any Group $\Gamma$ and any subset $S \subseteq \Gamma$, there exists a rearrangeable set $S' \subseteq S$ such that $|S'| = |S| - o(|S|)$. To do this we build upon the work of Bucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan focusing on the following problem of independent interest in Graph Theory. If $G$ is a d-regular properly colored directed graph does there exist a rainbow path of length $d - 1$? We establish an asymptotic version of this, proving that $G$ contains a rainbow path of length $d - o(d)$. This solves two problems given by Bucić, Frederickson, et al. and proves the above result on rearrangeable subsets of groups by considering the Cayley graph of $\Gamma$ with (not necessarily generating) set $S$.

We show that the systolic area of every nonpositively curved closed surface $M$ other than the torus is at least $1$, with equality if and only if $M$ is isometric to a square flat Klein bottle. The proof focuses on the Klein-double $4\mathbb RP^2$ and exploits Weil's isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.

An Eulerian orientation of a $4$-regular undirected graph (simple or multigraph) $G=(V,E)$ with $n=|V|$ vertices is an assignment of directions to its edges such that every vertex $v \in V$ has the same indegree and outdegree. In the present article, we improve the bounds on the number of Eulerian orientations for certain classes of connected, loopless $4$-regular graphs. The previous bound is due to M. Las Vergnas (1983) and is exactly $9\cdot 2^{n-3}$, which is a sharp bound for a certain family of multigraphs with $n\geq 4$. Here, we show that the number of Eulerian orientations for all biconnected $4$-regular multigraphs is at most $2^n+2$, which is also sharp. We exhibit families of graphs that attain this maximum value. For simple graphs, we prove an upper bound of $\mathcal{O}(3^{n/2})$ in the biconnected case and $\mathcal{O}(6^{n/3})$ for the separable case. Additionally, we provide a divide-and-conquer algorithm that leverages structural properties to compute the exact number of Eulerian orientations for separable graphs without exhaustive enumeration. Finally, we analyze the effect of standard inductive construction operations, used to generate $4$-regular graphs from smaller ones, as shown by F.Bories et.al. (1983) for simple graphs and by G.Ding et.al. (2003) for multigraphs, on the number of Eulerian orientations.

We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our approach does not rely on differentiating RNP-targets, and uses instead decomposability bundles and a careful blow-up analysis. We also present decomposability bundles in a way which avoids the mention of Alberti representations and generalizes the approach of Alberti–Marchese to measures in RNP-spaces. We moreover study fragment-wise differentiability into RNP-targets, give a new ${\rm Lip}-{\rm lip}$-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces $(X,\mu)\subset\ell^2$ whose Gromov–Hausdorff tangents are Hausdorff limits of $r^{-1} (X-x)$ in $\ell^2$ as $r\to 0$.

Let $n\ge3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctan\lambda_i(D^2u)=\theta(x) \] under the critical and supercritical phase condition $\theta\ge (n-2)\pi/2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.

We record two results in the theory of vector and Banach lattices related to order adherence. First, we give a negative answer to Gao and Leung's question on whether the $uo$-adherence of sublattices must be order closed. Second, we present a self-contained counterexample showing that a Banach lattice with a weakly Fatou norm need not admit any equivalent lattice norm with the Fatou property.

Mosquito-borne diseases such as dengue, Zika, and yellow fever impose a substantial global health burden, motivating genetic control strategies that replace wild mosquito populations with disease-refractory ones. Maternal-effect dominant embryonic arrest (Medea) is a gene drive in which the offspring of a Medea-carrying mother die unless they inherit the Medea allele, producing biased inheritance capable of driving a linked refractory trait to high prevalence. We develop and analyze a continuous-time compartmental model of Medea dynamics in Aedes aegypti that tracks mosquito abundance by life stage and genotype, and that generalizes the drive mechanism to allow both imperfect Medea-killing and imperfect rescue. We characterize the biologically relevant equilibria and derive their local stability conditions, together with genotype-specific reproduction numbers and the basic reproduction number (R0) governing invasion from low frequency. Bifurcation analysis reveals bistability, and hence a release threshold that must be exceeded for Medea to take over, even though R0<1 at baseline. Imperfect killing and imperfect rescue reshape this bifurcation structure in distinct ways: killing leakage governs the invasion threshold, while rescue efficiency governs the composition of the resulting population. Sensitivity analysis identifies the fitness coefficient and killing leakage as the dominant drivers of both threshold and coverage outcomes, with mosquito demographic parameters affecting absolute abundances but not genotype proportions. Simulations of batched release programs inform efficient field deployment and show that reversing an established drive is substantially more costly than achieving forward invasion, though co-releasing both sexes accelerates reversal considerably more than it accelerates invasion.

Laza and the second author have classified all possible symplectic automorphism groups of smooth cubic fourfolds. There are $34$ such groups, and by Koike, there are $48$ families of cubic fourfolds with speficied symplectic automorphism group. For $42$ among those families, we complete the classification of all possible automorphism groups. The approach of this paper is mainly lattice theoretic, starting from known results about the period map of cubic fourfolds by Voisin, Hassett, Looijenga and Laza. We use a lattice-enumeration algorithm implemented in OSCAR.

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean–Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

Using some recent examples of Anghel in dimension $2$, we prove that for any $n \ge 3$ there exist many smooth $n$-dimensional varieties $X$ with a very ample line bundle $H$, such that there is no Ulrich bundle for $(X,H)$. Moreover, we find many new examples in dimension $2$.

Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes \(p<q<r\), we characterize the square-period case: an unsupported factor \(\Phi_{pqr}\) occurs in a tiling of \(\ZZ_{(pqr)^2}\) if and only if \(r\in\langle p,q\rangle\), and every such tile lies in a single residue class modulo \(r\). Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is \(180\); if the order has three distinct prime factors, it is \(900\). An Apéry-set construction gives examples for every triple at period \(p^2q^2r^3\).The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.

Let $\mathbf{U}$ be a quantum group associated with a symmetric Cartan datum, let $\dot{\mathbf{U}}$ be its modified form, and let $\dot{\mathbf{B}}$ be the canonical basis of $\dot{\mathbf{U}}$. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in $\dot{\mathbf{U}}$ with respect to $\dot{\mathbf{B}}$ belong to $\mathbb{N}[v,v^{-1}]$. We introduce the thickening realization, which relates $\dot{\mathbf{B}}$ to the canonical basis of the negative part of a larger quantum group $\tilde{\mathbf{U}}^-$. More precisely, it identifies the relevant structure constants in $\dot{\mathbf{U}}$ with the structure constants in $\tilde{\mathbf{U}}^-$. As consequences, we prove, for arbitrary symmetric Cartan datum, $\dot{\mathbf{B}}$ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by $\dot{\mathbf{B}}$.

Let $N=m+1$ and consider the thin ellipsoid \[ \Omega_\varepsilon=\{(y,x_N)\in\mathbb R^m\times\mathbb R:\ |y|^2+\varepsilon^{-2}x_N^2<1\}. \] We prove that in every sufficiently large dimension, for every sufficiently small $\varepsilon>0$, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity $f_\varepsilon$ for which the extremal solution in $\Omega_\varepsilon$ is an unbounded $H^1_0$ solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity $f(t)=e^t$, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.

We study sparse Mellin spectral recovery from geometric samples. Dilation covariance is shown to generate the Hankel structure directly, rather than only after reduction to a classical exponential-sum model. For complex exponents, incommensurate sampling scales remove the logarithmic aliasing that persists at a single scale. A second recovery channel is obtained from the minimal Hankel pencil: contour winding counts spectral nodes regionally, while Rouché-type bounds certify isolated nodes and unresolved clusters under noise. The associated contour margin has an explicit degeneration rate as the sampling ratio approaches one. Numerical experiments show that the auxiliary scale improves both identifiability and resolution, and that certified contour counts can remain reliable beyond the regime of accurate pointwise recovery.

Every real vector bundle of rank at least seven over $CP^3$, or every real vector bundle of rank at least eight over $CP^3\#CP^3$, admits a complete metric of nonnegative sectional curvature on its total space. The corresponding unit sphere bundles also admit nonnegative sectional curvature. These are fixed-rank results: no additional trivial summand is required, and there is no restriction on any characteristic class. The proof combines low-dimensional bundle classification with the Grove–Ziller lifting theory over $S^4$ and $CP^2$.

In the 1980s Lyubich and Freire-Lopes-Mañé proved that for any rational function of degree d \geq 2, both preimages and periodic points equidistribute to the unique measure of maximal entropy log(d). Their results provide a fundamental understanding of the dynamics of iterated rational functions, and have since been generalized to many different contexts, including classes of higher-dimensional polynomial and rational maps. In the current paper we depart from the algebraic category and aim to prove analogous statements for transcendental functions in the complex plane, which have infinite topological entropy. We introduce two different methods for constructing invariant measures in the transcendental setting, namely via embedded symbolic dynamical systems and via transfer operators associated to suitably chosen weights. In the latter case we isolate three properties of the weights -normality, tightness, and irreducibility- which together imply convergence to an invariant measure. We provide examples for each method, given by three classes of transcendental entire functions: disjoint-type maps, strongly polynomial-like maps, and a class of maps inspired by Baker's construction of multiply connected wandering domains and by Bishop's construction of Julia sets of Hausdorff dimension 1, which we call Baker-Bishop maps. For each of these classes we prove that with respect to carefully chosen weights, preimages equidistribute to an invariant mixing probability measure of infinite entropy. For Baker-Bishop maps and disjoint-type maps we also prove equidistribution of periodic points. In contrast to the rational setting, the measures we construct are not unique: by varying the weights one obtains infinitely many distinct measures.

We study the virtually generating graph of pro-$p$ groups. We show that various classes of pro-$p$ groups have connected virtually generating graph and we bound its diameter in these cases; e.g.\ compact subgroups of analytic groups over local fields and the Nottingham group.

We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the “general” setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly–El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.

We study transversely projectively flat regular foliations on compact Kähler manifolds.

We study the one-dimensional elephant random walk with two memory channels introduced by Saha [Phys.\ Rev.\ E 106, L062105 (2022)] with memory parameter $p\in(0,1)$. Maulik, Roy and Sadhukhan [arXiv:2509.10225] proved recurrence for $p\le11/16$ and transience for $p>7/8$, leaving the range $11/16<p\le7/8$ open. We close this gap by proving that $p=11/16$ is the exact recurrence–transience threshold and by determining how fast the walk escapes from the origin throughout the transient regime. For $11/16<p<7/8$, we also show that the scaling limit obtained by Maulik, Roy and Sadhukhan is almost surely nonzero. At $p=7/8$, we prove that the walk is transient with zero asymptotic velocity and escapes at the scale $n/\sqrt{\log n}$.

In this paper, we investigate the constrained capacity of discrete Fourier transform (DFT)-precoded faster-than-Nyquist (FTN) signaling over frequency-selective channels with finite-alphabet inputs. With a cyclic prefix (CP) and cyclic suffix (CS), the FTN and multipath induced intersymbol interference (ISI) is decomposed into parallel eigenchannels, whose gains are jointly determined by the folded FTN spectrum and the channel frequency response. Based on this decomposition, we derive the constrained capacity for finite-alphabet constellations and formulate a mismatched decoding achievable information rate for DFT-precoded FTN signaling without CP/CS, quantifying the finite-block rate loss caused by imperfect diagonalization. We find that even under mismatched decoding, FTN significantly improves upon Nyquist transmission.

We prove an all-allocation global target-risk theorem for independent Gaussian sources that share a scalar nuisance and an unknown contact coordinate. For a fixed immersed nuisance curve with finitely many multiple fibres and pairwise nonparallel branch tangents, squared target risk is comparable to a primary estimation floor plus a target-gap-weighted Gaussian discrimination profile. Independent localization, finite branch selection and target-class refitting give an estimator attaining this comparison across every nonnegative integer allocation. A full-box specialization has positive-definite primary Fisher information everywhere and globally identifies its target, yet models with identical derivatives of every order along a critical hypersurface have different polynomial risk exponents. We construct a finite certified estimator and quantify the numerical accuracy needed to preserve rare branch decisions and acquisition windows. A polynomial tangency family both demonstrates the geometric boundary and quantifies its repair: known auxiliary gain, source contact order and target vanishing order determine a risk law uniform through zero gain. For a shared spherical direction, a separately proved composite-testing result transfers through an unknown contact coordinate using only counted observations. These theorems distinguish the resources needed for local estimation, discrimination between parameter regions and nuisance alignment, and yield acquisition thresholds under explicit scalar-observation costs.

Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.

For a constructible sheaf $\mathcal{F}$ on a smooth scheme $X$ over a perfect field, the singular support $SS(\mathcal{F})$ and the characteristic cycle $CC(\mathcal{F})$ are defined on the cotangent bundle. In analogy with the construction of Kashiwara–Schapira via microlocalization in the transcendental setting, we construct the microlocal versions ${ SS}_\mu(\mathcal{F})$ and $CC_\mu(\mathcal{F})$ of $SS(\mathcal{F})$ and $CC(\mathcal{F})$ respectively, and prove \[ SS_\mu(\mathcal{F})= SS(\mathcal{F}),\qquad CC_\mu(\mathcal{F})= {\rm cl}CC(\mathcal{F}), \] where $\operatorname{cl}$ is the cycle class map. As a consequence of the second equality, we show that the cohomological characteristic classes are the cycle classes associated with the corresponding characteristic cycles, as conjectured in (Invent.~Math.~207: 597–695, 2017). As further applications, we give new proofs of the index formula and Bloch's conductor formula in the proper case, independently of Abe's homotopical approach via $\infty$-categories.

The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.

In this paper, we resolve the long open problem of the diffusion approximation for a generally distributed processor sharing queue with reneging. In particular, we first extend the known fluid limit for a single class processor sharing queue with reneging to the multiclass proportional processor sharing queue with reneging. We prove uniqueness of fluid model solutions and convergence of fluid scaled models to fluid model solutions. Next, we show that, under diffusion-scaling, the models are tight, and that subsequential limits can be uniquely characterized by a certain system of SDEs driven by a Gaussian sheet on $\R_+^2$ with a nontrivial covariation structure.

For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $\gamma_2(T)\le Cr$, and $\gamma_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]$ and $\ell_p$. The lower bound follows from a multilevel product principle applied to a scale-separated product of Euclidean simplices. The same construction gives the sharp convexification profile of $\ell_p^r(\ell_2^{2^{128r}})$ for $2<p\le\infty$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.

We initiate the study of nonlinear wave equations on the Witten bubble spacetime (also known as the "bubble of nothing"), which is an $\mathrm{SO}(3,1)\times \mathrm{U}(1)$-symmetric solution to the Einstein vacuum equations in $(4 + 1)$ dimensions. This spacetime was introduced by Witten to model the semiclassical instability of the Kaluza–Klein spacetime $\mathbf{R}^{3+1}\times S^1$ (which is classically stable by work of Huneau–Stingo–Wyatt (arXiv:2307.15267)). We prove a small-data global existence result without symmetry assumptions and an improved decay result for $\mathrm{U}(1)$-symmetric solutions to a class of semilinear equations satisfying a version of the null condition. Key to the proof are novel estimates for the linear wave equation, which has been studied in the physics literature by Bhawal and Viveshwara and in the mathematics literature by Bachelot (arXiv:1601.03682).

This paper proposes and studies a likelihood-ratio test of the null hypothesis that weak stochastic transitivity (WST) does not hold. This is the reverse of the formulation commonly used in the literature, where WST is set as the null hypothesis. We show that, even when the number of items grows with the number of comparisons per pair, the uniform size converges to the nominal level, with the critical value determined by a chi-bar-square distribution. We further establish that the Type II error converges uniformly to zero under a sufficient signal-strength condition. Simulation studies demonstrate good finite-sample Type I error control and show that power increases with the sample size and signal strength.

We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite $p$-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the a priori time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition $H\in C_b^\gamma$, $\gamma>p$. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional $p$-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).