Papers

In this paper, we establish global-in-time Strichartz estimates for Klein-Gordon equations on non-trapping asymptotically hyperbolic manifolds. Our result is a positive-mass counterpart of the global Strichartz estimates of Sire, Sogge, Wang, and Zhang for shifted wave equations, and recovers the same range of admissible pairs. The main new feature is the low-frequency analysis, where the positive mass changes the spectral phase and leads to a different dispersive mechanism. Combining the resulting low-frequency estimates with microlocalized high-frequency estimates yields the full global Strichartz estimates. As applications, we obtain small-data global existence and scattering for power-type and certain derivative semilinear equations, as well as small-data global existence for a class of smooth semilinear equations without imposing a null condition.

The rank of the tensor of $3\times 3$ matrix multiplication over the field with two elements is at most $23$ by Laderman's algorithm, and Rudich and Rousseau recently proved that it is at least $22$. We prove that it equals $23$. Hence Laderman's algorithm uses the fewest multiplications among all bilinear algorithms over $\mathbb{F}_2$ and among all bilinear algorithms with integer coefficients. The proof uses the substitution method in the form developed by D'Ambrosio, Wang and Yang et al., which bounds the number of first factors of a decomposition that lie in a given subspace of the space of first factors. We sharpen these bounds for $111$ of Wang's $496$ classes of subspaces, constrain the ranks of the first factors through a $27\times 27$ flattening of the tensor, and list, up to symmetry, all sets of $22$ first factors that satisfy these constraints. An exact completion search shows that none of them is the set of first factors of a decomposition. The computation emits certificates, which are checked in the Lean 4 proof assistant. In one formal proof the largest checks are evaluated as compiled code, and in a second one the kernel evaluates every check, so that its final theorem depends only on the three standard axioms of Lean. The Lean proofs build on the formalization of Rudich and Rousseau and import their table of bounds and their reduction for block algorithms.

We represent (weighted-)selection-elliptical distributions as an affine combination of the $q$ selection variables plus an elliptical term whose direction alone is independent. This form suffices for $q+2$ fund separation via first-order stochastic dominance, inter alia relaxing Simaan's (1993) three-fund assumptions.

We study the approximation by a finite volume scheme of the stationary Gross-Pitaevskii equation with angular momentum rotation. We prove that, under standard mesh regularity assumptions, the discrete global minimizers converge to the continuous one as the mesh diameter tends to zero. Moreover, under a spectral gap assumption on the Hessian of the discrete energy, we prove that solutions to a normalized continuous gradient flow converges exponentially fast to such discrete minimizers. Additionally, we show that this previous discrete spectral gap assumption can be derived from similar hypotheses in the continuous setting. We support our theoretical results with numerical simulations, which broaden the scope of the analysis.

In this paper, we study a family of weighted Gagliardo–Nirenberg interpolation inequalities associated with Grushin–type operators on $\mathbb{R}^n \times \mathbb{R}^m$. Specifically, we examine how to control the weighted norm $\||x|^\beta u \|_{L^B}$ by balancing fractional derivatives $D_x^s u$ with weighted derivatives $|x|^b D_y^r u$. Using two–parameter scaling, spatial translations, and Littlewood–Paley decompositions, we map out exact parameter conditions and demonstrate that the scaling exponents are both necessary and optimal. For multi-parameter weighted extensions, we leverage Khintchine-type random sums to show why $Q \ge 2$ is strictly required. When $x$ and $y$ are radially symmetric, two key advantages emerge: the allowable parameter range for the weights expands, and compact embeddings of the energy space into $L^p$, $2 < p < B$, are completely restored. We establish this compactness through two complementary tools, geometric covering arguments and a generalized three–norm version of Strauss's lemma, that yields explicit pointwise weighted decay estimates. Finally, in the scale–invariant case $\beta=0$, a concentration–compactness analysis proves that the inequality's sharp constant is achieved by an optimal function, with minimizing sequences remaining precompact up to the natural three-parameter group of symmetries.

Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of $f$, we prove that the complement of its postcritical divisor is taut unless $f$ is a monomial power map or admits a nontrivial equivariant rational fibration with a polarized divisorially postcritically finite base map. We establish analogous alternatives for polarized PCF endomorphisms of rationally connected projective manifolds and, for Kobayashi hyperbolicity, on normal rationally connected projective varieties. In dimension two, the non-taut case can be described, up to iteration and rational semiconjugacy, by monomial power maps and skew products with monomial fiber maps.

We study the relation between equal-weight circle quadrature, Gauss–Chebyshev quadrature, and planar Minkowski reconstruction. The regular $N$-point grid is an $\mathbb S^1$-design of strength $N-1$; a half-step rotation of the $2M$-point grid projects to the classical $M$-point Gauss–Chebyshev rule. A common family of quadrature-error functionals gives exact identities for integration error, mixed-area error, and support-function error. For $N\ge3$, the centered Minkowski polygon $P_N$ with perimeter $2\pi$ has the exact Hausdorff distance from the unit disk $B^2$ \[ d_H(P_N,B^2)=1-\frac{\pi}{N}\cot\frac{\pi}{N} =\frac{\pi^2}{3N^2}+O(N^{-4}). \] With Steiner centering, we obtain $d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu)$ for balanced, nondegenerate probability measures, where $W_1$ uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity $W_1(\nu_N,\sigma)=\pi/(2N)$ yields a general $O(N^{-1})$ bound for the regular grid. The sharper $O(N^{-2})$ rate follows from the sparse discrepancy spectrum and the inverse multiplier $(1-k^2)^{-1}$. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.

A lazy cellular automaton $\tau:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $\tau$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.

We consider long-range percolation (LRP) on $\mathbb{Z}^d, d \ge 1$ with connection probabilities $\mathbf{p}(x, y) \approx \frac{\beta}{\|x-y\|^s}$ for $\beta>0$ and $s > \min(2d, d+2)$. Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length $n$ is of order $n^2$. To establish the upper bound, we show that a $d$-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.

We introduce a new and tractable class of non-Gaussian isotropic random fields on the sphere, which we call, for brevity, Grey Spherical Random Fields (GSRFs). The proposed framework provides a flexible extension of the classical Gaussian theory, while preserving strong isotropy and embedding Gaussian random fields as a limiting special case. The construction is based on random spherical harmonic coefficients obtained as Gaussian mixtures driven by multivariate subordinators evaluated at independent random times (with Mittag-Leffler or more general distribution). This representation allows us to derive explicit characteristic functions, mixed moments and cumulants of arbitrary order. We also investigate asymptotic high-frequency behaviour and identify general conditions under which suitably normalized spherical harmonic coefficients converge to Gaussian scale mixtures, recovering several important models, including stable, Gamma and inverse Gaussian cases, as particular examples. Proving that GSRFs emerge as scaling limits of multivariate continuous-time random walks, with heavy-tailed waiting times, provides a microscopic foundation for the proposed models. We further obtain quantitative bounds measuring the deviation from Gaussianity of the spherical harmonic coefficients. The resulting framework substantially broadens the class of analytically tractable isotropic random fields on the sphere, while preserving explicit probabilistic representations and flexible dependence structures.

In this paper, for a finite group $G$ and a positive integer $k$, we introduce the function $$\psi''_k(G)=\dfrac{\psi_k(G)}{|G|^{k+1}}, \quad \text{where} \quad \psi_k(G)=\sum_{g \in G}o(g)^k,$$ as the $k$-th generalization of the function $\psi''$, introduced by M. Tărnăuceanu. Different thresholds for cyclic, abelian, nilpotent and (super)solvable groups in terms of $k$ are obtained. It is shown that these thresholds often outperform similar thresholds derived from other parameters like $\psi'$ and $\psi''$. Moreover, considering $\psi''_k$ as a $(0,1]$-valued function, we study its injectivity, surjectivity and density.

We show that integers act invertibly on a derived scheme $S$ if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $\tau$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, \tau y)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $\tau$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher–Wenzel and Bloch–Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $\tau$: critical points of the relevant function correspond to $\tau$-twisted idempotents, and its Hessian is governed by $\tau$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.

For $0<s\leq 1/2$ and $d\geq 1$, the $\widetilde H^s$-projection onto piecewise linear finite element spaces on quasi-uniform meshes of a bounded Lipschitz domain in $\mathbb{R}^d$ with a uniform exterior ball condition is almost stable in the maximum norm. The mesh domain need not coincide with the original domain.

For the quadratic family $f_c(z)=z^2+c$, we study the non-escaping locus $\KM=\{(c,z):c\in\M,\ z\in K_c\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\M$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\partial K_c$. We prove that $\partial\KM$ has full Hausdorff dimension $4$. The set $\KM$ is locally connected at every $(c,z)$ with hyperbolic $c\in\M$. At parameters on $\partial\M$, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of $\KM$ at $(c,z)$. However, local connectedness fails at some $(c,z)$ with $c$ real parabolic and $z\in K_c\setminus J_c$, even though both the parameter space and the filled Julia fiber are locally connected there.

In this paper, we present a one-parameter extension of some supercongruences related to Van Hamme's $\mathrm{(E.2)}$ and $\mathrm{(F.2)}$ supercongruences. To deduce these results, we first establish the corresponding $q$-supercongruences modulo the third power of a cyclotomic polynomial. Our proofs rely on the creative microscoping method devised by Guo and Zudilin~\cite{guo2019q}, together with Jackson's ${}_{6}\phi_{5}$ summation formula.

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

We consider weakly reinforced Pólya urns on a countable graph of uniformly bounded degree. The vertices have positive bounded firing rates, which need not be bounded away from zero. For every reinforcement exponent $\alpha\in[0,1)$ and every deterministic choice of finite positive initial weights, the normalized edge weights converge almost surely, coordinate-wise, to a deterministic vector independent of the initial weights. We identify this vector as the unique non-vanishing equilibrium of the limiting equation. This proves the homogenization conjecture of Couzinié and Hirsch, including its formulation for non-uniform firing rates in the MATRIX open-problems report. The main step is a Liouville theorem for the limiting equation, established by a logarithmic-norm comparison of complete solutions using the row-sum balance at each vertex.

In this note, we show that a locally conformally flat free boundary minimal hypersurface in $B^n$ must be either an equatorial ball or a critical catenoid for all $n \geq 4$. As a consequence, we conclude that the equatorial ball is unique among locally conformally flat free boundary minimal hypersurfaces in $B^n$ with the topology of $B^{n - 1}$.

In this paper, we establish several $q$-supercongruences for basic hypergeometric series modulo the third and fifth powers of a cyclotomic polynomial. These results extend several existing supercongruences related to the $\mathrm{(A.2)}$ and $\mathrm{(C.2)}$ supercongruences of Van Hamme. Our proofs employ the creative microscoping method developed by Guo and Zudilin \cite{guo2019q}, together with Watson's ${}_{8}\phi_7$ transformation formula.

We consider two-level hybrid Schwarz domain-decomposition preconditioners for finite element discretizations of the high-frequency Helmholtz equation (with wavenumber $k$). To date, all existing $k$-explicit theory for these preconditioners considers coarse spaces in the $\textit{asymptotic regime}$, i.e., where the Galerkin solution is $k$-uniformly quasi-optimal. We consider the situation where the fine and coarse spaces consist, respectively, of piecewise polynomials of degree $p$ on meshes of width $h$ and $H_c$. We prove results for the coarse space in the $\textit{preasymptotic regime}$, under the familiar mesh threshold that $(H_c k)^{2p} \rho(k)$ is sufficiently small, where $\rho(k)$ is the norm of the solution operator (such that $\rho(k)\sim k$ for the Helmholtz equation in free space). We prove that, in this regime, the fixed-point iteration with this preconditioner converges in a number of iterations that grows at most like $\log k$. We give numerical experiments demonstrating this theory. Furthermore, these experiments indicate that our theoretical results are sharp, in that the number of iterations required for these preconditioners to converge with $(H_c k)^{2p} \rho(k) \gg 1$ grows rapidly with $k$.

Zador's celebrated theorem is a cornerstone of optimal quantisation: it establishes both the weak limit of the empirical distribution of an optimal $n$-point quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. In large dimension, however, observing this asymptotic behaviour requires an astronomically large sample size. We prove that, for spherically symmetric target distributions, optimisation over all spherically symmetric distributions is a convex problem and derive an equivalence theorem that both characterises global optimality and yields a constructive algorithm. We show that, for moderate $n$, random quantisers uniformly distributed on a sphere of suitably chosen radius $R$ perform exceptionally well and, over a broad range of values of $n$, are numerically certified to be optimal among all random quantisers. Their expected distortion has an explicit integral representation that can be evaluated to arbitrary precision, and we prove concentration across random quantisers: the distortion variance tends to zero as $n\to\infty$ for fixed $d$. For $s=2$, both the optimal radius and the associated minimum expected distortion admit exact expressions. For general $s$, the optimal radius can be determined efficiently, and extreme-value theory provides useful approximations when $n$ grows with $d$. Depending on this growth rate, $R$ either converges to zero or approaches a positive limit that is independent of $s$.

We study the multivariate convolution model with additive independent noise, multidimensional signal and noise, aiming to recover the signal's cumulative distribution function from contaminated observations. The noise is ordinary smooth with anisotropic regularity, and the signal belongs to an anisotropic Nikol'skii density class. We extend to the multivariate setting an approximate minimum \(L^1\)-distance estimator based on integrated kernel density estimation, and establish matching upper and lower bounds for the \(L^1\)-risk over anisotropic Nikol'skii classes. To our knowledge, this is the first minimax rate, not a sharpening of existing bounds. Although the lower-bound scheme is classical, the test-function construction is novel: it distinguishes no active component from at least one active component. The rate is governed by a censoring mechanism through the positive part, a noise filter in a harmonic-mean representation of the exponent. We further propose a fully data-driven, rate-adaptive procedure selecting an optimal bandwidth vector over the full Nikol'skii scale. On Nikol'skii product density classes, the \(L^1\)-distance between cumulative distribution functions and the \(1\)-Wasserstein distance share the same minimax rate, though not equivalent in general, via coordinate-wise decoupling of the \(1\)-Wasserstein cost on product measures. These findings reveal a link between the two distances, whose full understanding remains an open question.

Near-term quantum approaches to combinatorial optimization are limited by qubit counts, circuit fidelity, sampling cost, and the difficulty of encoding constraints, while machine learning is increasingly used to configure and control quantum optimization workflows. We call such workflows adaptive: decisions conventionally fixed in advance, from formulation and penalties to shot budgets, backends, and whether to invoke a quantum processor at all, are made by learned policies that respond to the instance, the progress of the solve, or the hardware. This review examines three paradigms, AI for quantum optimization, quantum for AI-driven optimization, and AI-quantum co-optimization, and organizes the literature by the decision being learned rather than by application. A structured search identified a 119-paper corpus, of which 67 papers form an evidence core coded in detail. The evidence is considerably stronger for AI-assisted quantum optimization than for the reverse direction: learning already reduces quantum evaluations, improves initialization, supports decomposition and penalty control, and mitigates noise, whereas evidence that quantum computation improves learned optimizers remains largely confined to small-scale simulation. Experimental controls are thin: of 52 studies in the three paradigms, 22 include no classical baseline, the quantum contribution is fully isolated in 10 of 24 where an ablation applies, and the median largest experiment uses 16 qubits. We introduce an M0-M5 evidence hierarchy, from simulation to practical quantum advantage, and find no broadly convincing result at the highest level. We argue that scaling is increasingly a systems problem: the question is not only whether a problem fits on a quantum processor, but how classical and quantum resources should be allocated across the optimization process.

For a $t$-graph $F$ and an integer $r\ge t$, the \emph{$r$-augmentation} $F(r)$ is the $r$-graph obtained by adding the same set of $r-t$ new vertices to every edge of $F$. We investigate the behaviour of the \emph{Turán density $\pi(F(r))$}, which is the asymptotically maximum edge density of a large $F(r)$-free $r$-graph, as a function of $r$. Let $H_k^r$ be the $r$-augmentation of the complete $(k-1)$-graph on $k$ vertices; equivalently, $H_k^r$ is the (unique up to isomorphism) $r$-graph with $r+1$ vertices and $k$ edges. This paper determines the Turán density of each $r$-graph $H_k^r$ within a constant factor and estimates the lower bounds on $\pi(H_k^r)$ coming from the circular construction of Sidorenko. In particular, it is shown that if $k/\log r\to\infty$ then $\pi(H_k^r)=(1+o(1))k/r$. Also, we consider augmentations of graphs (that is, the case $t=2$) and prove that $\pi(F(r))\ge(\log r)^{-1-o(1)}/r$ for every 3-chromatic graph $F$, which comes close to the known upper bound $1/r$. Note that, for all other graphs $F$, the function $\pi(F(r))$ is known to be either identically 0 or uniformly bounded away from 0.

We answer the following question by Frittaion, Nemoto, and Rathjen positively: "Is [independence of premise] an admissible rule of $\textsf{CZF}$ or any other familiar constructive/intuitionistic set theory $T$?" To be more precise, we show that whenever $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) derives a statement of the form $\lnot\psi \to \exists y\ \varphi(y)$ where $y$ does not occur in $\psi$, then $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) also derives $\exists y\ (\lnot\psi \to \varphi(y))$. Our proof method is a variant of the famous Friedman-Dragalin $A$-translation adapted to the context of set theory in a hereditary manner.

We propose and analyze two second-order basis-update & Galerkin (BUG) time integration methods for dynamical low-rank approximation on Tucker tensors and tree tensor networks. Both are built from a sequential pre-computation sweep, after which the differential equations for all basis matrices and connecting tensors are evolved by a Galerkin method. Further, the proposed methods are rank-adaptive by construction. The first, the second-order parallel BUG integrator, solves all these differential equations fully in parallel, which is favorable on parallel architectures, followed by a sequential augmentation and truncation step. The second, the second-order augmented BUG integrator, gives up full parallelism but conserves norm and energy for Schrödinger equations and dissipates energy for gradient flows up to the truncation tolerance. For both integrators, we prove a second-order error bound that is robust with respect to small singular values of the matricization of connecting tensors. Numerical experiments for radiative transfer and quantum spin systems validate the theoretical findings.

Radio Resource Management (RRM) is a fundamental challenge in 6G wireless networks, particularly under dynamic user demands, inter-cell interference, and heterogeneous QoS constraints. Centralized optimization solutions are often infeasible in practice due to the lack of system-wide state information, dynamic conditions, and signaling delays, making distributed learning-based approaches attractive. However, conventional multi-agent reinforcement learning (MARL) struggles with scalability and constraint satisfaction in such highly dynamic environments. We introduce a scalable two-phase hybrid learning framework to address the RRM challenges where orthogonal-frequency division multiplexing (OFDM) domain knowledge is explicitly incorporated into the MARL pipeline. In our proposed two-phase learning framework, the first phase allocates a minimum resource to satisfy users' QoS requirements based on channel statistics, avoiding the inefficiencies of training policies under hard QoS constraints. Subsequently, a multiagent system is employed in the second phase to optimally allocate the remaining resources for system throughput maximization. By exploiting the structural interference information, we propose a scalable MARL algorithm which decomposes the original learning problem into smaller subproblems that can be handled independently, thereby avoiding exponential growth of the action space without compromising performance. Extensive simulations in realistic scenarios with 50MHz bandwidth and different numerologies show that our method significantly outperforms existing optimization and learning baselines, offering up to 40% throughput improvement, and 100% constraint satisfaction with minimal resource usage.

Standard weight decay treats each weight matrix as a vector and ignores its spectral structure. We introduce spectral weight decay, a post-step decoupled nuclear-norm update that applies additive rather than multiplicative spectral shrinkage. We connect the update to approximate proximal descent and show that its sensitivity to update order can exceed that of conventional $\ell_2$ weight decay near rank deficiency. Across LLaMA models with $124$M to $500$M parameters, spectral weight decay lowers effective rank and improves SVD-LLM compression at matched validation loss. At $500$M and a $4\%$ distortion budget, it reaches $1.89\times$ compression and $1.18\times$ GPU inference speedup, compared with $1.14\times$ and $1.01\times$ after standard weight decay. Under fixed-horizon training with $60\%$ label noise, it also improves final mean clean-test accuracy over matched $\ell_2$ regularization by up to $17.8$ points on MNIST and $4.6$ points across four BERT-base tasks. Code is available at https://github.com/brain-lab-research/SpectralWD.

We study the combinatorics of the intervals below quasi-Coxeter elements in the absolute order of finite Coxeter groups. In particular, we prove that these intervals are EL-shellable, except possibly for quasi-Coxeter elements of types $E_7$ and $E_8$, and we compute their zeta polynomials. We also show that a total order of the reflections yields an EL-labeling if it does so on all rank-two subintervals, which proves a conjecture of Williams for Coxeter elements. In order to obtain these results, we prove them for the infinite families, and we treat the exceptional types by computer. In addition, we study the related interval monoids, and we find a counterexample to cancellativity for every proper quasi-Coxeter element of type $D_n$ ($n\ge5$), $E_6$, $E_7$, $E_8$; in particular, these monoids do not embed in the corresponding interval groups. This gives a negative answer to a question posed by Baumeister, Holt, Neaime, and Rees.