Papers

We define metrics in space that are natural counterparts of the hyperbolic metric in plane domains, using the characterization of the hyperbolic metric due to Beardon and Pommerenke. We obtain inequalities for these metrics under quasiconformal homeomorphisms between domains in space that have at least two boundary points. We discuss the failure of the existence of such estimates for non-homeomorphic quasiregular mappings.

In Diaconis and Saloff-Coste (1996), the authors introduced the simple “transvection" walk on $\mathrm{GL}_n(\mathbb F_2)$: at each step, choose two distinct rows and add one to the other. In Ben-Hamou (2025), the author recently proved that this walk has mixing time $O(n^2\log n)$. Inspired by applications in cryptography (see Sotiraki (2016)), Ben-Hamou and Peres (2018) conjectured that the first $k$ columns of this walk mixed in $O(nk \log(n))$ steps. Our main result is a proof of this conjecture uniformly in $n$ and $k.$ Our proof is based on a local-to-global entropy estimate, in the spirit of block factorization results such as Caputo et al (2015), Caputo et al (2021). In our setting, the kernels that correspond roughly to the block kernels of Caputo et al (2021) do not have uniformly large log-Sobolev constants, and so naively applying these techniques does not improve over Ben-Hamou (2025). We avoid these bad blocks by combining our entropy estimates with a burn-in argument similar to classical drift-and-minorization arguments of Rosenthal (1995). This method may be of broader interest, and so we illustrate it by proving an analogous result for a family of product-replacement algorithms on the Heisenberg group.

In this paper, we study the behavior of Ekedahl-Oort strata under natural embeddings between the good reductions modulo $p$ of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties, a prototypical setting for the construction of special cycles in the Kudla program. In each case, we determine the EO stratum containing the image of a given EO stratum under the embedding. We also compute discrete invariants of these Shimura varieties, including their $p$-ranks and $a$-numbers; in the GSpin case, these are obtained via the Kuga-Satake embedding.

The relative entropy between two states is a key concept in quantum information theory and quantum field theory. In the setting of quantum field theory, its computation requires the handling of relative modular Hamiltonians, which are typically very difficult to compute explicitly. In this paper, we exploit locality properties of general algebraic QFTs to estimate relative modular Hamiltonians between two states, $\omega$ and $\tilde{\omega}$, and hence also their relative entropy, in terms of the modular Hamiltonian of a reference state $\hat{\omega}$, which might be better understood. For suitable pairs of states we can estimate the relative modular Hamiltonian for the algebra of a region $V_2$ from above, resp. from below, in terms of the modular Hamiltonian of $\hat{\omega}$ on a larger region $V_3$, resp. a smaller region $V_1$. Pairs of states and choices of regions which are susceptible to our scheme are related to the presence of superluminal signalling in the sense of Sorkin's paradox. If $\omega=\hat{\omega}$, then there exists a unitary that maps $\omega$ to $\tilde{\omega}$ on $V_3$ and that does not allow superluminal signalling from the spacelike complement $V_3'$ to $V_2$, resp. from $V_1$ to $V_2'$, if our upper, resp. lower, bound applies. To investigate the strength of our estimates we consider coherent states for CCR systems, focussing particularly on free scalar fields. Our estimates apply even if the relative modular Hamiltonian cannot be computed exactly. For sufficiently regular excitations we recover an exact result by squeezing. Our method thus yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian cannot be computed exactly. For massless fields we establish the analogous result also for double cone regions. These results indicate that our estimates do not lose too much information.

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{\mu_+,\mu_-}(\hgs)$ is obtained from the Drinfeld current presentation of twisted quantum loop algebras by shifting the Cartan–Drinfeld currents $\phi_i^\pm(z)$ according to a coweight pair $(\mu_+,\mu_-)$. We prove that it admits a triangular decomposition and that, up to isomorphism, they depend only on the total shift $\mu=\mu_+ + \mu_-$. For each shift $\mu$, we define a category $\mathcal O_\mu$ of representations of $\U_q^\mu(\hgs) = \U_q^{0,\mu}(\hgs)$ and prove a rationality theorem for the Cartan currents: on every weight space, the two currents $\phi_i^+(z)$ and $\phi_i^-(z)$ are expansions of the same rational operator-valued function, whose degree is prescribed by $\alpha_i(\mu)$. As a consequence, we classify the simple objects of $\mathcal O_\mu$ by rational $\ell$-weights of the corresponding degrees. We then construct a deformed Drinfeld coproduct and use it to define a fusion product on the direct sum $\mathcal{O}^{sh}$ of the categories $\mathcal O_\mu$. This fusion product is compatible with $q$-characters. We also classify finite-dimensional simple modules in $\mathcal{O}^{sh}$ in terms of dominant rational $\ell$-weights, with a separate treatment of type $A_{2n}^{(2)}$. Finally, we construct restriction representations relating representations of twisted quantum affine Borel algebras to representations of shifted twisted quantum affine algebras, and establish a $q$-characters formula for simple finite-dimensional representations of shifted twisted quantum affine algebras in terms of the $q$-characters of the corresponding simple representations of the twisted quantum affine Borel algebra $\U_q(\bs)$.

We study order parameters in one-dimensional quantum lattice models with finite invertible or non-invertible symmetry. We investigate what properties a string operator must satisfy in order to acquire a non-vanishing expectation value in a given gapped phase. We deduce that multiplets of string order parameters organise into a Lagrangian algebra in the Drinfel'd centre of the symmetry category. In particular, we highlight the role of the multiplication rule as governing the fusion of the twisted sector local operators that constitute the string operator in the infrared limit. Our derivations exploit the tensor network approach to the classification of gapped phases and its reformulation in terms of module categories over the symmetry category. Within this framework, a gapped phase is associated with a pattern of spontaneous symmetry breaking wherein a Morita class of algebras of topological lines is preserved in the ground state subspace. The crux of the proof is to show that the expectation value of any string operator explicitly depends on the tube algebra module associated with the Lagrangian algebra, which is realised as the full centre of the corresponding module category. Finally, we demonstrate that these techniques extend to phases of symmetric mixed states.

Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.

We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of “mock” Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.

This paper investigates the new notion of $2$-word-$\pi$-repre\-sentable graphs: the nodes of the graph correspond to the letters of the two words and there exists an edge between two nodes if the projections of any two letters of both words are equal. The benefit of not only using one word for a representation as introduced by Kitaev and Pyatkin is that every graph is $2$-word-$\pi$-representable. We present an algorithm that returns two representing words for any graph. Aside, we show that every permutation graph is representable by two $1$-uniform words and give constructions how graph operations on $2$-word-$\pi$-representable graphs can be realised on their representing words which give further insights into the representation of cographs.

Quadratically regularized optimal transport (QOT) is an alternative to entropic regularization that yields sparse couplings and avoids numerical instabilities due to exponential scaling. From an optimization viewpoint, the dual QOT objective is concave but features a positive part function which prevents strong concavity and reduces smoothness of optimizers. Consequently, standard arguments for linear convergence of algorithms do not apply. In this paper, we nevertheless establish a quantitative curvature property for the QOT dual. Under mild assumptions covering both continuous and semi-discrete transport problems, we prove a local error bound and a Polyak-Lojasiewicz (PL) inequality, with explicit constants depending only on the problem primitives. These results are obtained by functional-analytic techniques exploiting that near the optimum, the argument of the positive part function is positive on the interior of the support of the optimal coupling. As applications, we derive linear convergence of the gradient ascent, coordinate ascent, and coordinate gradient ascent algorithms on the dual problem, with explicit contraction rates.

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with minimum degree to be $k$-factor-critical. A graph $G$ is fractional $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a fractional perfect matching. Clearly, the fractional $k$-factor-criticality of a graph is a necessary property for a graph to be $k$-factor-critical. Jia, Fan and Liu [Discrete Appl. Math. 386 (2026) 255-263] proposed a tight sufficient condition in terms of the spectral radius for a graph with fractional $k$-factor-criticality to be $k$-factor-critical. A natural question arises: can we derive analogous sufficient conditions by incorporating the minimum degree parameter of graphs? We first establish a lower bound on the size to ensure that a $(k+1)$-connected graph with fractional $k$-factor-criticality is $k$-factor-critical, where $k$ is a positive integer with $k\geq1$. Moreover, we provide a sufficient condition in terms of the spectral radius for a $(k+1)$-connected graph with fractional $k$-factor-criticality to be $k$-factor-critical. Our results generalize the result of Jia, Fan and Liu to $(k+1)$-connected graphs. Furthermore, our spectral conditions apply to a broader family of connected graphs compared with the results of Fan and Lin, as well as Jia et al.

In latent-position random graph models (LPMs), latent vertex positions $U_{1},\ldots,U_{n}$ are sampled from some distribution on a latent space $\Omega$, then edges of an observed graph $G = ([n],E)$ are sampled with some probability $\mathbb{P}[(i,j) \in E ]=w(U_i,U_j)$ that depends on the unobserved latent positions. LPMs are ubiquitous in the statistical analysis of networks, offering models that have good empirical performance, strong theoretical guarantees, and tractable algorithms. The special case $\Omega = [0,1]$ is important, as it corresponds to graphs with temporal or preference-based structure. In this paper, we study three problems related to LPMs with latent space $[0,1]$: ordering the vertices according to the latent positions, estimating the generating graphon $w$, and testing whether an observed graph $G$ could have come from an LPM with state space $[0,1]$. Our results on the ordering problem greatly generalize two observations of Janssen/Smith (2022): (i) for some families of graphons, the best estimate of the ordering converges much faster than the usual statistical rate of $\frac{1}{\sqrt{n}}$, and (ii) this occurs even though, for the same families of graphons, the best estimate of the latent positions still occurs at the usual $\frac{1}{\sqrt{n}}$ rate. As a main consequence, we develop a computationally-efficient graphon-estimation algorithm and show that it has the same convergence rate as the non-explicit optimal algorithm of Gao et al (2015). We also derive and analyze a testing procedure.

Let $q$ be an odd prime power and $\chi_q$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. In this paper, we study the arithmetic properties of the product $$R_q(\chi_q)=\prod_{0<k<(q-1)/4}\left(J_q(\phi_q,\chi_q^k)+J_q(\phi_q,\chi_q^{-k})\right),$$ which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix $$\left[\phi_q(s_i+s_j)\right]_{1\le i,j\le (q-1)/2},$$ where $\phi_q$ is the unique quadratic multiplicative character of $\mathbb{F}_q$, and $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

This paper addresses a novel Riesz–Kolmogorov theorem on quasi-Banach function spaces under mild conditions and the extrapolation of multilinear compact operators in the context of weighted variable Lebesgue spaces. We establish the latter result via our Riesz–Kolmogorov theorem which yields a weighted interpolation theorem for multilinear compact operators in the variable Lebesgue setting. In proving this, we also show a weighted interpolation theorem in mixed-norm variable Lebesgue spaces. By means of our extrapolation result, we obtain new weighted compactness estimates for the commutators of multilinear $\omega$-Calderón–Zygmund operators, multilinear fractional integrals and multilinear Fourier multipliers on weighted variable Lebesgue spaces. Our work generalizes several recent ones, including but not limited to those of Cao, Olivo and Yabuta in the setting of multilinear operators acting on the classical weighted Lebesgue spaces as well as the previous result by the authors in the setting of bilinear operators and variable Lebesgue spaces.

The goal of this paper is to describe the birational geometry of the blowup of $\mathbb{P}^n$ at $n+4$ points in very general position. To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a correspondence between configurations of $n+4$ points in the projective spaces $\mathbb{P}^n$ and $\mathbb{P}^2$. We first prove that the blowup $X$ of $\mathbb{P}^n$ at $n+4$ general points is isomorphic to a certain Gieseker moduli space of rank $2$ vector bundles on the surface $S$ obtained by blowing up $\mathbb{P}^2$ at the $n+4$ Gale dual points. We then study the variation of these moduli spaces as we vary the polarization $L$ on $S$, and translate this variation into a partial Mori chamber decomposition of $\overline{Eff}(X)$, describing to some extent the birational geometry of $X$.

This work studies spectral properties of Schrödinger operators in the context of aperiodic order, using weighted Delone sets to explore the interplay between the underlying dynamics and spectral properties. We study parameter-dependent families interpolating between periodic and aperiodic regimes, whose spectra form so-called spectral butterflies. These reflect fractal and self-similar structures of the spectra. We review existing results, introduce additional examples, and establish new connections between works in the literature. The framework is largely dimension-independent and extends to non-Abelian groups and more general settings.

We establish an oracle Bernstein-von Mises theorem for high-dimensional sparse generalized linear models under an outcome-independent spike-and-slab prior. The posterior consistently recovers the true active set and converges in total variation to the Gaussian law that would arise if this set were known in advance, even when the active dimension grows. The result holds for both ordinary and fractional posteriors, the latter with the expected variance inflation under tempering. In fixed-design logistic regression, a sufficient lower bound on the squared minimum signal is proportional to the logarithm of the ambient dimension divided by sample size. Separate likelihood controls for model selection and oracle approximation allow the Gaussian approximation to use a weaker curvature condition. We verify the conditions for six canonical and noncanonical models under fixed and random designs, including Poisson regression with potentially unbounded Fisher weights. The results also yield oracle-rate contraction, credible regions on the recovered support and pointwise frequentist coverage.

A theorem of Ding and Geiges states that every closed, connected contact $3$-manifold can be obtained from the standard tight contact $3$-sphere by contact $(\pm1)$-surgery along a Legendrian link. The literature also contains some examples of contact Kirby moves, i.e. explicit operations on front projections of Legendrian surgery links that change the surgery link but preserve the contactomorphism type of the surgered manifold. Among the most commonly used are cancelling pairs and contact handle slides; however, these moves alone are not sufficient to relate all contact surgery diagrams of contactomorphic contact manifolds. In this article, we introduce two new families of contact Kirby moves, called lantern moves and chain moves, and use them to give a complete set of contact Kirby moves. More precisely, we show that two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of planar isotopies, Legendrian Reidemeister moves, insertions or removals of standard cancelling pairs, the two standard contact handle slides, the standard lantern move, and the standard chain move. All these moves are explicit diagrammatic operations in the front projection. The proof follows an approach initiated by Avdek through his ribbon-move framework, which is rooted in the Giroux correspondence, and combines it with a presentation by Gervais of the mapping class group. We also discuss several consequences of the main theorem, illustrating the effectiveness of the contact Kirby calculus by recovering the invariance of Gompf's $d_3$-invariant purely diagrammatically and by deriving the topological Kirby theorem from contact-geometric methods.

We construct open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians, which are weakly unobstructed by a bounding cochain. This uses the foundational work of \cite{HH25,HH26} and the algebraic framework of \cite{ST21}. We prove the open WDVV relations and show that these invariants are independent of the choice of almost complex structure and under Hamiltonian isotopy. We also prove a relation between open Gromov-Witten invariants of cobordant Lagrangians.

Let $n\in \mathbb N\cap[2,\infty)$. In this article, we show that there exists a bounded $C^1$ domain $\Omega\subset \mathbb R^n$ such that, for any given $s\in(1,2)\setminus\{\frac32\}$, \begin{align*} \left[H_0^1(\Omega),H^2(\Omega)\cap H_0^1(\Omega)\right]_{s-1} =H^s(\Omega)\cap H_0^1(\Omega)=H_0^s(\Omega) \end{align*} with equivalent norms, but \begin{align*} \left[H_0^1(\Omega),H^2(\Omega)\cap H_0^1(\Omega)\right]_{\frac12} \subsetneqq H^{\frac32}(\Omega)\cap H_0^1(\Omega), \end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $\Omega$ \begin{align*} H^2(\Omega)\cap H_0^1(\Omega)\subsetneqq D(-\Delta_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-\Delta_D$ on $\Omega$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(\Omega)\cap H_0^1(\Omega))$ for any given $T\in(0,\infty)$.

We prove that the topological flow category $\mathcal{M}$ arising from a Morse-Smale pair $(f,\xi)$ on a smooth closed manifold $X$ is equivalent, as an $\infty$-category, to Lurie's $\infty$-category $\mathrm{Sing}_A(X)$ of exit paths in $X$ with respect to the stratification by the stable manifolds of $\xi$. The objects of $\mathcal{M}$ are the critical points of $f$, and for every pair of critical points, the space of morphisms of $\mathcal{M}$ between these is the space of possibly broken trajectories of $\xi$ connecting them; it can be identified up to homotopy with the space of unbroken ones. The latter maps naturally to the space of exit paths connecting these critical points; we prove this map to be a weak homotopy equivalence. Then, we combine these ingredients with several others to construct a zigzag of equivalences between the homotopy coherent nerve of $\mathcal{M}$, denoted $\mathcal{N}(\mathcal{M})$, and $\mathrm{Sing}_A(X)$. The $n$-simplices of $\mathcal{N}(\mathcal{M})$ are homotopy coherent diagrams of $n$ composable morphisms of $\mathcal{M}$; we introduce the notion of unbroken diagram, yielding an $\infty$-subcategory of $\mathcal{N}(\mathcal{M})$, which we refer to as the flow coherent nerve of $\mathcal{M}$. The simplices of the latter give rise to stratified maps out of a family of stratified cubes, into $X$. We organize this family into a functor from the category of finite ordered sequences of critical points, to the category of $A$-stratified topological spaces, and we prove a comparison result with the usual stratified geometric realization functor. We finally use a theorem of Tanaka that associates a functor to a semi-simplicial map between $\infty$-categories that satisfies certain conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in [Discrete Math. 306 (16) (2006) 1997-2004] that every tree $G$ satisfies $\chi_{r}(G)=\min\{r,\Delta(G)\}+1$. It is known that every cycle graph $C_{n}$ with order $n$ has $\chi_{L,r}(C_{n})=\chi_{r}(C_{n})$. The main results are the following: $(1)$ If $G$ is a tree, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; $(2)$ Let $G$ be a unicyclic graph which is not isomorphic to the cycle $C_{n}$. If $n\neq 5$ and $r\geq3$, then $\chi_{L,r}(G)=\min\{r,\Delta(G)\}+1$; otherwise, $\min\{r,\Delta(G)\}+1\leq\chi_{L,r}(G)\leq\min\{r,\Delta(G)\}+2$.

The goal of this note is to show that the left $\chi$-coalgebra, which is an additional structure on one of the coefficients used in the construction of the cyclic operator for the cyclic sets that generalises the twisted nerve of a group by Loday is an instance of a general theory of left $\chi$-coalgebras. It is also shown that in the general case, the resulting cyclic operator requires that the other coefficient needs to be equipped with a crossed $G$-sets structure.

We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in which the evolution of a Riemannian metric is driven by transport together with a source term. Choosing the $L^2$-metric to penalize the transport vector field and the Ebin metric to penalize the source component yields a new Riemannian metric on the manifold of Riemannian metrics, which we call the Wasserstein–Ebin metric. Our main result shows that the volume map defines a Riemannian submersion from the Wasserstein–Ebin metric to the Wasserstein–Fisher–Rao metric on the space of smooth densities. In addition, we construct a Riemannian submersion from the automorphism group of the tangent bundle onto the space of Riemannian metrics, providing a generalization of Otto's geometric description for the Wasserstein metric to the setting of the Wasserstein–Ebin metric. To propose a static formulation of unbalanced optimal Riemannian metric transport, we introduce two Kullback–Leibler-type divergences on the space of Riemannian metrics: one inspired by matrix information geometry, and another related, through the volume map, to the classical Kullback–Leibler divergence on densities. Establishing a link between the static and dynamic formulations remains an open direction for future work.

Win measures, including the win ratio, win odds, net benefit, and desirability of outcome ranking, are increasingly used in randomized clinical trials with multiple hierarchical ordinal endpoints. In practice, however, one or more component endpoints may have missing data. The standard pairwise-comparison approach, which treats pairs with missing outcomes as ties, can produce biased estimates, even if the data are missing completely at random. Although inverse probability of censoring weighting methods have been developed for censored survival endpoints, corresponding methods for addressing missing hierarchical ordinal endpoints are not yet available. To address this gap, we develop inverse probability weighted (IPW) and augmented IPW (AIPW) estimators for win measures with hierarchical ordinal endpoints subject to missing data, allowing missingness to depend on treatment assignment and baseline covariates. The IPW estimator corrects bias by reweighting complete observed outcomes using joint non-missingness probabilities involved in estimating the joint cell probabilities that define the win measures. The AIPW estimator additionally incorporates outcome modeling, potentially improving estimation precision and achieving double robustness. For inference, we derive closed-form variance estimators for both methods based on influence functions. Simulation studies show that the standard approach can be substantially biased, whereas the proposed IPW and AIPW estimators exhibit small bias and near-nominal coverage under their respective identifying and model conditions. Applications to the SCOUT-CAP and ACTT-1 trials illustrate the practical utility of the proposed methods. An R package WinMO is provided for implementation.

We present new convergence estimates for the iterated penalty method applied to structure-preserving discretizations of linear generalized saddle point systems. The method may be viewed as an Uzawa iteration on an augmented Lagrangian formulation of the system. As a by-product, we obtain sharper stability estimates for penalized/perturbed saddle point problems. Three model finite element applications show agreement with the theory.

We use the high-low method and wavepacket pruning to prove new small-cap decoupling estimates for the moment curve in $\mathbb{R}^4$. As an application, we verify a conjecture of Demeter regarding the $L^{12}$ square-root cancellation of exponential sums associated with the moment curve in $\mathbb{R}^4$. This provides a continuum of square-root cancellation estimates that connects the Vinogradov MVT in $\mathbb{R}^3$ with a result of Bourgain, related to improving the best-known estimate for the Lindelöf hypothesis.

We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{\Theta(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko.

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[ S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \exists\lambda\in\mathbb{C}, f^{\circ m}(\lambda)=g^{\circ n}(\lambda)=c(\lambda)\right\rbrace \] is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case $m=n$. We also obtain partial results on recurrence sets for rational maps. These results are related to higher-dimensional questions of dynamical Mordell–Lang type of rank $\leq2$.

We prove that the Cauchy problem for the model hyperbolic operator in $ \R^{4} $ \[ Q=-D_t^2+2xD_tD_y+D_x^2+x^3D_y^2+D_z^2+z^2D_y^2 \] is not locally solvable at the origin, in the Gevrey $s$ class if $s>6$.