We develop a toric topological framework for studying the cohomology of Vietoris–Rips complexes $VR(Q_n;r)$ of hypercube graphs. Using total domination invariants and spectral methods, we establish general lower bounds on connectivity, which lead to infinite families of counterexamples to Shukla’s conjecture, and derive first global upper bounds on coconnectivity. Our approach interprets Vietoris–Rips complexes via Stanley–Reisner rings, moment-angle complexes, and Tor algebras, allowing global topological information to be extracted from combinatorial data. In a second direction, we construct explicit cohomology classes using the Koszul resolution and show that they are decomposable products of $1$-dimensional classes, and that their representatives can be combinatorially realized as the boundary of cross polytopes, positively answering the question posed by Adams and Virk. We introduce ghost vertices as a new tool for detecting, extending, and proving linear independence of cohomology classes.

We introduce conformal transformations in the synthetic setting of metric spaces and Lorentzian (pre-)length spaces. Our main focus lies on the Lorentzian case, where, motivated by the need to extend classical notions to spaces of low regularity, we provide the first consistent notion of conformal length, and analyse its fundamental properties. We prove that the conformal time separation function $\tau_{\Omega}$ (and the causal structure it induces) yields a Lorentzian pre-length structure if the original space is intrinsic and strongly causal. This allows us to construct a notion of conformal transformation between spaces within this class, yielding an equivalence relation. As applications, we show that the conformal length functional agrees with the standard conformal length of (strongly causal) spacetimes. We also prove conformal invariance of angles and causality conditions, give a characterization of global hyperbolicity via finiteness of $\tau_{\Omega}$ for all conformal factors, and establish the behaviour of the Lorentzian Hausdorff measure defined by McCann and Sämann under conformal changes. Moreover, we apply the same methods to the metric case, which is of interest in its own right. This is exemplified by proving an analogue of the Nomizu–Ozeki theorem for metric length spaces, which has the advantage that the resulting complete space is conformally related to the original space.

Any quaternionic Kähler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped with a Killing vector field X with nowhere vanishing quaternionic moment map carries an integrable almost complex structure J1 that is a section of the quaternionic structure $\mathcal Q$. Using the HK/QK correspondence, we study properties of the almost Hermitian structure $(g_{\bar N},\tilde J_1)$ obtained by changing the sign of J1 on the distribution spanned by X and $J_1X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure $\tilde J_1$ is integrable. We do also show that the integrability of $\tilde J_1$ implies that $(g_{\bar N},\tilde J_1)$ is conformally Kähler in dimension 4, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two planes, we construct a third canonical Hermitian structure $(g_{\bar N},\hat J_1)$. Finally, we give a complete local classification of quaternionic Kähler fourfolds for which $\tilde J_1$ is integrable and show that these are either locally symmetric or carry a cohomogeneity 1 isometric action generated by one of the Lie algebras $\mathfrak{o}(2) \lt imes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$.

Goal of this paper is to study classes of Cauchy–Dirichlet problems which include parabolic equations of the type

\begin{equation*}u_t -\Delta u= a(x,t)f(u)\quad\hbox{in}\ \Omega\times(0,T)\end{equation*}

$\Omega\subset\mathbb{R}^n$

$T\in(0,+\infty]$

$a$

$f$

$u$

$a$

$u$

$f(u)=1$

$f(u)=u^q$

$q\in(0,1)$

$f(u)=u$

$f(u)=u\log(u^2)$

$f(u)=\frac{u^2}{1+u}$

$f(x,u)=a(x)u-u^2$

Some exact results provide a different approach to the existing literature, as well as several generalizations. Moreover, some quantitative results are also valid and new in the elliptic setting $-\Delta u=a(x)f(u)$.

In this paper, we consider minimal group actions of countable groups on compact Hausdorff spaces by homeomorphisms. We show that the existence of a point with finite stabilizer imposes strong restrictions on the dynamics: a residual set of points then has stabilizers conjugate to the same almost normal subgroup of the acting group. Under certain conditions on the acting group, such an action also has no essential holonomy. If the acting group is residually finite, we show that every group action with finite almost normal stabilizers with no essential holonomy arises as an almost finite-to-one factor of an essentially free action.

We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.

Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau$, and let $A$ be an expansive dilation on $\mathbb{R}^d$. In this paper, the authors first investigate a class of local operator-valued anisotropic Hardy spaces, denoted by $\mathbf{h}_p^A(\mathbb{R}^d,\mathcal{M})$, and identify their dual spaces. This further develops the seminal work of T. Mei [MR2327840: Mem. Amer. Math. Soc., (2007)]. In addition, we also show that $\mathbf{h}_1^A(\mathbb{R}^d,\mathcal{M})$ and $\mathrm{bmo}_A(\mathbb{R}^d,\mathcal{M})$ serve as natural endpoint spaces for the noncommutative $L_p$-spaces $L_p(L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M})$, and these endpoint spaces are then used to establish the $L_p$-estimates for certain operators via interpolation. Finally, as applications, an atomic decomposition for local Hardy spaces and Hardy–Lebesgue estimates for a local class of noncommutative Calderón–Zygmund operators are also established. To the best of our knowledge, these results are new in the operator-valued setting even for diagonal anisotropic dilations (when $A:=\mathrm{diag}(\lambda_1, \lambda_2, \ldots, \lambda_d)$ with $|\lambda_j| \gt 1$, $j=1,2,\ldots,d$).

The Buchsbaum–Eisenbud–Horrocks conjecture has attracted the attention of many researchers working in Commutative Algebra and Algebraic Geometry in the last fifty years. Quite recently, a variant of this conjecture has been formulated by Lima-Pereira, Nuño-Ballesteros, Orefice-Okamoto, and Tomazella in their study of complete intersection singularities. The purpose of this paper is, on the one hand, to exhibit some cases where this conjecture holds. On the other hand, we show that the conjecture fails in general by exhibiting some counterexamples. Finally, we formulate a conjecture that can be regarded as a local variant of Buchsbaum–Eisenbud–Horrocks’ one.

For an étale groupoid, we define a pairing between the Crainic–Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is $C^*$-algebra theory. The Elliott invariant of a $C^*$-algebra is defined in terms of $K$-theory and traces; it is fundamental in the long-running programme to classify simple $C^*$-algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the $C^*$-algebraic Elliott invariant of the groupoid $C^*$-algebra: this includes irrational rotation algebras and the $C^*$-algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui’s HK conjecture for the relevant groupoids.

We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on the calculus of the Denham–Suciu fibre sequence, the Iriye–Kishimoto decomposition of the polyhedral product constructed from a collection of pairs of cones and their bases, and the Hilton–Milnor decomposition. As a corollary, we show that the Goodwillie calculus of these polyhedral products converges integrally and diverges in $v_h$-periodic homotopy unless the simplicial complex is a full simplex.

We consider a general system of hyperbolic-parabolic balance laws, which contains both second-order dissipation represented by a viscosity matrix and lower-order damping or relaxation. As dictated by physics, the viscosity matrix and the Jacobian matrix of the lower-order term are usually rank deficient. We study the Green’s function of its linearization around a constant equilibrium state. The Green’s function is constructed using heat kernels along equilibrium characteristic directions and delta functions along other predetermined directions. Detailed error estimates are given in the space-time pointwise sense. Our main result is then applied to two different models of polyatomic gas flows. The first one is a model of translational and vibrational non-equilibrium flows with a general setting in the fundamental thermodynamic relations. The second one is a two-temperature Navier–Stokes type model for polyatomic gas flows, proposed by Aoki et al., with specific equations of state. For each model, we give an explicit formulation of Green’s function for the Cauchy problem.

Let $(-A,B,C)$ be a linear system in continuous time $t \gt 0$ with input and output space ${\mathbb C}$ and state space $H$. The scattering (or impulse response) functions $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x = \int_x^\infty e^{-tA}BCe^{-tA}\,dt$ on $H$, and obtains solutions of the Kadomtsev–Petviashvili PDE. Pöppe’s semi-additive operators are identified with orbits of a shift action on integral kernels, and Pöppe’s bracket operation is expressed in terms of the Fedosov product. The paper shows that the Fredholm determinant $\det (I+R_x)$ gives an effective method for numerical computation of solutions of $KP$.

We present a simplified approach to the stable Hopf invariant. We provide short elementary proofs of the Cartan formula, the composition formula, and the transfer formula. In addition, when $\pi$ is a discrete group, we show how to extend these results to the stable category of $\pi$-spaces. We also consider the extent to which the stable Hopf invariant is unique.

In this paper, we obtain new lower bounds for the evolution of the radius of analyticity of solutions to two initial value problems (IVPs) with initial data belonging to the class of analytic functions $H^{\sigma,s}(\mathbb{R})$ defined via a hyperbolic cosine weight. First, we consider the IVP for the modified Korteweg–de Vries (mKdV) equation. For this problem, we prove that the evolution of the radius of analyticity $\sigma(T)$ of the solution admits an algebraic lower bound $cT^{-\frac 12}$ for some $c \gt 0$ and given arbitrarily large $T \gt 0$. Next, we analyse the IVP for the mKdV equation with generalized dispersion (mKdVm) and a damping term. For this problem, we guarantee the local well-posedness in $H^{\sigma,s}(\mathbb{R})$ and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity $\sigma(t)$ as time goes to infinity. The outcome of this paper concerning the mKdV equation represents an improvement on that achieved by the authors’ previous work in [New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations. J. Evol. Equ. 24 (2024), no. 2, Art. 42]. As far as we know, the results for the mKdVm with damping are new.

The main objective of this article is to present a detailed structure of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is the best possible.

We study automorphisms and invariants for the algebra ${\mathbb O}$ of octonions and octonionic slice regular functions $f:{\mathbb O}\to{\mathbb O}$.

This paper examines the stability of Couette flow in the two-dimensional incompressible magnetohydrodynamics system with only vertical dissipation. Our analysis departs from previous works by avoiding coordinate transformations and instead exploiting the special structure of the magnetic stream function. A key point is that the equation for the magnetic stream function $\phi$ avoids unfavourable linear terms present in the magnetic field formulation, which allows us to uncover new stability mechanisms specific to the anisotropically dissipative regime. Making use of this special structure, we establish global uniform regularity of solutions near background magnetic fields and provide a rigorous resolution of the associated stability problem.

We show that the existence of non-zero tropical forms of degree at least two implies that the tropical Chow group of points of an integral affine manifold is infinite-dimensional. This can be seen as a tropical analogue of classical results of Mumford and Roitman for Chow groups of smooth (complex) projective algebraic varieties. We also show that the existence of tropical 1-forms on integral affine surfaces does not imply infinite dimensionality by considering the case of a tropical Klein bottle.

We prove a quantitative isoperimetric inequality for nearly spherical domains in the Bergman ball in $\mathbb{C}^2$. We prove a Fuglede-type theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball, and it provides the first result on the isoperimetric phenomenon in the Bergman ball.

This paper addresses the simultaneous homogenization and dimension reduction of thin composite plates reinforced with rigid substructures, within the framework of non-linear elasticity. The reference (undeformed) configuration consists of a periodically perforated elastic matrix, where the holes are occupied by rigid inclusions. We first establish a decomposition of the deformation for such composite structures. Under the assumption that the thickness parameter $\delta$ is asymptotically smaller than the periodicity $\varepsilon$, we derive a reduced asymptotic model as both parameters tend to zero. Using rescaled unfolding operators, we characterize the limiting behaviour of the Green–St. Venant strain tensor. Through $\Gamma$-convergence, we obtain the homogenized limit energy and establish the existence of a minimizer. The resulting limit model is of constrained Kármán type, where the limiting displacement satisfies a first-order infinitesimal isometry constraint.