We study metric spaces that in some sense thin out at infinity. We define and investigate a measure of sparsity that is a quasi-isometry invariant, and introduce an analogue of topological ends for sparse spaces that is also invariant under quasi-isometries. We study some 51F30examples arising in various contexts.

We prove a singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition $\mathsf{TCD}^e_p(0,N)$, yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.

Multispecies spatial data arise in many applications where interactions between different entities are central to system behaviour, including biomedical imaging, geospatial analysis, and species ecology. Despite their importance, relatively few quantitative tools exist to capture such interactions. In this work, we propose magnitude-based features for the analysis of multispecies spatial data. Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale. We develop global and local magnitude feature vectors and demonstrate their utility on synthetic tumour microenvironment data, and in tissue microarray data from human colorectal cancer samples. Locally, the method identifies distinct neighbourhood types and reveals spatial heterogeneity; in the model, this includes radial patterns associated with different qualitative outcomes of the simulations, while in the real-world data it reflects the importance of tertiary lymphoid structure-like interactions between B and T cell populations. Globally, the approach recovers known classifications of long-term simulation outcomes across parameter regimes in synthetic data, and suggests important roles for CD4+ T cells and CD163+ macrophages in distinguishing patients with favourable Crohn's like reactions from unfavourable diffuse immune infiltration. Together, these results suggest that magnitude-based features provide a powerful and flexible tool for the analysis of multispecies spatial data.

We show that if $M$ is a metric space of Nagata dimension at most $d$, then there exists an atlas on $M$ modeled on $\mathbb R^d$ such that every Lipschitz map $f:M\to Y$ (with values in an arbitrary Banach space $Y$) can be uniformly approximated by maps that are affine, and thus $\mathcal{C}^1$-smooth, with respect to this atlas. The construction relies on random metric partitions and stochastic retractions inside Lipschitz-free spaces. As an application, we introduce approximate continuous upper gradient $X$-structures (ACUG $X$-structures) on metric spaces and prove that every space of finite Nagata dimension carries an ACUG structure modeled on a superreflexive Banach space. Finally, adapting a proof due to Bourgain, we show that if $M$ has an ACUG superreflexive-structure, then the Lipschitz-free space $\mathcal{F}(M)$ has Pelczyński's property (V*). In particular, at least in the compact case, our result recovers all previously known examples of metric spaces $M$ for which $\mathcal{F}(M)$ has property (V*).

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\pi/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/\Gamma$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $\Gamma$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\pi/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

We present a constructive version of the Brøndsted-Rockafellar theorem in general geodesic metric spaces. Applications include a constructive form of the Caristi theorem and quantitative relations between metric slope error bounds and the global growth of functionals.

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.

We prove that the conjectured capillary Blaschke-Santaló inequality holds for any unconditional, strictly convex capillary hypersurface when $θ\in \left(0, \tfracπ{2}\right)$. Moreover, for $θ\in \left(\tfracπ{2}, π\right)$, we show that the capillary volume product has no finite upper bound.

Robertson and Seymour proved that for every finite tree $H$, there exists $k$ such that every finite graph $G$ with no $H$ minor has path-width at most $k$; and conversely, for every integer $k$, there is a finite tree $H$ such that every finite graph $G$ with an $H$ minor has path-width more than $k$. If we (twice) replace ``path-width'' by ``line-width'', the same is true for infinite graphs $G$. We prove a ``coarse graph theory'' analogue, as follows. For every finite tree $H$ and every $c$, there exist $k,L,C$ such that every graph that does not contain $H$ as a $c$-fat minor admits an $(L,C)$-quasi-isonetry to a graph with line-width at most $k$; and conversely, for all $k,L,C$ there exist $c$ and a finite tree $H$ such that every graph that contains $H$ as a $c$-fat minor admits no $(L,C)$-quasi-isometry to a graph with line-width at most $k$.

A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch separation and valence at most $n$, contains a quasisymmetrically ''universal'' element, that is, an element of this class into which every other element can be embedded quasisymmetrically. We also show that every quasiconformal tree with uniform branch separation quasisymmetrically embeds into $\mathbb{R}^2$. Our results answer two questions of Bonk and Meyer from 2022, in higher generality, and partially answer one question of Bonk and Meyer from 2020.