Techniques from topological data analysis (TDA) have proven effective in studying time-dependent data arising in dynamic systems, such as animal swarming behavior and spatiotemporal patterns in neuroscience. While early algorithms leveraged efficient updates to persistence diagrams for dynamic data, they struggled to distinguish behaviors that are isometric at each fixed time but differ qualitatively. This limitation was addressed by Kim and Mémoli, who introduced a spatiotemporal persistence framework for dynamic metric spaces, resulting in multiparameter persistence modules. However, these modules pose computational challenges. To address this, we build on insights from Gómez and Mémoli, who observed that the homology of Rips complexes over size (2k+2) point subsets of a metric space–termed principal curvature sets–is both tractable and informative. We extend this idea to dynamic settings by introducing dynamic curvature-set persistent homology, applying the spatiotemporal framework of Kim and Mémoli to curvature sets. We prove that the resulting multiparameter persistence modules are interval-decomposable: in fact, they possess a stronger property we term antichain-decomposable. Utilizing this property, we present a new algorithm to efficiently compute the erosion distance d_E (due to Patel) between arbitrary antichain-decomposable modules (including, but not limited to modules produced by our construction). Additionally, our construction is stable with respect to a generalized Gromov-Hausdorff distance between time-dependent datasets proposed by Kim and Mémoli. This enables a robust computational pipeline for distinguishing dynamic data, as demonstrated in experiments with the Boids model, where we successfully detect parameter changes.

For n≥2, the n-th curvature set of a metric space X is the set consisting of all n-by-n distance matrices of n points sampled from X. Curvature sets fully capture the shape of datasets and they can be regarded as a geometric analogue of configuration spaces. In this paper we carry out a geometric and topological study of the curvature sets of the unit circle S1 equipped with the geodesic metric. Via an inductive argument we compute the homology groups of all curvature sets of S1 and, to provide a more detailed analysis, we also find a compatible simplicial complex structure.

We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put ζ_n:=arccos(-1n+1), the common geodesic distance between distinct vertices of a regular simplex with n+2 vertices inscribed in 𝕊^n. We prove that d_GH(𝕊^n,𝕊^n+1)=ζ_n/2 (n≥1), resolving a conjecture of Lim, Mémoli, and Smith. All cases n≥4 were previously open. This equality is established by explicitly constructing a family of correspondences ℛ_n⊆𝕊^n+1×𝕊^n, whose distortion matches the known quantitative Borsuk-Ulam lower bound ζ_n. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences ℛ_n yields new bounds for spheres of nonconsecutive dimensions, including lim_m→∞ d_GH(𝕊^m,𝕊^m+d(m)) = 4 whenever d(m)≥1, and d(m)=o(m).

A metric space Z gives rise to three natural classes of infinite-dimensional metric spaces associated to Z: p-Wasserstein spaces of probability measures on Z, nonlinear Lebesgue L^p-spaces of Z-valued maps, and p-Gromov-Wasserstein spaces of Z-valued kernels. The latter class, referred to as Z-Gromov-Wasserstein (Z-GW) spaces, extends the classical Gromov-Wasserstein framework from metric measure spaces to more general, possibly attributed, network-like structures, and unifies many GW-type distances that nowadays play a significant role in metric geometry, data science and machine learning. In this article we develop a unified metric-geometric theory of these three classes of spaces, with a particular focus on the Z-GW spaces. Our first main result identifies a fundamental submetry structure linking them: the nonlinear Lebesgue space maps via a submetry onto the Z-GW space, which in turn maps via a submetry onto the Wasserstein space. This structure provides a mechanism for transferring geometric information among the three spaces. We apply this framework to geodesics and Alexandrov curvature. For 1<p<∞, we prove that geodesicity of Z is equivalent to geodesicity of each of the three associated spaces; in the endpoint case p=1, all three associated spaces are geodesic, even when Z is not. We also characterize geodesics in the Z-GW space as generalized interpolations, extending a known characterization in the classical setting due to Sturm. Finally, we give a complete classification of Alexandrov curvature bounds for these spaces in terms of the curvature of Z. Thus, while the main focus of the paper is a new metric-geometric theory of Z-GW spaces, the submetry framework also extends classical theorems for Wasserstein and Gromov-Wasserstein spaces and yields new geometric consequences for nonlinear Lebesgue spaces.

For 0≤ m≤ n, let c_m,n be the infimum of scales at which the Vietoris–Rips filtration of the round sphere 𝕊^m admits a continuous odd map from 𝕊^n. A quantitative Borsuk–Ulam theorem gives c_m,n/2≤ d_GH(𝕊^m,𝕊^n), and it was asked whether equality always holds. Using a synchronized product-measure lift of the spherical join, we construct continuous odd maps between Vietoris–Rips metric thickenings with target scale equal to the maximum input scale. Iteration gives c_m+d,n+d≤ c_m,n for every integer d≥0. More generally, the finite join law adds m_i+1 and n_i+1 separately and bounds the resulting c-value by max_i c_m_i,n_i. For 0≤ r<π, the pairs (k,ℓ) with c_k-1,ℓ-1≤ r are closed under addition, so Fekete's lemma gives a limit for the maximal admissible ℓ/k as k→∞. This structure, exact values, and projective-code estimates give finite and asymptotic bounds. If 1≤ m_jc_m_j,n_j/2 for all sufficiently large j, producing infinitely many counterexamples.

In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to M\"obius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations.

Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology.

We introduce Orthogonal M\"obius Inversion $\mathsf{OI}$, a concept analogous to M\"obius inversion on finite posets, which is applicable to order-preservings functions from a finite poset to the Grassmannian $\mathsf{Gr}(V)$ of an inner product space $V$. This notion critically relies on the inner product structure on $V$ enabling it to capture much finer information than standard integer-valued persistence diagrams. Orthogonal Inversion is a special case of the broader concept of Orthomodular Inversion, where the target space is any orthomodular lattice, which we also identify. We apply Orthogonal Inversion in order to construct a "non-negative" persistence diagram for any given multiparameter filtration $\mathsf{F}$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$. This is done by applying it to the birth-death spaces of $\mathsf{F}$. Analogously to $1$-parameter classical persistence diagrams, these multiparameter Grassmannian persistence diagrams offer straightforward interpretability. Specifically, to a segment $(b, d) \in \mathsf{Seg}(P)$, (1) the Grassmannian persistence diagram canonically assigns a vector subspace of $C_{\rho}^K$ consisting of cycles that are born at $b$ and become boundaries at $d$ and (2) this assignment is exhaustive at the homology level. Finally, we relate our Grassmannian persistence diagrams to the recently introduced notion of M\"obius homology, thus enhancing its interpretability through the lens of our framework.

In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations.

We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module M indexed by a bifiltration of n simplices in O(n^3) time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has O(n^4) runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of M from O(n^4) to O(n^3). In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space Z, we define a Z-network to be a measure space endowed with a kernel valued in Z. We introduce a method for comparing Z-networks by defining a generalization of GW distance, which we refer to as Z-Gromov-Wasserstein (Z-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the Z-GW distance defines a metric on the space of Z-networks which retains desirable properties of Z, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

. In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset X of a Banach space Y, we analyze the topological features arising in the family N,pX & Abreve; Yq of nested neighborhoods of X in Y and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in Y, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the P8-principal components of X, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space X: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Cech and Vietoris-Rips extinction times of X by relating X to its convex hull and to its tight span, respectively.

We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results; one can quantify how seriously issues like sampling bias, noise, limited data, and approximations might change a given problem by bounding how much these modifications can move the problem under the Risk distance. With the distance established, we explore the geometry of the resulting space of supervised learning problems, providing explicit geodesics and proving that the set of classification problems is dense in a larger class of problems. We also provide two variants of the Risk distance: one that incorporates specified weights on a problem's predictors, and one that is more sensitive to the contours of a problem's risk landscape.

For an arbitrary finite group G, we consider a suitable notion of Gromov Hausdorff distance between compact G-metric spaces and derive lower bounds based on equivariant topology methods. As applications, we prove equivariant rigidity and finiteness theorems, and obtain sharp bounds on the Gromov Hausdorff distance between spheres.

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the p-edit distance for p∈ [1,∞], and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the p-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the p-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts.

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality.

We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^d-1 and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance.

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers $k\geq 0$ and $n\geq 1$ we consider the dimension $k$ Vietoris-Rips persistence diagrams of \emph{all} subsets of a given metric space with cardinality at most $n$. We call these invariants \emph{persistence sets} and denote them as $\mathbf{D}_{n,k}^\textrm{VR}$. We establish that (1) computing these invariants is often significantly more efficient than computing the usual Vietoris-Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris-Rips persistence diagrams, and (3) they enjoy stability properties. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy's inequality. We also identify a rich family of metric graphs for which $\mathbf{D}_{4,1}^\textrm{VR}$ fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris-Rips persistence diagrams using Mayer-Vietoris sequences. These yield a geometric algorithm for computing the Vietoris-Rips persistence diagram of a space $X$ with cardinality $2k+2$ with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. We consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category-theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result, we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants, for example the notion of spread introduced by M Katz. As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to the homotopy type of the Vietoris-Rips complexes of spheres, which follow from work of Katz, and characterization (rigidity)results for spheres in terms of their Vietoris-Rips persistence barcodes, which follow from work of F Wilhelm. Finally, we establish a sharp version of Hausmann's theorem for spheres which may be of independent interest

We study notions of persistent homotopy groups of compact metric spaces together with their stability properties in the Gromov-Hausdorff sense. We pay particular attention to the case of fundamental groups, for which we obtain a more precise description. Under fairly mild assumptions on the spaces, we proved that the classical fundamental group has an underlying tree-like structure (i.e. a dendrogram) and an associated ultra-metric.