Compers 2025 Abstracts

The identification of dynamics from time series data is a problem of general interest. It is well established that dynamics on the level of invariant sets, the primary objects of interest in the classical theory of dynamical systems, is not computable. We recall a coarser characterization of dynamics based on order theory and algebraic topology and prove that this characterization can be identified using approximations that can be realized by feedforward neural networks.

[Abstract]

Mapper graphs are a widely used tool in topological data analysis and visualization, offering discrete summaries of the shape and connectivity of high-dimensional data. They can be viewed as approximations of Reeb graphs, with nodes capturing local neighborhoods and edges reflecting overlaps. A natural way to compare two mapper graphs is the interleaving distance, quantifying how much they need to be “stretched” to align. But this distance is NP-hard to compute. In this talk, I’ll introduce a loss function that upper bounds the interleaving distance and is computationally feasible. Using a categorical formulation, we optimize this loss via integer linear programming to estimate graph similarity. Preliminary results on small examples show great promise, suggesting this approach could make rigorous mapper graph comparison practical and scalable.

In some applications of topological data analysis one might consider only zero-dimensional homology. The zero-dimensional homology of a space is conceptually and computationally simpler than higher dimensional homology since it can be characterized as the linearization of the set of path-connected components of the space. In particular, the zero-dimensional homology modules of a filtration indexed by a poset P will in general form a proper subcategory of rep P (to the contrary, add Im H_i is dense for i greater than zero). We show that when the indexing poset P is a rooted tree, only finitely many indecomposables can be obtained as zero-dimensional homology of a filtration. In fact, all these indecomposables are reduced rooted tree modules in the sense of Kinser. Moreover, we give an algorithm for the efficient decomposition of rooted tree modules, generalizing the so-called elder rule used in TDA for linear quivers. This is a report on joint work with Riju Bindua and Luis Scoccola.

In this talk, I will introduce a new framework, poset cocalculus, which is a variant of functor cocalculus that is defined for functors from a poset to a model category. The motivation for this framework lies in the relevance for multipersistence of functors from posets to the model category of chain complexes over a field, as any multipersistence module is the homology of such a functor. I will present several applications of poset cocalculus, including showing that the Vietoris-Rips filtration is the degree 2 approximation of the Čech filtration, and that the interval decomposability of a middle-exact bipersistence module is a consequence of the fact that any degree 1 bipersistence module is interval decomposable.

The p-presentation distance is an l^p-type generalization of the interleaving distance defined for multiparameter persistence modules and merge trees. Despite recent NP-hardness results for the computation of presentation distances, certain fundamental aspects of these distances are still poorly understood. For example, is infimum in the definition of the p-presentation distance actually attained? By appealing to linear optimization, we answer this question in the affirmative for p=1 and provide bounds on the size of compatible presentations and the length of a sequence realizing the 1-presentation distance. We conjecture that the infimum is attained for arbitrary p, and state several open problems which, if solved, would result in partial or full resolution of the conjecture. This is joint work with Mike Lesnick.

In a k-critical bifiltration, every simplex enters along a staircase with at most k steps. Examples with k>1 include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a k-critical bifiltration into a 1-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes minimal free resolutions dimension-wise and connects them with lifts and homotopies. These maps are computed by finding paths in the degree-wise resolutions viewed as path graphs. The algorithm is simple to describe and generally performs well in practice as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because the resulting chain complex can contain a dense boundary matrix arising from long paths. Our second algorithm replaces the degree-wise path graphs with ones that maintain short paths. This leads to almost linear runtime and output size. The algebraic structure of this second algorithm can be used to compute free resolutions of chain complexes coming from trifiltrations. However, in this more general case, the running time is cubic.

Joint with Håvard Bjerkevik and Fabian Lenzen. The ability to decompose multiparameter persistence modules in smaller - at best indecomposable - parts would speed up the computation of other (additive) invariants and generally help in their analysis. Unfortunately, decompositions are unstable. Bauer and Scoccola (2022) even showed that every persistence module is arbitrarily close to being indecomposable. Since the release of AIDA we know that this is not only a theoretical limitation: The decomposition of the Persistent Homology of density-scale bifiltrations often contains large components. The pruning construction introduced by Bjerkevik (2025) shows promise to solve these problems. We will give an overview of its construction and devise an algorithm to compute it.

A flat-injective presentation of a persistence module M characterizes M as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be easily represented by a single monomial matrix, completely describe the persistence module, and can be used as starting point to compute other invariants of it, such as the rank invariant, persistence images, and others. If M has finite total dimension, then it has a finite free resolution F_• and a finite injective resolution I^•, and a flat-injective presentation of M is given by the morphism φ: F₀ → I⁰. We present an algorithm to compute a monomial matrix representing φ from the matrices representing F_• (or I^•), using a simple correspondence between free and injective resolutions of M. The algorithm, of which a Julia implementation is available, has cubic run time in the total rank of F_•.

Joint with Marc Fersztand. The skyscraper invariant is a filtration of the classical rank invariant introduced by Fersztand, Jacquard, Tillmann, Nanda (2024), which is stable with respect to the interleaving distance. It is defined via the Harder-Narasimhan filtration of the module along each central charge. In this talk we will give an overview of this invariant, explain its advantages, and introduce an algorithm that efficiently computes it for practically appearing density-scale bifiltrations. The code is published at https://github.com/JanJend/Skyscraper-Invariant

Topological Data Analysis (TDA) provides a powerful framework for uncovering structure in complex data through its underlying shape. This talk highlights recent advances in visualizing maps between high-dimensional spaces to uncover correlations across datasets and in extending TDA to settings where representative sampling is infeasible. Key developments include integrating TDA with machine learning to analyze infinite or unstructured data domains while enhancing the interpretability of learning outcomes. A central focus is the application of these methods to knot theory, where the rapid growth in knot complexity makes the space of knots and their invariants a natural testbed for mathematical big data. This is joint work with P. Dlotko and D. Gurnari.

Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C^2. We recently proved that any manifold with positive reach can be approximated arbitrarily well by a C^\infty manifold without significantly reducing the reach. This result implies that nearly all theorems established for C^2 manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C^2, for free! I will present this result in more detail and touch upon the techniques we used to prove it --- partitions of unity and smoothing with convolution kernels.

Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, a generalization of metric trees whose decomposition does not have a “prime” component. Their close relationship with trees makes totally decomposable spaces attractive in the search for spaces whose persistent homology can be computed efficiently. We study the subclass of circular decomposable spaces, finite metrics that resemble subsets of S^1 and can be recognized in quadratic time. We give an O(n^2) characterization of the circular decomposable spaces whose Vietoris-Rips complexes are cyclic for all distance parameters and compute their homotopy type using well-known results on VR_r(S^1). We extend this result to a recursive formula that computes the homology of certain circular decomposable spaces that fail the previous characterization. Going beyond totally decomposable spaces, we identify an O(n^3) decomposition of in terms of the blocks of the tight span of X, and use it to induce a direct-sum decomposition of the homology of VR_r(S^1).

Let S^n be the n-sphere with the geodesic metric and of diameter π. The intrinsic Čech complex of S^n at scale r is the nerve of all open balls of radius r in S^n. In this talk, I will explain how to control the homotopy connectivity of intrinsic Čech complexes of spheres at each scale between 0 and π in terms of coverings of spheres. The upper bound on the connectivity, which is sharp in the case n=1, comes from the chromatic numbers of Borsuk graphs of spheres. The lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of S^n. These bounds imply the new result that for n≥1, the homotopy type of the Čech complex of S^n at scale r changes infinitely many times as r varies over (0,π).

Many problems in TDA have solutions in homological — and indeed, linear — algebra. However, it's challenging to harness these solutions, computationally. Matrices are often large, having millions or billions of rows and columns. They are indexed by simplices, rather than integers. They have coefficients in abstract fields and require exact numerical accuracy. They have unusual sparsity patterns. This talk will introduce an open-source library to address some of these problems, Open Applied Topology (OAT). OAT is a high-performance linear algebra solver with a user-friendly front end. It allows the user to perform mathematical operations including matrix/vector addition, multiplication, and factorization (R=DV, RU=D, U-match), and to compute persistence diagrams, barcodes, (optimal) (co)cycle representatives, and induced maps. Users can easily link the library to new types of chain complexes (simplicial, cubical, etc.), and to Python libraries such as SciPy. In sum, OAT is a user-friendly tool for matrix algebra in TDA. If time remains, we will examine broader trends in TDA software development across the research landscape (government, private, academic), and opportunities for future growth and cooperation.

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a "nice" Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on G-means clustering which searches for the optimal number of clusters in k-means by iteratively applying the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly faster than a previous iterative method.

We study the intersection problem for families of non-homogeneous ellipsoidal regions in Euclidean space and its relevance to geometric and topological data analysis. The condition for intersection is reduced to a single convex optimization problem whose minimum value determines the critical radius at which all ellipsoids meet. The associated set function governing this radius satisfies the axioms of an LP-type problem, which gives us many algorithms to profile for constructing efficient implementations. Beyond its algorithmic interest, the framework provides a principled replacement for isotropic Euclidean balls in persistent homology, capturing local anisotropy and curvature while suppressing spurious topological features.

The generation of a filtered simplicial complex is a foundational and critical part of the persistent homology pipeline. We will discuss recent work, in collaboration with Ulrich Bauer, on fast generation of Vietoris-Rips complexes. Zomorodian's algorithm consumes edges in order and builds up all the cofaces at once. Bauer's groundbreaking software Ripser in turn, enumerates simplices dimension by dimension in a (reverse) lexicographic order, as cofacets of the previous dimension's simplices, and leaves the sorting to a later stage of the algorithm. As an alternative approach, we propose generating simplices dimension by dimension as cofacets of the previous dimension, but in a way that guarantees simplices appear directly in filtration order.

This work explores how Topological Data Analysis (TDA) can be applied to art recognition. Using persistent homology, we analyze 100 paintings from ten artists—both Western and Indian—by transforming each image into grayscale, color channels, and edge-detected versions. These representations capture topological and geometric features unique to each artist’s style. Bottleneck and Wasserstein distances are used to compare persistence diagrams, revealing distinct topological signatures across artists. A permutation test confirms that these topological differences are statistically significant. The infra-artist analysis shows that even minor or non-significant features carry stylistic meaning when considered together, while the intra-current study demonstrates that TDA can differentiate artists within the same visual movement. The same framework, when applied to AI-generated art, reveals clear structural and stylistic differences from human-created works.

The Gromov–Hausdorff distance measures the similarity between two metric spaces by embedding them (isometrically) into a common ambient metric space. Since its introduction, it has found numerous applications beyond pure mathematics. The prevalence of directed networks in data science has recently brought increased attention to spaces endowed with a notion of direction. In this work, we introduce an analogue of the Gromov–Hausdorff distance for metric d-spaces that we define as the collection of metric spaces equipped with a directed structure. We refer to this new distance as the directed Gromov–Hausdorff distance and denote it by ⃗d_GH. Our construction of ⃗d_GH involves a new notion of isometry on d-spaces called d-isometry and a variant of the Hausdorff distance computed with respect to an extended metric called the zigzag distance. Additionally, by exploring different formulations of the classical Gromov–Hausdorff distance, we introduce two more distances between directed spaces: the directed distortion distance and the d-correspondence distortion distance. Interestingly, while these distances are equivalent to the Gromov–Hausdorff distance in the undirected setting, we show that this equivalence does not hold for directed metric spaces. We also derive some properties of the newly introduced distances. (This is a joint work with Lisbeth Fajstrup, Brittany Terese Fasy, Wenwen Li, Tatum Rask, Francesca Tombari and Ziva Urbančič).

Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance, which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but (as conjectured in Gasparovic et al.) we prove that it is true for the intrinsic bottleneck and interleaving distances realized by infinitesimal path length in merge tree space. Our proof suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).

The talk is based on the papers [1,2,3,4] extending Topological Data Analysis (TDA) to a wider area of Geometric Data Science, which continuously parametrizes moduli spaces of real objects under practically important equivalences. The key example is a cloud A of unordered points under isometry in R^n. For all standard filtrations (Vietoris-Rips, Cech, Delaunay) of complexes on A is invariant under isometry (any distance-preserving transformation). Hence, persistence can be considered a partial solution to the following problem: design an invariant I on clouds of m unordered points satisfying the following conditions. (a) Completeness: any clouds A,B in R^n are related by rigid motion if and only if I(A)=I(B); (b) Realizability: the invariant space {I(A) for all clouds A in R^n} is explicitly parameterized so that any sampled value I(A) can be realized by a cloud A, uniquely under motion in R^n; (c) Bi-continuity: the bijection from the space of clouds to the space of complete invariants is Lipschitz continuous in both directions in a suitable metric d on the invariant space; (d) Polynomial-time: the invariant I, the metric d, and a reconstruction of A in R^n from I(A) can be obtained in a polynomial time in the size of A, for a fixed dimension n. The talk will outline a full solution to this problem, which makes sense for any data (embedded graphs, meshes, or complexes) and relations (dilation, affine, or projective maps) instead of clouds and rigid motion. [1] V.Kurlin. Complete and continuous invariants of 1-periodic sequences in polynomial time. SIAM Journal on Mathematics of Data Science, 2025. [2] P.Smith, V.Kurlin. Generic families of finite metric spaces with identical or trivial 1-dimensional persistence. J Applied and Computational Topology, v.8, p.839–855 (2024). [3] D.Widdowson, V.Kurlin. Recognizing rigid patterns of unlabeled point clouds by complete and continuous isometry invariants with no false negatives and no false positives. Proceedings of CVPR 2023, p.1275-1284. [4] Y.Elkin, V.Kurlin. Isometry invariant shape recognition of projectively perturbed point clouds by the mergegram extending 0D persistence. Mathematics, v.9(17), 2121 (2021).

This talk is based on our recent work arXiv:2506.21488, which establishes connections between three of the most prominent metrics on persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.

Persistent homology (PH) is one of the main methods used in Topological Data Analysis. An active area of research in the field is the study of appropriate notions of PH representatives, which allow to interpret the meaning of the information provided by PH, and is thus an important problem in the application of PH, and in the study of its interpretability. Computing optimal PH representatives is a problem that is known to be NP-hard, and one is therefore interested in developing context-specific optimality notions that are computable in practice. Here we introduce time-optimal PH representatives for data that grows over time, i.e., where new data points may be added but not deleted as time passes, allowing one to extract representatives that are close in time in an appropriate sense. We illustrate our methods on point clouds arising from embeddings of univariate time series, and on temporal networks, using synthetic data, as well as time series arising from climate models, and temporal networks originating from money-laundering models. We show that our methods provide optimal PH representatives that are better suited for these types of problems than existing optimality notions, such as length-optimal PH representatives.

In this talk I will use the framework of poset-stratified spaces to study games, where reward can be both discrete and continuous. Following work by Yuliy Baryshnikov, I will show how certain video games naturally give rise to stratified spaces. Surprisingly, when modern neural nets are trained to play these same video games, a similar stratification structure can be observed in their latent representations. Our methods follow recent work by Michael Robinson and others on using Volume Growth Laws to detect non-manifold structure in the token space for LLMs. We expand and strengthen Robinson’s analysis by considering non-textual data and prove a realization result for volume growth in a stratified space. This is joint work with Brennan Lagasse, Ngoc B. Lam, Gregory Cox, David Rosenbluth, and Alberto Speranzon.

The barcode of a filtration and its representative cycles encode rich information, but are computationally expensive. Methods that update the barcode and cycles often strive for a general approach, inurring computational overhead, or effect particular changes in specific filtrations, limiting generalizability. We provide an implementation in C++ to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), executes provably minimal changes to the factorization of the boundary matrix, and also intrinsically updates the representative cycles. We show that the complexity of our algorithm, when removing a fixed number of simplices, is lower than recomputing the barcode from scratch, with both theoretical and experimental methods. This is joint work with Barbara Giunti. Funding is provided by the Latvian Council of Science grant 1.1.1.9/LZP/1/24/125.

A Sudden Stratospheric Warming (SSW) is an extreme wintertime circulation event in the Northern Hemisphere. There have been many technical definitions of SSW's proposed over the last 60 years, but none have been widely accepted. By using Persistent Homology to detect and categorize the shape of Polar Vortices, we introduce a new constant-free definition that uses a scale of intensity rather than a cut-off. This approach is very promising as it seems to agree with all previous ones when different intensities are considered.

Persistent Topological Laplacians (PTLs) are both the extension of the graph Laplacian to filtered simplicial complexes and a discrete version of the Hodge Laplacian. Their spectra provide multiscale geometric and topological information that cannot be detected by other methods, such as persistent homology. This additional information comes at a significant computational cost, making large scale data analysis challenging. To improve the scale and broaden the impact of PTL applications, we introduce several computational techniques as part of the open-source Persistent Topological Laplacian Software (PETLS), including a novel algorithm using the topology of the underlying complex.

Traditional analyses of brain functional connectivity using fMRI data rely on Pearson correlation between brain regions: a static, time-averaged measure. While informative, this approach overlooks rich temporal dynamics. Here, we leverage the edge time series—the framewise product of fMRI activity between brain region pairs, whose temporal mean recovers the Pearson correlation—to capture fine-grained neural coordination (Faskowitz et al., 2020; Zamani Esfahlani et al., 2020; Owen et al., 2021). To our knowledge, we are the first to apply the Mapper algorithm—a topological data analysis (TDA) method (Singh et al., 2007)—to edge time series, enabling a visualization of brain-wide co-fluctuation patterns over time, which clinicians can use directly. This approach fills a key methodological gap in capturing the shape of edge-level dynamics across spontaneous (resting) and task-evoked conditions. We also find that mesoscale properties of the resulting Mapper graphs are significantly associated with individual differences in self-reported psychopathology.

We present the foundations of the theory of persistent cohomology operations and construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from these are strictly sharper than those obtained using persistent homology.