Special Session 122: Topological Data Analysis Theory, Algorithms, and Applications

Hausdorff vs Gromov-Hausdorff distances

Henry Adams University of Florida USA Co-Author(s):

Abstract: Tools from topology can bound or compute quantities arising in metric geometry. An example geometric quantity is the Hausdorff or Gromov-Hausdorff distance between two metric spaces (or datasets). Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. An example topological tool is the nerve lemma: a good cover of a space faithfully encodes the shape of that space. When X is a sufficiently dense subset of a closed Riemannian manifold M, we can use the nerve lemma to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved, and even obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) when M is the circle.

Higher-order connectomics of human brain function

Enrico Amico University of Birmingham England Co-Author(s):    Enrico Amico

Abstract: Traditional models of human brain activity often represent it as a network of pairwise interactions between brain regions. Going beyond this limitation, recent approaches have been proposed to infer higher-order interactions from temporal brain signals involving three or more regions. However, to this day it remains unclear whether methods based on inferred higher-order interactions outperform traditional pairwise ones for the analysis of fMRI data. In this talk I will introduce a novel approach to the study of interacting dynamics in brain connectomics, based on higher-order interaction models. Our method builds on recent advances in simplicial complexes and topological data analysis, with the overarching goal of exploring macro-scale and time-dependent higher-order processes in human brain networks. I will present our preliminary findings along these lines, and discuss limitations and potential future directions for the exciting field of higher-order brain connectomics.

Studying 2-Parameter Persistent Homology via Directed Topology

Robin Belton Vassar College USA Co-Author(s):    Robyn Brooks, Brittany Terese Fasy, Lisbeth Fajstrup, and Elizabeth Vidaurre

Abstract: Although multi-parameter persistence modules contain more information than single-parameter persistent homology, they are more difficult to understand, visualize, and integrate into a data analysis pipeline. We propose a new construction for relating the parameter space of a 2-parameter persistence module to a directed space. We discuss how this construction relates the set of 1-d persistence diagrams along non-decreasing paths for a particular 2-parameter persistence module, to the space of directed paths in the parameter space of that module. From this construction, we can use directed collapses to simplify the parameter space while still maintaining information to understand the module.

ZZFormer: A Sliding Window Zigzag Persistent Homology Transformer for Repetitive Biological Sequences

Dhananjay Bhaskar University of Wisconsin-Madison USA Co-Author(s):    Kritika Kumari, Levi Meng Hong Svaren

Abstract: Repeated patterns occur in biological sequences, neural activity, and in tokenized representations of data across many domains. Modeling such repetition is central to sequence analysis, but remains challenging for modern learning methods. While transformer architectures can capture long-range dependencies, they do not explicitly account for recurrence or cyclic structure, which are key features in many sequences. We introduce a topological framework for learning on sequences with repetition using zigzag persistent homology. As a sliding window moves along a sequence, this construction detects and tracks repeated patterns through the appearance and persistence of loops. Building on this idea, we develop ZZFormer, a transformer architecture that incorporates zigzag topological structure directly into token embeddings, enabling end-to-end learning on variable-length sequences. By encoding repetition explicitly, ZZFormer provides a way to incorporate topological structure into sequence models. This framework applies broadly to problems involving repeated or cyclic patterns. We demonstrate its effectiveness on tasks involving repetitive DNA, including classification of transposable element superfamilies and orders.

TopoFormer: Topology Meets Attention for Graph Learning

Baris Coskunuzer University of Texas at Dallas USA Co-Author(s):    Md Joshem Uddin, Astrit Tola, Cuneyt Gurcan Akcora

Abstract: In this talk, we show that reshaping topological ideas to fit modern deep learning pipelines can substantially improve graph learning. We present \textbf{TopoFormer}, a lightweight framework that injects topological inductive bias into transformer based models by converting each graph into a short sequence of topology aware tokens. The core module, \textbf{Topo-Scan}, slices a graph using node or edge filtrations to produce an ordered token sequence that captures multi scale structure, from local motifs to global organization, in a form naturally suited to attention. Unlike classical persistent homology pipelines, Topo-Scan is parallelizable, avoids costly persistence diagram computations, and integrates cleanly into end to end training. We provide stability guarantees for the proposed encodings and demonstrate state of the art performance on graph classification and molecular property prediction benchmarks, matching or surpassing strong GNN and topology based baselines while remaining scalable and compute predictable.

Dynamics meets topological data analysis

Pawel Dlotko Warsaw University Poland Co-Author(s):

Abstract: \begin{abstract} Topological data analysis (TDA) is usually associated with point clouds and scalar filtrations, but many problems in dynamics naturally come with richer objects: finite trajectory segments, delay-coordinate reconstructions, or vector fields known either analytically or only through sampling. In this talk I will discuss a collection of recent TDA-inspired methods that shift the focus from the topology of static data to the comparison of dynamical systems themselves \cite{Carlsson2009,EdelsbrunnerHarer2010}. On the one hand, I will present tests for topological conjugacy constructed from finite orbit samples and time series, including a formulation based on Takens reconstructions, which makes it possible to assess whether two observed processes encode the same underlying dynamics \cite{Takens1981,DlotkoLipinskiSignerska2024}. On the other hand, I will describe new descriptors for sampled and continuous vector fields---including begin--end point embeddings, densities of directions, Euler characteristic curves, and Euler characteristic profiles---designed to compare dynamics directly at the level of trajectories or vector fields, detect qualitative changes such as bifurcations, and remain computationally feasible in data-driven settings \cite{MarszewskaSignerskaRynkowskaDlotkoInPrep}. Taken together, these methods show that ideas originating in TDA can be turned into practical tools for the quantitative and qualitative analysis of dynamics well beyond the standard point-cloud setting: from finite samples of trajectories to sampled or continuous vector fields, and from discrete observations to analytically defined systems. \end{abstract} \begin{thebibliography}{99} \bibitem{Carlsson2009} G.~Carlsson, \newblock Topology and data, \newblock {\em Bulletin of the American Mathematical Society} 46(2):255--308, 2009. \bibitem{EdelsbrunnerHarer2010} H.~Edelsbrunner and J.~Harer, \newblock {\em Computational Topology: An Introduction}, \newblock American Mathematical Society, 2010. \bibitem{Takens1981} F.~Takens, \newblock Detecting strange attractors in turbulence, \newblock in {\em Dynamical Systems and Turbulence, Warwick 1980}, Lecture Notes in Mathematics 898, Springer, 1981, pp.~366--381. \bibitem{DlotkoLipinskiSignerska2024} P.~D{\l}otko, M.~Lipi{\`n}ski, and J.~Signerska-Rynkowska, \newblock Testing topological conjugacy of time series, \newblock {\em SIAM Journal on Applied Dynamical Systems} 23(4):2939--2982, 2024. \bibitem{MarszewskaSignerskaRynkowskaDlotkoInPrep} M.~Marszewska, J.~Signerska-Rynkowska, and P.~D{\l}otko, \newblock Topological Characteristics for the Analysis of Vector Fields: New Stable Characteristics for Discrete and Continuous Dynamical Systems, \newblock in preparation.

Probabilistic Statements for Mapper Graphs

Halley Fritze University of Michigan USA Co-Author(s):    Enrique Alvarado, Robin Belton, Nicholas Della Pesca, Aine Doherty

Abstract: The Mapper algorithm is a popular exploratory data analysis tool for visualizing the underlying graphical structure of a dataset. The Mapper algorithm involves several user specified parameters that make it possible for any graph to be a Mapper graph. We investigate the question of how likely we are to see specific types of subgraphs within a Mapper graph, when data points are randomly sampled from a continuous probability density function. We provide both theoretical and experimental results for how likely we are to get cyclic subgraphs within a Mapper graph. Additionally, we discuss how we can generalize these probabilistic results and use them as a starting point for a statistical inference framework with Mapper graphs.

Generalised Wasserstein Metrics on Persistence Diagrams via Banach Sequence Ideals

Fernando Galaz-Garcia Durham University England Co-Author(s):    Mauricio Che, Fernando Galaz-Fontes

Abstract: Persistence diagrams may be thought of as countable multisets of points in a metric space $X$, considered relative to a distinguished nonempty closed subset $A\subset X$. Their comparison is usually based on the bottleneck distance or on $p$-Wasserstein distances. In this talk, I will present a common framework in which these distances arise as instances of a more general construction involving a normalized permutation-invariant Banach sequence ideal $E$, with the norm of $E$ used as the cost function in the corresponding optimal transport problem between persistence diagrams. This yields a family of spaces of persistence diagrams parametrized by the Banach sequence ideal $E$, extending earlier definitions of diagram metrics. I will explain natural assumptions on $E$ under which the resulting matching formula defines a metric and the associated spaces of persistence diagrams are complete, separable, and geodesic. In this way, the framework enlarges the class of metrics available for applications while preserving the main structural properties of the classical diagram spaces. The bottleneck and $p$-Wasserstein distances are recovered as the special cases $E=\ell^\infty$ and $E=\ell^p$.

How do biological neural networks learn topologically structured data?

Chad Giusti Oregon State University USA Co-Author(s):    Nikolas Schonsheck

Abstract: Populations of neurons in the brain encode complex information through the aggregate activity -- so called \emph{spikes} -- of neurons in those populations. Dimensionality reduction methods are commonly used to qualitatively investigate such population activity, driving much modern understanding of the function of neural systems. However, when the represented information carries non-trivial topology, these methods can obscure salient structure. Topological methods have been successfully employed to characterize non-linear organizing principles of neural population activity. As these work without dimensionality reduction, we can use them to investigate fundamental questions about population behavior. For example, is propagation of nonlinear coordinate systems a generic feature of biological neural networks, or must this be learned? If learning is necessary, what mechanisms can produce it? In this talk, we will apply recent developments in TDA to demonstrate that simple Hebbian spike-timing dependent plasticity reorganizes feed-forward networks to correctly propagate toroidal coordinate systems. We also observe the emergence of geometrically non-local receptive field types, postdicting several experimentally observed phenomena. These observations provide quantitative support for the hypothesis that simple biologically plausible plasticity mechanisms suffice explain topological organization observed in real neural systems.

Shapes, Spaces, Simplices, and Structure: Geometry, Topology, and Machine Learning

Bastian Grossenbacher-Rieck University of Fribourg Switzerland Co-Author(s):

Abstract: A large driver contributing to the undeniable success of deep learning models is their ability to synthesize task-specific features from data. For a long time, the predominant belief was that ``given enough data, all features can be learned.`` However, as large language models are hitting diminishing returns in output quality while requiring an ever-increasing amount of training data and compute, new approaches are required. One promising avenue involves focusing more on aspects of modeling, which involves the development of novel \emph{inductive biases} such as invariances that cannot be readily gleaned from the data. This approach is particularly useful for data sets that model real-world phenomena, as well as applications where data availability is scarce. Given their dual nature, geometry and topology provide a rich source of potential inductive biases. In this talk, I will present novel advances in harnessing multi-scale geometrical-topological characteristics of data. A special focus will be given to show how geometry and topology can improve representation learning tasks. Underscoring the generality of a hybrid geometrical-topological perspective, I will furthermore showcase applications from a diverse set of data domains, including point clouds, graphs, and higher-order combinatorial complexes.

The Shape of Orbits in the Circular Restricted Three-Body Problem: Opportunities for Topological Data Analysis

Davide Guzzetti Auburn University USA Co-Author(s):    Davide Guzzetti

Abstract: Within an unperturbed central-body gravitational field, Keplerian orbital elements provide both a coordinate representation and an intuitive geometric description, enabling clear visualization of orbit properties and intuitive design of spaceflight solutions. In contrast, for the Circular Restricted Three-Body Problem (CR3BP), no compact and expressive parameterization exists that plays an analogous role to Keplerian elements or other two-body coordinate sets. This limitation creates a disconnect between commonly used state representations and salient orbit features that are readily observable yet not rigorously encoded. Topological data analysis (TDA) offers a promising pathway to bridge this gap by augmenting coordinate descriptions with topological, potentially geometry-aware, signatures and distance metrics that capture the intrinsic shape of orbits. Such representations may enable a more faithful characterization of nonlinear dynamical behavior and provide a principled basis for comparing underlying structures and signals linked to orbit motion. Extending these ideas across models of varying fidelity, as well as to observational sensor data streams, opens new opportunities for enhanced space domain awareness and orbit motion inference in cislunar environments.

Quantum computing and persistence in topological data analysis

Ryu Hayakawa Kyoto University Japan Co-Author(s):    Ryu Hayakawa

Abstract: It has been recently revealed that surprising connections exist between topological data analysis and quantum computing. In this talk, I will show a quantum algorithm for estimating persistent Betti numbers, which is a central quantity in TDA. Moreover, I will show a complexity theoretic evidence that quantum computing exponentially outperforms classical computing for certain problems in TDA.

New Method for Analyzing The Hole-Structure of a Crystal: Merge Tree for Periodic Data

Teresa Heiss-Synak the Australian National University Australia Co-Author(s):    Herbert Edelsbrunner

Abstract: Periodic data is abundant in materials science; for example, the atoms of a crystalline material repeat periodically. Additionally, periodic boundary conditions are used in many simulations, for example in molecular dynamics simulations of materials, to remove boundary effects. However, it is unclear how to deal with the periodicity of the data when computing topological descriptors, like the merge tree or persistent homology, which track connected components or holes at different length scales. A classical approach is to compute the respective descriptor simply on the torus. However, this does not give the information needed for many applications and is even unstable under certain types of noise. Therefore, we suggest decorating the merge tree gained from the torus with additional information, describing for each connected component on the torus how many components of the infinite periodic space map to it. As there are often infinitely many, we describe their density and growth rate inside a growing sphere. The resulting periodic merge tree and its induced periodic 0-th persistence barcode carry the desired information and satisfy the desired properties, in particular: stability and efficient computability (under mild assumptions, the running time is of order $(n + m)log(n)$, where $n$ and $m$ are the number of vertices and edges per fundamental domain).

Robustness of persistent homology when noising and denoising 3D images

Jonathan Jaquette New Jersey Institute of Technology USA Co-Author(s):    Ebru Dagdelen, Aakash Karlekar, Manav Arora, Matthew Illingsworth, Jonathan Jaquette, Linda J. Cummings, Lou Kondic

Abstract: The classical bottleneck stability theorem tells us that two functions which are close in the sup norm will yield persistence diagrams which are close in the bottleneck distance; that $C^0$ perturbations may only produce small changes in the lifespans of generators. There is no guarantee, however, on the number of generators which may be produced by such a perturbation. Indeed, millions of such generators may arise when studying noisy datasets, presenting an obstacle to both (1) the actual computation of persistent homology (PH) of large 3D images, and (2) any analysis of PH which incorporates the number of generators. As such, it is often necessary to denoise the data before computing its PH. In this talk, we analyze the PH of synthetic 3D images of porous media in the presence of spatially uncorrelated noise, and perform a comparative analysis of various topological measures (e.g. bottleneck distance, Wasserstein distance, persistence statistics, persistence images, etc.) to assess their robustness to the noising and denoising process.

Detecting Stochasticity in Discrete Signals via Persistent Homology

Firas Khasawneh Michigan State University USA Co-Author(s):    Sunia Tanweer

Abstract: Distinguishing deterministic chaos from stochastic dynamics in discrete time series remains a fundamental and challenging problem, particularly in the presence of noise and limited data. Prior work has explored subjective metrics like entropy as a framework for capturing structural invariants between such signals---motivating the search for a robust, theoretically grounded method. In this work, we develop a nonparametric, theory-driven method based on excursion statistics of stochastic processes. Building on classical results on the persistent homology for continuous semimartingales, we exploit a universal scaling law relating the number of barcodes of size $\eps$ to the quadratic variation of the signal. This yields a simple and interpretable diagnostic: stochastic diffusion processes exhibit an inverse-squared scaling in barcodes/excursion counts, while deterministic systems---periodic or chaotic---systematically violate this behavior. Using this principle, we construct a data-driven classification framework that operates directly on a single observed time series without embedding, symbolic transformations, or model training. We demonstrate the effectiveness of the method across a wide range of systems, including canonical stochastic diffusions, deterministic chaotic maps and flows, noisy hybrid systems, and real-world datasets including financial time series and audio signals.

Interleaving distance as a Galois-edit distance

Woojin Kim KAIST Korea Co-Author(s):    Won Seong

Abstract: The concept of \emph{edit distance}, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By contrast, the \emph{interleaving distance}, introduced in the 2000s within the study of persistent homology, has become a foundational metric in topological data analysis. In this work, we show that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as a so-called \emph{Galois-edit distance}. The key lies in clarifying a connection between the Galois connection and the interleaving distance, via the established relation between the interleaving distance and free presentations of persistence modules. In addition to offering new perspectives on the interleaving distance, we expect that our findings will facilitate the study of stability properties of invariants of multi-parameter persistence modules. As an application of the edit formulation of the interleaving distance, we present an alternative proof of the well-known bottleneck stability theorem.

Digitalizing the Euler Characteristic Transform

Henry KIrveslahti Univerrsity of Southern Denmark Denmark Co-Author(s):    Xiaohan Wang

Abstract: The Euler Characteristic Transform of Turner et al. Is a TDA-inspired tool that can be used for statistical shape analysis. This transform enjoys many desirable mathematical properties that make it very appealing for applications. However, the transform is typically discretized, and this process complicates the maximal exploitation of the beautiful theory behind it. In this talk, we present how this transform can be represented in a truly lossless way in the spirit of digitalization of the Kendall Shape Space. We demonstrate that this digital representation is practically feasible to compute and can be made auto-differentiable to benefit from synergies with deep learning. We will also outline some directions for theoretical research that would allow us to tackle more ambitious applied problems with this new representation. This is joint work with Xiaohan Wang.

The shadow of Vietoris-Rips complexes in limits

Atish J Mitra Montana Technological University USA Co-Author(s):    Sushovan Majhi, Kazuhiro Kawamura

Abstract: The Vietoris-Rips (VR) complex of a metric space at a certain scale is an abstract simplicial complex where each k-simplex corresponds to (k+1)-point sets with diameter less than that scale. The pioneering work by Hausmann established that any closed Riemannian manifold is homotopy equivalent to its Vietoris-Rips complex for sufficiently small scales. This fundamental result naturally motivated the finite reconstruction problem of a manifold with a finite Hausdorff close sample, which was answered by Latschev. For any abstract simplicial complex whose vertex set is a Euclidean subset, its shadow is the union of the convex hulls of its simplices. We consider the homotopy properties of the shadow of VR complexes, along with the canonical projection map from the VR complex to its shadow. The study of the geometric/topological behavior of this projection map is a natural yet non-trivial problem, and the map may have many ``singularities`` which have been partially resolved only in dimensions up to 3. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from shape theory, we show that in the limit when the scale approaches zero and the sample becomes increasingly dense, the limit map behaves well with respect to homotopy/homology groups when the space is an ANR (absolute neighborhood retract) and admits a metric that satisfies some regularity conditions. This results in Hausmann-type and Latschev-type reconstruction theorems in the limit.

Building Canopies for the Decomposition of Persistence Bundles

Elizabeth Munch Michigan State University USA Co-Author(s):    Elizabeth Munch, Barbara Giunti

Abstract: Static persistence, that is the persistent homology computed for a single filtration function $f:K \to \mathbf{R}$, has been highly utilized throughout topological data analysis. The first work looking at the persistent homology for a parameterized family of functions, $f_{p}:K \to \mathbf{R}$, came in the form of vineyards by Morozov et al, where the parameterization was over some interval, i.e. $p \in [a,b]$. More recently, there has been a growing interest in parameterized persistence where $p \in B$ for some more complex base space. For example, Turner et al introduced the Persistent Homology Transform (PHT), which studies the persistence of a family of functions on an embedded space $|K| \subset \mathbf{R}^n$ parameterized over $p \in \mathbf{S}^{n-1}$. The most general form of this came with the introduction of persistent homology bundles by Hickok, for arbitrary base space $B$. This parameterized input data holds interesting structure; of note is the discovery of monodromy in these bundles, a point in the persistence diagram might not come back to the same place while tracing out a closed loop in the base space. In this talk, we will introduce the concept of a \textit{canopy}: a topological space with some additional structural data reminiscent of a bundle which stores the representatives of persistence classes over the parameterizing base space $B$. Canopies are particularly useful since they are well defined, even in the presence of monodromy. We show that for nice enough parameterized families of functions, we can construct such a canopy and give structure theorems for the types of issues that can give rise to monodromy.

Balancing Geometry and Density with Fermat Metrics: Graphs, Clustering, and Applications

James M Murphy Tufts University USA Co-Author(s):

Abstract: The curse of dimensionality renders statistical and machine learning in high dimensions intractable without additional assumptions on the underlying data. We consider geometric models for data that allow for mathematical performance guarantees and efficient algorithms that break the curse. Specifically, this talk considers a family of data-driven metrics that balance between density and geometry in the underlying data, known as Fermat distances. We consider discrete graph operators based on these metrics and prove performance guarantees for clustering with them in the spectral graph paradigm. Fast algorithms based on Euclidean nearest-neighbor graphs are proposed and connections with continuum operators on manifolds are developed.

Learning Generative Models on Persistence Diagram Space via Policy-Induced Markov Processes

Farzana Nasrin University of Tennessee Knoxville USA Co-Author(s):    Farzana Nasrin and Ernesto Ugona

Abstract: Persistence diagrams (PDs) are central objects in topological data analysis, capturing multiscale geometric structure in a stable and interpretable form. While statistical methods for analyzing collections of PDs are well developed, learning probability distributions on diagram space remains challenging due to variable cardinality and geometric constraints. We propose a generative framework for persistence diagrams based on stochastic dynamics on diagram space. Specifically, we construct Markov processes driven by local edit operations---addition, relocation, and deletion of points---combined with projection mechanisms that enforce diagram validity. We show that the resulting process is irreducible, aperiodic, and geometrically ergodic, and therefore admits a unique stationary distribution. To learn from data, we parameterize the transition dynamics via policies and design reward functions that approximate distributional discrepancies between the stationary distribution and empirical diagram distributions. This establishes a connection between policy optimization and distribution matching on non-Euclidean spaces. Experiments on synthetic and real datasets demonstrate that the proposed approach captures key structural properties of diagram distributions, including persistence statistics and rare topological events.

Autonomous Applications of Topologically-Informed Data Driven Analytics & Fusion

Paul T Schrader Air Force Research Laboratory (AFRL) USA Co-Author(s):    Paul T. Schrader

Abstract: This decade`s `digital wild` is inundated with high rate, voluminous combinations of physics- and human-based data. The challenge is further exacerbated by rapidly accelerating autonomous architectures lacking explainability, compromised/poisoned heterogeneous data, and managing denied/degraded/intermittent/low-bandwidth environments. These and other prohibitive symptoms weaken user certainty while concurrently increasing compute complexities and reducing edge processing compatibility, resulting in failed support of (near) real-time decision speed. Addressing these challenges motivates rigorous scientific creativity though multi-disciplinary approaches, particularly leveraging the rich mathematical properties of a network`s upstream data and their fused aggregates. Computational and algebraic topology implemented by topological data analysis (TDA), both classical and emerging, provide access to these properties and a path to modality/model agnostic fusion. This talk exhibits one such ongoing success, an efficient and scalable TDA Machine Learning (TDAML) algorithm and examines several case studies recently developed including applications in surveillance, time-series forecasting, data assurance, and materials/systems inspection. After a brief overview introducing TDA, an overarching dissection of the TDAML along with a survey of its initial object detection/classification studies and emerging implementation family are presented. Finally, future directions are discussed for the TDAML leveraging arbitrary physics- and human-based data analytics/fusion towards reliable and deployable situational/state/systems/material/cyber awareness. Distribution Statement A. Approved for public release: distribution is unlimited. AFRL-2026-0970.

Using topology to inform embedding

Michael Small University of Western Australia Australia Co-Author(s):    Eugene Tan

Abstract: Embedding via time delay reconstruction is a fundamental step in reconstructing deterministic dynamical systems from time series data. In this talk I will review two techniques, due to topological data analysis )TDA) that allow one to better construct a time delay embedding and to be evaluate performance of models of dynamics in that reconstructed space. First, TDA is used to select a geometrically tuned value of time delay for the reconstruction. Second, TDA applied to reconstructed attractors will be used to provide a metric to evaluate model performance. Time permitting I will introduce new methods to use information theory to optimise the modelling process with reservoir computing.

Relative commutative algebra of multigraded modules

Anastasios Stefanou University of Bremen Germany Co-Author(s):    Fritz Grimpen, Matthias Orth

Abstract: A fundamental problem in applied algebraic topology is the efficient computation of minimal presentations and minimal free resolutions of homology modules arising from multigraded torsion-free complexes. While substantial progress has been made in the bigraded setting, the general multigraded case remains largely open. After a brief introduction to the algebraic framework of applied algebraic topology, I will present joint work with Fritz Grimpen on the construction and minimization of free--injective presentations of multigraded modules. I will then discuss ongoing work with Matthias Orth and Fritz Grimpen on developing relative Groebner basis techniques for submodules of quotients of free modules, and in particular for submodules of injective modules. These methods provide new tools for computing minimal resolutions of such modules, including, as special cases, modules given either as homology of chain complexes or via free--injective presentations.

Topological model selection: a case-study in tumour-induced angiogenesis

Bernadette Stolz Max Planck Institute of Biochemistry Germany Co-Author(s):    RA McDonald, HM Byrne, HA Harrington, T Thorne

Abstract: Topological data analysis (TDA) offers powerful tools for studying biological phenomena. In this talk, I will present a recent application to spatial and dynamic biomedical data. Specifically, I will discuss a case study of topological model selection in tumour-induced angiogenesis, the process in which blood vessel networks are formed during tumour growth. While many mathematical models of tumour-induced angiogenesis exist, significant challenges persist in objectively evaluating and comparing their outputs. We develop a flexible pipeline for parameter inference and model selection in spatio-temporal models. Our pipeline identifies topological summary statistics which quantify spatio-temporal data and uses them to approximate parameter and model posterior distributions. We validate our pipeline on models of tumour-induced angiogenesis, inferring four parameters in three established models and identifying the correct model in synthetic test-cases.

Nested sequential inference for hotspots in noisy images with cubical persistent homology

Andrew M Thomas University of Iowa USA Co-Author(s):    Ranjan Maitra

Abstract: Given an image that primarily consists of noise, one may wish to perform segmentation of the image into regions of signal deemed ``hotspots``. Of equal importance is quantifying the statistical signficance of each of these estimated hotspots, also known as peaks or regions of interest. In this paper, we develop a topological data analysis method---which we deem ``imphr``---to sequentially estimate hotspots and assess the statistical significance of each subsequent noise hypothesis in the image domain outside the union of the iteratively identified regions of interest. Our method employs newly developed advanced statistical methodology from the sequential testing and selective inference literatures to ensure both a highly rigorous and computationally efficient algorithm. We apply said algorithm to simulated data, and compare it with various methods for hotspot detection and spatial clustering in the literature; we also demonstrate its performance on real-world vibrothermography and fMRI data. In sum, the imphr algorithm not only provides rigorous theoretical guarantees and estimated hotspots with meaningful topology, but also yields performance as good and in many cases better than existing methods.

Using Persistent Homology to Analyze Access to Heterogeneous-Quality Resources and Heterogeneous-Severity Nuisances

Sarah Tymochko College of the Holy Cross USA Co-Author(s):    Gillian Grindstaff, Abigail Hickok, Jerry Luo, Mason Porter

Abstract: Ideally, all public resources (e.g. hospitals, grocery stores, vaccination clinics, etc.) should be distributed in a way that is fair and equitable to everyone. However, this is not always the case. Quantifying how much (or little) access individuals have to certain resources is a complex problem. Previous work has shown that tools from topological data analysis (TDA) can be useful in determining holes in the locations of resource locations based on geographic locations and travel times [Hickok et al., Persistent homology for resource coverage: a case study of access to polling sites, SIAM Review, 2024]. Some resources may necessitate incorporation a notion of quality. We develop a framework for studying the coverage of resources of heterogeneous quality. As a case study, we look at public parks to determine how equitably green space is distributed. This topological framework for studying the coverage heterogeneous resources can be extended in many application areas. For example, it could be used to analyze the coverage of nuisances, or sites that one wants to avoid.

TDA-driven parameter inference in an agent-based model of zebrafish patterns

Alexandria Volkening Purdue University USA Co-Author(s):    Yue Liu

Abstract: Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the example of pattern formation in zebrafish, which are named for their dark and light stripes. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells, making agent-based models a natural approach for describing cell behavior. However, stochastic, microscopic models are often not analytically tractable using traditional techniques, and parameter inference in biologically detailed, spatial agent-based models faces significant challenges. With this motivation, here I will describe how we are combining techniques from topological data analysis and approximate Bayesian inference to quantify structure in messy, cell-based patterns and infer parameter values.

Topological decoding of grid cell activity via path lifting to covering spaces

Iris Yoon Wesleyan University USA Co-Author(s):    Jared Yao

Abstract: High-dimensional neural activity often reside in a low-dimensional subspace, referred to as neural manifolds. Grid cells in the medial entorhinal cortex provide a periodic spatial code that are organized near a toroidal manifold, independent of the spatial environment. Due to the periodic nature of its code, it is unclear how the brain utilizes the toroidal manifold to understand its state in a spatial environment. We introduce a novel framework that decodes spatial information from grid cell activity using topology. Our approach uses topological data analysis to extract toroidal coordinates from grid cell population activity and employs path-lifting to reconstruct trajectories in physical space. The reconstructed paths differ from the original by an affine transformation. We validated the method on both continuous attractor network simulations and experimental recordings of grid cells, demonstrating that local trajectories can be reliably reconstructed from a single grid cell module without external position information or training data. These results suggest that co-modular grid cells contain sufficient information for path integration and suggest a potential computational mechanism for spatial navigation.

Topological optimization with birth and death cochains

Ling Zhou Duke University USA Co-Author(s):    Thomas Weighill

Abstract: We introduce the notion of birth and death cochains as generalized versions of birth and death simplices in persistent cohomology. We show that birth and death cochains (unlike birth and death simplices) are always unique for a given persistent cohomology class. We use birth and death cochains to define birth and death content as generalizations of birth and death times. We then demonstrate the advantages of using that birth and death content as loss functions on a variety of topological optimization tasks with point clouds, time series and scalar fields. We close with a novel application of topological optimization to a dataset of arctic ice images.