Advances in topological data analysis (TDA) require efficient algorithm designs that can extract algebraic structures hidden in the data. This talk centers around the theme that special geometric/combinatorial constructions underlying the algebra can facilitate designing efficient algorithms in TDA. As examples of this premise, we present three (four time permitting) cases: (i) how a geometric viewpoint on zigzag persistence in terms of the Mayer-Vietoris pyramid helped designing a fast algorithm for computing zigzag persistence from an input zigzag filtration, (ii) how the special structure of two-dimensional grid ($\mathbb{Z}^2$) helped designing an efficient algorithm for computing the generalized rank (rank of the limit-to-colimit map) for $2$-parameter persistence, (iii) how combinatorics helped designing efficient algorithms for multiparameter persistence, and (iii) how combinatorial multivectors representing dynamical systems helped designing efficient algorithms for computing algebraic summaries such as Connection Matrices and Conley-Morse barcodes.

We define a new framework that unifies the filtration and mapper approaches, and present efficient algorithms to compute it. Termed the box filtration of a point cloud data (PCD), we grow boxes (hyperrectangles) that are not necessarily centered at each point (in place of balls centered at points). We grow the boxes non-uniformly and asymmetrically in different dimensions based on the distribution of points. We present two approaches to handle the boxes: a point cover where each point is assigned its own box at start, and a pixel cover that works with a pixelization of the space of the PCD. Any box cover in either setting automatically gives a mapper of the PCD. We show that the persistence diagrams generated by the box filtration using both point and pixel covers satisfy the classical stability based on the Gromov-Hausdorff distance. Using boxes also implies that the box filtration is identical for pairwise or higher order intersections whereas the VR and Cech filtration are not the same. We demonstrate through examples that the box filtration can produce more accurate results to summarize the topology of the PCD than VR and distance-to-measure (DTM) filtrations. We also present results for a recent pipeline to construct box mappers of PCDs that scales efficiently to large dataset sizes and dimensions.

This is joint work with Prashant Gupta and Enrique Alvarado. A paper is available at https://arxiv.org/abs/2404.05859.

Recent advances in topological data analysis (TDA) and artificial intelligence (AI) provide new ways to study large collections of structured data through relations and their associated simplicial complexes. In this talk, we investigate datasets arising from collections of knot invariants by combining graph-based methods with Dowker and multiway Dowker constructions. Relations among classical and homological invariants give rise to simplicial complexes whose topology captures dependencies that are not visible through pairwise comparisons alone. We illustrate how these relational models reveal geometric structure associated with concordance invariants such as the knot signature and Rasmussen's s-invariant, and discuss how higher-order relations provide a new perspective on the organization of invariant datasets.

Mapper is a widely used technique in topological data analysis for representing high-dimensional datasets as graphs that reveal their underlying topological structure. By capturing connected components, branches, loops, and other global features, Mapper has been successfully applied across a wide range of scientific domains.

Part I of the tutorial reviews the mathematical foundations of Mapper, including its relationship to the Reeb graph, stability and convergence theory, statistical guarantees, and recent advances that have established a theoretical framework for the method. We also highlight emerging applications of Mapper in machine learning, particularly for analyzing high-dimensional latent representations from deep learning.

Part II provides a hands-on introduction to two interactive software tools, led by Dhruv Meduri and Xinyuan Yan. Mapper Interactive is an open-source platform for constructing, visualizing, and exploring Mapper graphs from general high-dimensional datasets, enabling users to investigate topological structures alongside the underlying data. Explainable Mapper integrates Mapper with explainable AI (XAI) techniques to interpret machine learning models by relating topological structures in learned representations to model predictions and feature attributions.

By combining recent theoretical advances with practical software tools, this tutorial equips participants with both a principled understanding of Mapper and hands-on experience applying it to the exploration, analysis, and interpretation of complex high-dimensional data.

I will discuss methods for analyzing complex datasets based on adding geometric/topological/measure-theoretic attributes to merge trees and Reeb graphs generated from the data. This will be a two-part talk/tutorial, where I intend to simultaneously explain the underlying theory and illustrate it via interactive code examples. This will be based on a few papers which approach these ideas from different perspectives, which are joint work with Justin Curry, Haibin Hang, Washington Mio, Osman Okutan, and Florian Russold. Interactive notebooks associated to the talk will be available here: https://github.com/trneedham/Graph-Based-Methods-In-TDA.

In this talk, I will introduce the notion of labeled interleaving between two labeled Reeb graphs. There exist several distances defined for Reeb graphs, including the interleaving distance. However, these are generally NP-hard to compute even for the case where the Reeb graph is a tree. Inspired by the labeled interleaving between merge trees, we study the corresponding notion of labeled Reeb graphs and the labeled interleaving distance. I shall start by reviewing the main concepts: thickening and smoothing of Reeb graphs and the interleaving distance between Reeb graphs. We then study the behavior of nodes under smoothing, and then I will describe what it means for an interleaving map to be label-preserving for Reeb graphs. Labeled interleaving maps allow us to define the labeled interleaving distance between two labeled Reeb graphs. Our main theorem is that the ordinary interleaving distance can be recovered with some labeling of all essential nodes. Moreover, I will talk about the algorithms to compute the labeled interleaving distance for Reeb graphs. In particular, in the case that the Reeb graph is a tree, the labeled interleaving distance can be computed in polynomial time.

Mapper is a construction from topological data analysis that turns a dataset \(\hat{X}\) into a graph, or more generally, a (simplicial) complex intended to summarize the large-scale structure of the dataset. Its construction is done with three additional ingredients: a lens function from \(\hat{X}\) to a lens space \(Z\), a cover of \(Z\), and a clustering algorithm on \(\hat{X}\). Mapper was originally introduced as a data-driven approximation to the Reeb graph (more generally, space) of \(X\), the space that our data \(\hat{X}\) was sampled from. Previous work has used notions from category theory to study the convergence of the mapper graph to the Reeb graph when the lens function takes values in Euclidean space and the diameters of the cover elements uniformly tend to zero.

In this talk, we present recent results from an ongoing project with Bala Krishnamoorthy, where we use tools from metric geometry to study the convergence of mapper complexes to Reeb spaces. Our framework applies when the the underlying space \(X\) comes equipped with a pseudometric and the lens function maps takes values in an arbitrary metric space.

Although metrics on mapper graphs have been studied, the distinguishing feature of our metric is that it is derived from a natural metric on the Reeb space. Under mild assumptions on the underlying metric spaces and lens function, the topology induced by this natural metric agrees with the quotient topology on the Reeb space. We will discuss additional advantages of this Reeb space metric, and show how it can be approximated from a finite sample \(\hat{X}\) of \(X\). We describe how it can be used to equip the Mapper complex with a metric and then show how they Gromov-Hausdorff convergence of the metric mapper complex to the metric Reeb space as the diameters of the covers uniformly go to zero. Time permitting, we will conclude by discussing how we will be using this framework to create computationally efficient algorithms for selecting optimal parameters for mapper.

The Mapper algorithm is a fundamental tool in exploratory topological data analysis. Derived from the nerve construction, Mapper graphs can contain additional information about clustering density when considering the higher-dimensional skeleta. To capture1-dimensional topology, we construct the Mapper graph's 2-dimensional complex: 2-Mapper. We also introduce a new algorithm implementation of Multiscale Mapper to choose 2-Mapper parameters that reflect persistent Betti-1 information. With this tool, we study how cover choice affects 2-Mapper output, and introduce new hypothesis testing methods to understand stability of 1-dimensional cycles under parameter perturbations.

Open-source software (OSS) development is a central topic at the intersection of IS and organizational research. Research in this area has consistently shown that OSS development processes are dynamic, non-deterministic, and emergent. In this presentation, I introduce a new method and a new theory for studying how OSS development processes change over time. The new topological data analysis (TDA) approach, Temporal Mapper, allows us to theorize the OSS development process as a complex dynamical system. Specifically, it enables us to visualize and conceptualize the emergence of recurrent patterns of action in the OSS development process. Due to a lack of data and methods, early work on organizations as complex dynamical systems was largely metaphorical. Recent advancements in topological data analysis (TDA), such as Temporal Mapper, make it possible to conduct rigorous empirical studies, especially in areas like OSS where detailed data are available. These methodological advancements create new opportunities to revisit earlier scholarly interests in viewing organizations as complex dynamical systems.

Neural networks learn high-dimensional representations that capture rich semantic and structural relationships, yet understanding these representations remains a fundamental challenge for explainable AI. This work will present how Mapper graphs can serve as interactive visual abstractions that help domain experts explore, interpret, and compare neural representation spaces. First, I introduce Explainable Mapper, a framework for interpreting Mapper graphs of language model representations. By combining Mapper-based exploration with LLM-assisted explanation and perturbation-based verification, the framework enables users to automatically generate and validate hypotheses about the linguistic properties encoded in Mapper elements, including nodes, edges, paths, connected components, and trajectories. Second, I present TopoAlign, which extends the interpretation of a single Mapper graph to the comparison of Mapper graphs constructed from representations of different models, layers, training epochs, and modalities. Through topology-aware alignment and coordinated visualizations, TopoAlign enables users to compare global structures, identify corresponding local subgraphs, and inspect fine-grained alignment patterns, such as feature splitting, merging, and disentanglement. Together, these two works illustrate how visual analytics with Mapper graphs supports the understanding of neural representations from explanation to comparative alignment.

The field of neuroscience produces complex, multimodal datasets for revealing insight into the nervous system, and ultimately, the brain. A recent experimental trend in social neuroscience is simultaneously recording brain and behavior in naturalistic settings. Here, brain activity is captured via internal recording, and behavior is tracked using computer vision via video recording. Both brain and behavior recordings produce complex time series data, where extracting meaning relationships remains a key challenge. The currently available data analysis tools implement time averaging techniques, omitting the moment-to-moment dynamics within time series data. Therefore, new data analysis tools are needed to capture and characterize the fine-grained changes occurring in complex, high-dimensional time series data. In this talk, I will present my work using Temporal Mapper, a topological data analysis method, to represent complex time series data as transition networks. Here, we applied Temporal Mapper to characterize the moment-to-moment dynamics between freely moving ferrets. We found that brain and behavioral states during social interaction are less stable compared to non-social behavior. Our future work will develop cross-scale mappings between brain and behavioral transition networks.