Theory and Applications of
Topological Data Analysis

In this talk, we will describe how to compute the Minimal Intersection Radius (MIR) of growing, non-homogeneous ellipsoids in arbitrary ambient dimension, using a geometric method and techniques from convex optimization. We will also show how to phrase the statement as an LP-type problem, where the computation from convex optimization acts as a certificate. We will then compare different implementations of these computations on various random inputs. Lastly, we provide a comparison with similar (but crucially different) problems that appeared in the literature and show that finding the MIR is, in general, not equivalent to finding the minimal enclosing ellipsoid.

Understanding community structure in large-scale networks is a fundamental problem in network science with applications in social networks, biological systems, and information diffusion. Community detection in large-scale social networks requires balancing structural accuracy and computational scalability. Curvature-based approaches, particularly Ollivier-Ricci curvature (ORC), have demonstrated strong ability to identify inter-community bridges and structural bottlenecks, but optimal-transport-based curvature computation is expensive and limits applicability to large graphs.

We propose a two-stage curvature-guided pipeline that integrates Lower Ricci Curvature (LRC) as a fast combinatorial preprocessing step with ORC and discrete Ricci flow applied on a reduced graph. The LRC stage efficiently filters weak edges using triangle-based local structure, significantly reducing graph size before performing transport-based curvature computation. We evaluate the framework on SNAP Facebook ego-networks, the SNAP YouTube social network, and synthetic networks. Results show that the combined LRC-ORC pipeline achieves accuracy close to ORC-only preprocessing while substantially reducing computational cost, suggesting that integrating combinatorial and geometric curvature notions yields a scalable and structurally principled approach for community detection in complex networks.

Joint work with Christina Bergonzo and Anthony Kearsley

A wide range of medical treatments depend on artificially produced monoclonal antibodies. Similar amino acid sequences can generate biomolecules which adopt different shapes in 3-dimensional space. Molecular dynamics simulations are a valuable tool for revealing potential arrangements of the atoms in these proteins. Topological data analysis detects and quantifies structural features which are not easily measured by classical data analysis techniques. We discuss results from using topological data analysis to explore molecular dynamics simulations of monoclonal antibodies, with a focus on the NIST monoclonal antibody (NISTmAb) reference material. We describe matrix summaries which can be used for visualization as well as subsequent classification, clustering, or machine learning tasks, and discuss methods for including additional data along with spatial location when generating topological summaries.

In this talk, I will introduce the Rips-type ellipsoid complex — a geometrically informed variant of the classical Rips complex. Instead of using Euclidean balls, we construct the complex from ellipsoids whose axes align with tangent directions estimated from the data via PCA. I will present a stability result showing that small perturbations of a point cloud lead to controlled changes in the resulting barcodes. Finally, I will describe experiments on datasets from Turkeš, Montúfar, and Otter [TMO22], where ellipsoid barcodes outperform both the Rips and alpha complexes in classification tasks.

The Mapper algorithm is a popular tool for visualization and data exploration with applications ranging from discovering new types of breast cancer to classifying plants. However, it is difficult to use because the user needs to input values for several parameters, and the "best" values are unknown. In this talk, I'll discuss how my collaborators, students, and I have approached parameter selection in this setting, focusing on studying the inverse problem, proving probabilistic statements, and developing parameter selection algorithms.

The Weisfeiler–Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry — meshes with identical connectivity but different embeddings are indistinguishable. In this talk, we introduce the Geometric Simplicial Weisfeiler–Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. We show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), this yields a geometric expressivity characterization together with an approximation framework.

Abstract to be announced.

Mapper graphs are a widely used tool in topological data analysis and visualization, offering insight into the shape and connectivity of complex data as discrete approximations of Reeb graphs. Given a high-dimensional point cloud 𝕏 equipped with a function f : 𝕏 → ℝ, a mapper graph summarizes the topological structure induced by f, where each node represents a local neighborhood and edges connect overlapping neighborhoods.

Our focus is the interleaving distance for mapper graphs, which quantifies their similarity by measuring the extent to which they must be "stretched" to become comparable — computing this distance is NP-hard in general. We present a categorical formulation of mapper graphs and introduce the first computational framework for estimating their interleaving distance via an associated loss function. We formulate the problem as an integer linear program to find an optimal assignment, and demonstrate that on small examples our optimized bound matches the exact interleaving distance. We also show results from an image benchmarking experiment where pairwise mapper loss values are used for image classification, illustrating the practical potential of this approach.