Shape, Dynamics, and Biology: Topological Data Analysis in Action

Topology interfaces with data science in two complementary ways: it provides robust, deformation-invariant descriptors that quantify the shape and structure of data, and it enables visualizations of high-dimensional data that preserve topological features. Grounded in algebraic and combinatorial topology, these ideas yield practical tools for data analysis when paired with efficient computation.

In the first part of this talk, I will survey topological methods developed in my group for analyzing complex and multi-valued data, including constructions based on the Euler characteristic and techniques inspired by phylogenetics. I will emphasize non-stationary settings where data come with dynamics- either as trajectories (ordered point clouds) or as vector fields- and discuss how TDA-based invariants can capture dynamical structure and support reliable classification of different dynamics. I will also touch on recent developments in using topological summaries for statistical hypothesis testing, particularly nonparametric

goodness-of-fit testing in high dimensions, and on interpretable visualization strategies based on Mapper and its generalizations.

In the second part, I will present two applications of these tools to multiscale biological organization. First, I will show how persistent-homology descriptors of super-resolution microscopy quantify differentiation-dependent reorganization of CTCF nuclear clusters, revealing how RNA-binding proteins and the lncRNA Pantr1 contribute to chromatin-topology maturation and strengthening of architectural loops. Second, I will discuss Topological Malignant Regions (TopMR), a persistent-homology framework capturing multiscale malignant morphology while embedding malignant–immune spatial interactions into a unified topological interaction space. Together, these case studies illustrate how TDA links structural features to gene-regulatory stability and clinically relevant tumor–microenvironmental patterns.