Invited Speakers
Title:
Graph-based signatures in topological data analysis
Abstract:
Topological data analysis (TDA) is a growing field that extracts topological summaries from data by leveraging ideas from algebraic topology as well as tools from computational geometry. Rather than measuring distances precisely, TDA asks coarser questions: how many connected components does a dataset have? How many loops? How does shape change across scales? The answers are encoded in topological signatures, which are compact descriptors that are stable under noise and useful in many settings. While persistent homology is perhaps the most famous example, other particularly rich sets of descriptors are graph-based. For example, given a scalar function on a topological space, the Reeb graph captures how the level sets of that function connect and merge, yielding a graph whose structure reflects the topology of the underlying domain. The mapper graph is a practical, combinatorial approximation of the Reeb graph well-suited to point cloud data. Merge trees record the birth and death of connected components as a threshold sweeps through function values, forming a rooted tree that encodes hierarchical structure. Vineyards track how these persistence diagrams evolve as the underlying function changes continuously over time, yielding a dynamic, film-strip view of topological change. These objects sit naturally at the intersection of graph theory and topology, and raise compelling algorithmic questions. This talk will introduce several of these graph-based signatures, with attention to their combinatorial structure and the computational complexity of comparing and visualizing them, with the goal of giving graph drawing researchers both a foothold in TDA and a sense of where their expertise might open new doors.
Biography:
Dr. Erin Wolf Chambers is the Schnyder Mission Family Collegiate Professor in the Department of Computer Science and Engineering at the University of Notre Dame, with a concurrent appointment in the Department of Applied and Computational Mathematics and Statistics. Her research focus is on computational topology and geometry, with a more general interest in combinatorics and combinatorial algorithms. She serves on the CGWeek Steering Committees and on the SafeTOC organizing committee, as well as being an editor for the Journal of Applied and Computational Topology. Prior to joining Notre Dame, she was a professor at Saint Louis University from 2008-2024, where she served as Department Chair from 2022-2024. She received her PhD in Computer Science in 2008 and her MS in Mathematics in 2006 from the University of Illinois at Urbana-Champaign, and was a Visiting Research Professor at Saarland University in summer 2011.
