Colloquium

Reception at 3:30 p.m. in Petty 116

Nicolò Zava
Institute of Science and Technology Austria (ISTA)

4:00 p.m. – 5:00 p.m. in Petty 219

“An introduction to the role of the Gromov-Hausdorff
distance in computational topology”

Computational topology is a well-established field at the crossover of topology and computational geometry. It aims to transfer the power of topology for quantitative analysis to the setting of discrete approximations, typically finite data sets sampled from underlying objects. This analysis is usually performed by computing invariants of the spaces that extract patterns and features. The most important tool developed in this theory is persistent homology, summarized by persistence diagrams. Successful applications of these invariants to real-world datasets started a whole new field known as topological data analysis (TDA). Given the ubiquity of data in our modern world and their importance in data-driven science, a more interpretable, geometry-based approach to treating large amounts of data is crucial.

The Gromov-Hausdorff distance is a notion of dissimilarity between metric spaces introduced by Gromov to study the convergence of metric structures. Earlier notions can be found in Edward’s and Kadets’ works. In the past decade, it found applications in computational topology, where it provides a theoretical framework to shape recognition and dataset comparison. Studying the metric properties of this distance helps us understand the advantages and limits of specific invariants. Using notions from dimension theory, we can quantify the unavoidable loss of information.