Summer Intern — Persistence Theory and Its Application

This summer, I studied persistence theory and its applications to time-series analysis and knot theory under the supervision of Prof. Martin Frankland at the University of Regina.

The filtration on a simplicial complex

 

We begin with a brief introduction to persistent homology, covering filtered complexes, barcodes, and persistence diagrams. We then explore two distinct applications:

• Time-series analysis — Inspired by the work of Perea and Harer, we implement sliding-window embeddings and one-dimensional persistence to detect periodic behavior in data.
• Knot theory — Following the approach of Celoria and Mahler, we apply persistence-based invariants to investigate geometric properties of embeddings of knots.

Our main contribution lies in developing Python and MATLAB tools that make these methods computationally accessible—supporting visualization and quantitative analysis of topological features in both temporal and geometric datasets.

📌 Code available on GitHub: github.com/wiliiiii/Persistence_Homology

The full report is available below: