Abstract

Persistent homology has been proposed in recent years and made rapid development both in theory and applications. For instance, it can be combined with dynamical systems, time series analysis, and machine learning in theoretical approach. It has also been used to quantify a diurnal cycle in hurricanes, to classify rocks from their binarized 3D images and to detect periodic sound signals in the medical field.

In this talk, I will introduce this approach to recover topological features of a point cloud sampled from a topological space. Then I will use it to reconstruct the topological information of the orbit of a point in a dynamical system through Taken’s theorem and sliding window embeddings. I will also discuss its application to wheeze detection, which helps achieve a higher accuracy compared to other methods. The talk is based on works of Perea and of Emrani, Gentimis, and Krim.