Persistent Homology (PH) is one of the key methods in Topological Data Analysis (TDA) for capturing the intrinsic shapes and features of complex, high-dimensional data. In this seminar, we explore PH comprehensively from theoretical foundations to application techniques. Starting with the foundational aspects of PH, we describe the construction of persistent homology, elucidating how it captures topological features across various scales and remains stable under perturbations, making it a robust tool for data analysis.

 Besides the theoretical framework, we illustrate the practical application workflow of interdisciplinary research applying PH. From biology to material science, the application of PH provides novel methods to interpret the underlying topology of data.

 Finally, to further investigate the information hidden within persistent diagrams, we employ deep learning (DL) techniques to analyze the objects obtained from these diagrams. This integration with deep learning not only allows us to extract more topological information of data with advanced methods but also significantly enhances the interconnectivity of persistent homology with diverse academic disciplines.