Content

Objective

-This course bridges the gap between abstract algebraic topology and data science.
It covers the mathematical foundations of persistent homology, stability theory, and the algorith-
mic implementation of topological summaries. The course concludes with modern applications
in machine learning, including vectorization methods and topological deep learning.

Course Content

This doctoral course provides a rigorous introduction to Topological Data Analysis (TDA). It covers the mathematical foundations of simplicial complexes and algebraic homology, followed by the theory and computation of Persistent Homology. Key topics include the construction of filtrations (Vietoris-Rips, Cech), stability theorems (Bottleneck and Wasserstein distances), and efficient algorithms for computing topological invariants. The course concludes by integrating TDA with Machine Learning through vectorization methods (Persistence Landscapes, Images), the Mapper algorithm, and Topological Deep Learning applications

Course Learning Outcomes

1.Understand the mathematical foundations of simplicial homology and persistent homology.
2. Master the algorithms used to compute topological invariants from point cloud data.
3. Analyze the stability and statistical properties of topological summaries.
4. Apply TDA tools (Mapper, Persistence Landscapes) to real-world datasets and integrate
them with Machine Learning pipelines.

References

•Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction.
AMS.
• Zomorodian, A. J. (2005). Topology for Computing. Cambridge University Press. (Ex-
cellent for algorithms and complexity).
• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. (The standard
reference for rigorous Homology theory).
• Oudot, S. Y. (2015). Persistence Theory: From Quiver Representations to Data Analysis.
AMS.

Assessment System