The aim of the course is to provide an introduction to the theory of persistent homology and to its application in the context of data analysis. Topological data analysis is a recent and fast growing research area developing new topological and geometric tools to infer features of complex data.

This course aims to present the basics of the mathematical background of TDA. In particular the following topics will be discussed:

• simplices and simplicial complexes;
• boundary operators and simplicial homology;
• filtrations of a simplicial complex;
• Cech/Vietoris-Rips/alpha complexes;
• persistent homology and persistence diagrams;
• distances between persistence diagrams;
• stability of persistence diagrams;
• examples of TDA.

Software for computing persistence diagrams of complex data will be presented.