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B0321 - TOPOLOGICAL DATA ANALYSIS

Academic Year 2022/2023

                
                        Docente:
                        Patrizio Frosini
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/03
                    
                        Language:
                        English
                    
                        Teaching Mode:
                        Traditional lectures
                        
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 5827)

                            Teaching resources on Virtuale
                        
                                    Course Timetable
                                
from Sep 27, 2022 to Dec 21, 2022

Learning outcomes

At the end of the course, the student knows the main theoretical results and techniques used in topological data analysis (e.g., persistent homology, Mapper), and some examples of their application to data comparison and machine intelligence.

Course contents

Introduction to Topological Data Analysis (TDA).

Topological groups. If X is a compact metric space, then the group Homeo(X) of all homeomorphisms of X is a topological group that acts continuously on C⁰(X, R). Definition of the natural pseudo-distance associated with a subgroup G of Homeo (X). Main properties of the natural pseudo-distance.

Some reminder of simplicial homology and singular homology. Persistent homology and persistence diagrams. Comparison of persistence diagrams via the bottleneck distance. Stability of persistence diagrams.

Multiparameter persistence. Main definitions in multiparameter persistence. The foliation method. Monodromy in multiparameter persistent homology.

Non-expansive equivariant operators (GENEOs). Definition of GENEO. Some theoretical results about GENEOs. Links between GENEOs and TDA.

Applications of Topological Data Analysis. Applications of persistent homology. Applications of group equivariant non-expansive operators.

Readings/Bibliography

H. Edelsbrunner and J.L. Harer, Computational topology: An introduction, American Mathematical Society, 2010.

Teaching methods

Lecture of traditional type.

Assessment methods

Written exercises and oral examination.

Teaching tools

See the web page http://www.dm.unibo.it/~frosini/DIDMAT.shtml

An informal and concise description of what TDA is can be found in many videos on Youtube (e.g., at this link: https://www.youtube.com/watch?v=nG_Veme7bqw )

Office hours

See the website of Patrizio Frosini