B0321 - TOPOLOGICAL DATA ANALYSIS

Learning outcomes

At the end of the course, students acquire essential knowledge of geometric and topological methods for practical applications, with a focus on persistent homology and other tools from Topological Data Analysis. They are able to apply topological methods in various contexts.

Course contents

• Simplicial complexes and their realizations.
• Discrete Morse theory for simplicial complexes and chain complexes.
• Persistence modules and persistent homology: structure and stability theorems.
• Applications of topology to distributed computing and data visualization.

Prerequisites: basics of algebraic topology, programming and algorithms.

Recommended prerequisite courses: students are strongly encouraged to have attended at least the first part of the Algebraic Topology course, up to homology. It is not necessary to have taken the course Introduction to Machine Learning.

Readings/Bibliography

Main references:

D. Kozlov, Organized Collapse: An Introduction to Discrete Morse Theory

V. Nanda, Computational Algebraic Topology Lecture Notes

M. Lesnick, Notes on Multiparameter Persistence

F. Chazal, V. de Silva, M. Glisse, and S. Oudot, The structure and stability of persistence modules

 

Additional useful references:

A. Hatcher, Algebraic Topology

D. Kozlov, Combinatorial Algebraic Topology

H. Edelsbrunner and J. Harer, Computational topology: An introduction

M. Herlihy, D. Kozlov, and S. Rajsbaum, Distributed Computing Through Combinatorial Topology

T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms

M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature

Teaching methods

Lectures delivered at the blackboard.

Assessment methods

Oral exam on theory and exercises.

During the exam, the use of AI tools and any electronic device is prohibited.

Office hours

See the website of Giovanni Paolini