Math 4570 Applied Algebraic Topology

This is an introductory course on topological data analysis, with a focus on persistent homology.

References: 

• Introduction to Applied Algebraic Topology by Tom Needham

• A roadmap for computing persistent homology by Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod, and Heather A Harrington

• Topological pattern recognition for point cloud data by Gunnar Carlsson

• Introduction to Persistent Homology by Žiga Virk 

Topics: 

1. Linear Algebra

• Abstract Vector Spaces

• Basis and Dimension

• Linear Transformations and Matrix Representations

• Subspaces and Quotient Spaces

2. Topology

• Metric Spaces

• Topological Spaces

• Continuous Functions

• Homeomorphisms

3. Simplicial Complexes and Homology

• Geometric Simplicial Complexes

• Abstract Simplicial Complexes

• Chain Groups and Boundary Maps

• Homology Groups

4. Persistent Homology

• Filtered Simplicial Complexes

• Vietoris－Rips Complexes

• Persistent Homology

• Persistence Vector Spaces

• Structure Theorem For Persistence Vector Spaces