Course description: Topological data analysis is a modern subject which uses topological and geometric tools to study large and complex data. It is particularly useful for analyzing data sets where complexity arises not only from the data size but also from the underlying features and patterns. Currently these methods are used in a variety of applications, including biology and medicine.
The course will cover foundations of topological data analysis, including both theory and applications. Particular topics will include: combinatorial representation of data (graphs, simplicial complexes), introductory topology (topological spaces, manifolds, simplicial homology groups), basics of Morse theory and Reeb graphs, using the Vietoris-Rips construction to transform data into simplicial complexes, persistence homology and barcoding, topological data analysis for visualization of data (the mapper algorithm and other methods), dimension reduction, practical applications of topological data analysis.
Prerequisites: MATH 4651 or (3351 and 4040) , and instructor permission
“Computational Topology” by Edelsbrunner, Harer,
“Elementary Applied Topology” by Ghrist (available online)
“Topological Data Analysis for Genomics and Evolution: Topology in Biology” by Rabadan, Blumberg
“Topological Data Analysis with Applications” Carlsson, Vejdemo-Johansson