Topological Data Analysis

Topological data analysis is an approach to the analysis of datasets using techniques from topology. One of the key messages around topological data analysis is that data has shape and the shape matters. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. Topological data analysis provides a general framework to analyze such data in a manner that is insensitive to the particular metric and provides dimension reduction and robustness to noise. The application of topological techniques to traditional data analysis, which before has mostly developed on a statistical setting, has opened up new opportunities. This course is intended to cover the basics of computational topology that underlie such techniques along with the developments of generic techniques for various topology-centered problems.
The course will consist of two main parts. The first part covers fundamental concepts of computational topology, which include elementary topology, homeomorphisms, homotopy, complexes, (persistent) homology, Betti numbers, topological persistence, and Morse theory. The second part of the course focusses on applications of these techniques to the analysis of various types of data. Possible examples include shape comparison/recognition, surface reconstruction, image segmentation, geographic data analysis, computational biology, network analysis and machine learning.

At the end of this course students should be able to
• analyze topological properties of data and identify topological structure in problems in data analysis
• reason about basic topological concepts underlying topological data analysis
• compute topological properties like Betti number, topological persistence, homology cycles, Reeb graphs, Morse-Smale-complexes from data, and analyze the corresponding algorithms
• design algorithms for topological problems in applications dealing with data

You must meet the following requirements

• Completed none of the course modules listed below
• Topological Data Analysis (2IMG10)

• In the second part of the course we will additionally use research articles on recent applications of computational topology.