Konstantin Mischaikow
Applied and Computational Topology
This course is an introduction to Topological Data Analysis (TDA). Though still relatively new TDA has blossomed over the past two decades, thus we will only have time to touch on a few subjects and these topics will be decided in part by the interests of students in the course. While an introductory course in algebraic topology would be ideal, it is not absolutely necessary as the basic ideas of homology will be discussed as needed.
We will start the course by considering:
1. Methods for the efficient computation of homology groups, induced maps on homology, persistent homology, and Conley complexes. This is essential knowledge for working with large data sets.
The following topics will be covered (the depth of coverage will depend on the class)
A. Data driven nonlinear dynamics. We will discuss topological and statistical tools for extracting specific dynamical structures from time series data.
2. Persistent homology with a focus on persistence diagrams. We will view persistence as providing nonlinear method of reduction of data that preserves topological structures measured by homology.
We will discuss applications to a variety of topics.
4. Machine learning. We will explore the dynamics of machine learning via topological dynamics and/or the use of machine learning to reduce dimension of data for topological analysis.
The choice of topics to which these techniques will be applied will be determined in part by the interest of participants in the course, but will may include the problems arising from image processing,
protein structures, materials science, gene regulatory networks, fluid flow, social choice, and dense granular media.
will provide lecture notes
Undergraduate courses in topology and algebra.