MATH 640 Computational Algebraic Topology

This course covers foundational aspects of combinatorial algebraic topology with a view towards applications to computational data analysis. It will cover basic geometric-combinatorial constructions, and it will concentrate on the study of invariants associated to topological spaces, such as homology, Euler characteristic, Betti numbers, etc. The mathematical formalism will be as basic as possible and the course will focus on examples. The concept of cubical homology will be discussed and its applications to images. Some other invariants to understand the underlying topology of data sets will be discussed, such as persistent homology as well as other homology theories associated to data sets “approximating” a space. We will give an introduction to computational environments such as JavaPlex and CHomP, to obtain Betti numbers and bar-codes. Some examples to be discussed can include the invariants associated to conformation spaces of proteins, the space of natural images and other higher dimensional examples.

3 Credit(s)