The homology of data
Abstract:
Techniques and ideas from topology - the mathematical area that studies shapes - are being applied to the study of data with increasing frequency and success.
In this lecture series we will explore how we can use homology, a technique in topology that gives a measure of the number of holes of a space, to study data. The most well-known method of this type is persistent homology, in which one associates a one-parameter family of spaces to a data set and studies how the holes evolve across the parameter space. A more recent and less well-known technique is magnitude homology, which one can think of as giving a measure of the "effective number of points" of a metric space. In this course we will introduce the theoretical background for persistent and magnitude homology, and then dive into applications using software implementations and statistical analysis tools on real-world data sets.