A Topology Scavenger Hunt to Introduce Topological Data Analysis

Abstract

Topology at the undergraduate level is often a theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic topology. However, the last two decades have seen an explosion of growth in applied topology and topological data analysis, which are topics that can be presented in an accessible way to undergraduate students and can encourage exciting projects. For the past several years, the Topology course at Macalester College has included content from point-set and algebraic topology, as well as applied topology, culminating in a project chosen by the students. In the course, students work through a topology scavenger hunt as an activity to introduce the ideas and software behind some of the primary tools in topological data analysis, namely, persistent homology and mapper. This scavenger hunt includes a variety of point clouds of varying dimensions, such as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in 400D. The students' goal is to analyze each point cloud with a variety of software to infer the topological structure. After completing this activity, students are able to extend the ideas learned in the scavenger hunt to an open-ended capstone project. Examples of past projects include: using persistence to explore the relationship between country development and geography, to analyze congressional voting patterns, and to classify genres of a large corpus of texts by combining with tools from natural language processing and machine learning.