Math529: Introduction to Computational Topology

Topology studies how a shape or object is connected. In the past couple decades, there has been an increased interest in the development and use of topological methods for solving various problems in science and engineering. This new line of study is called Computational Topology, Topological Data Analysis (TDA), or Applied Algebraic Topology. Computational topology combines topological results with efficient efficient algorithms to analyze data and solve problems in many fields—biomedicine, phenomics, machine learning, computer graphics and image analysis, sensor networks, robotics, geography, and several others. For motivation, see my TEDx talk on how TDA helps to find hidden structures in data.

This course will present an introductory, self-contained overview of computational topology. There are no prerequisites, but mathematical sophistication at the senior undergraduate level and some familiarity with the use of computer packages such as Matlab or Python are expected. We will cover basic concepts from a number of areas of mathematics, such as abstract algebra, algebraic topology, and optimization. We will also look at algorithms and data structures, and efficient software for analyzing the topology of point sets and shapes.

While there is a recommended book, we will rely a lot on handouts and class notes. Material from several recent (and not-so recent) papers will also be covered. Since the main goal of this course is to expose the audience to this nascent interdisciplinary research area, evaluation will be done through homework (around 6 assignments) and a course project. No exams will be given.