Math 574--Introduction to
Computational Topology (Spring 2018)
Topology studies how a shape or object is connected. In the past few years, 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 or Applied Algebraic Topology. Computational topology combines topological results with efficient efficient algorithms to analyze data and solve problems in many fields -- computer graphics and image analysis, sensor networks, clustering, robotics, genetics, protein biochemistry, geography, and others. For a recent overview, check out the topics discussed in the workshop on Topological Data Analysis held at the IMA a couple years back.
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 homeworks (around 7-8 assignments of) and a course project. No exams will be given.