Strange Random Topology of the Circle

This course explores the peculiar behavior of "topological noise" in the context of the circle. By constructing a random Cech complex from the circle, intervals of filtration radii are identified where the complex is homotopy equivalent to bouquets of n-spheres with positive probabilities. The course delves into how higher Betti numbers can become arbitrarily large within these intervals. The learning outcomes include understanding the concept of topological noise and its implications on the topology of the circle. Students will gain insights into the construction of random Cech complexes and their homotopy equivalences. The course aims to deepen the understanding of topological data analysis beyond conventional wisdom. The course teaches skills in analyzing topological noise, constructing random Cech complexes, and interpreting homotopy equivalences. Students will develop the ability to identify and analyze intervals of filtration radii where specific topological properties emerge. The teaching method involves theoretical exploration, mathematical analysis, and probabilistic reasoning to uncover the strange topology of the circle. The course may include visual aids, examples, and demonstrations to enhance understanding. This course is intended for individuals interested in algebraic topology, topological data analysis, and probabilistic methods in mathematics. It is suitable for students, researchers, and professionals seeking to deepen their knowledge of topological concepts and their applications.