Often called squishy geometry or rubber sheet geometry, topology is the branch of mathematics dealing with the properties of objects that are conserved when an object is continuously deformed. Students first focus on point set topology through the lens of formal proof. The course covers the fundamental ideas of continuity, connectedness, and compactness. Later, students also learn about algebraic and geometric topology, which includes topics such as knot theory and manifolds. There is a focus on drawing accurate diagrams with concrete steps. Students also present proofs of theorems to the class multiple times and culminate the session with presentations about different applications of topology. Much of the challenge of the class comes from the abstractness of concepts and the struggle to visualize certain objects in 3-space.
Topology was founded as a class in 19.1 by instructor Jake Pichelmeyer. He was working in the field of topology for his Ph.D and had previously introduced some topological concepts into other courses he taught at CTY. Students enjoyed learning about topology, so he decided to develop a course focused on the subject.
LAN.19.1: Manifolds in the fifth dimension. Is somehow a new AI joke?
Reception by Students
Reactions to the course from its pilot run in 19.1 were largely but not entirely positive, with most complaints centering around the brutal difficulty of the class (and to a lesser extent Pichelmeyer's excessive timekeeping when it came to returning from meals or breaks). Covering large amounts of highly abstract and theoretical content from the upper undergraduate and graduate levels in only three weeks, it was agreed upon by students as well as the instructor and TA that it was the hardest class offered at CTY by far. This intense difficulty and complexity spawned many AI jokes, most of which involved a student entering Park Bench or a similar game and reciting the very common first line of a topological proof or definition "let X be a topological space", causing the student already in the center to flee in fear from the difficult math.
An oft-recited quote from Pichelmeyer on the first day of the class was:
The curriculum I've made here is ridiculous, it's silly, we won't get through all of it, the other instructors made fun of me for even trying to teach it in 3 weeks, and I'll be happy if you walk out of here with even 20% of the content.
Despite this difficulty, most students reported that they did in most part enjoy the class, due in no small way to Pichelmeyer's teaching. There was also a sense among everyone involved of being in it together in testing out this brand-new class.