# Topology

Topology
Math Course
Course CodeTOPO
Year Opened2019
Sites OfferedJHU
Sites OfferedLAN

## Course Description

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.

From the CTY Catalog:

Topology is the mathematical study of shapes and space that considers questions such as, “What objects that visually seem quite different share the same properties?” One of the major fields of mathematics, topology possesses wide-ranging applications and beautiful theorems with far-reaching consequences. This course will introduce you and your classmates to point-set topology as you delve into bizarre notions of "space" and develop skills with rigorous, proof-based mathematics. You’ll begin by tackling the core concepts of sets, topologies, and continuous mappings before moving on to topological invariants such as compactness, connectedness, and the separation axioms. With these tools in hand, you will explore how to deform shapes and spaces without altering their fundamental properties. This knowledge allows you to see why it took 100 years for mathematicians to prove Poincaré’s 1904 conjecture about the nature of a sphere. Finally, you’ll survey different applications of topology, such as how the study of knots influenced our understanding of proteins, or how the study of manifolds led scientists to a deeper understanding of the topological shape of the universe.

## Class History

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.

### Lancaster

LAN.19.1: Manifolds in the fifth dimension. Is somehow a new AI joke?

### Baltimore

JHU.22.1: The instructor for topology at JHU in 2022 was Sam McCrosson, who was nice and generally well-liked by students. The TA was Abraham, who was also great and well-liked. Like in previous years, the content of the course was difficult, with Sam stating at the end of the course that the class had covered about a semester and a week of college-level material. The last day of the course was on category theory instead of topology since Sam was studying it at the time and wanted to share his interest in the field with students. Very few students understood the category theory concepts Sam taught that last day, but they were considered interesting nonetheless.

One notable theorem introduced was Yoneda's lemma, which Sam claimed could be used to "prove everything in math that we care about." Overall, students enjoyed the course. One popular joke was to quote the line "It's like a belt" from the video Outside In that the class watched (said video was from the channel ssgelm, and it described the process of turning a sphere inside out). There was also one incident at the talent show in which a CTYer doing stand-up comedy referred to topology as "one of those dorky classes." The topology students didn't mind the comment much.

The instructor Sam also had many wild stories. For seven months, he lived out of a tent while visiting forty states on a motorcycle. During that time, he went to the bathroom only outside and in "low-end businesses," and he got most of his food out of dumpsters. Right before that, Sam bought a bus and planned to drive across the nation, only for the bus's wheels to fly off on the highway. He then sold the bus to the tow truck driver who picked it up. The bus also had no AC, so the portion of the journey that went through the desert was miserable. During that time in the desert, he had to fix the bus's exhaust pipe (which had detached from the bottom of the bus and begun dragging along the road) with a clothes hanger, a wrench, and duct tape. Both of these stories (the bus and motorcycle) appear to have happened during Sam's gap year.

Another story Sam told involved several important people in math saving a man who contracted hypothermia from swimming in a mountain lake. They apparently did this by having a group hug in only their underwear to exchange their body heat with him. Sam also detailed his advisor's goal to run up and down fourteen mountains in under 40 hours, which would be a world record, and would entail him running up and down mountains for 40 hours straight. That same advisor also played a strange practical joke on his friends during a backpacking trip to a mountain. While all his friends were asleep, he secretly climbed up the mountain and placed a coconut at the peak, one which he had carried all the way to the foot of mountain. He did not reveal to his friends that he was responsible for the coconut when they all found it the next day.

## 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 Nick 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.