A side note: I’m intending to publish a lot more frequently this year, aiming for twice a month at least. Any feedback is greatly appreciated.

Most people first encounter prerequisites in college. Your advisor will sternly tell you that you can’t take that engineering class without sitting through linear algebra first. At an unfortunate number of schools this is the end of the discussion. However, at many colleges, the professor has discretion over who can take their class. If you shoot them an email, explaining why you want to take their class, they might allow you to do so. Even if it’s not possible, you can still ask to audit/sit in. There’s no possible downside. Most professors are quite happy to see someone who’s actually interested in taking the class!

That said, there are two situations you can be in. You can be completely qualified to take the class, just lacking prior credits. Or you can be unqualified for the class but have no desire to take the prerequisite and you think it’ll probably be fine. This is how I took a graduate algebraic topology class in high school.

My background

In tenth grade I started taking undergrad math classes at my local university. I took linear algebra in the fall and abstract algebra in the spring. The linear algebra course I found extremely boring and unmotivated. I liked abstract algebra a lot better. In eleventh grade I took multivariable calculus (also very boring) and a complex analysis class. Sometime in the spring I was pondering what classes to take next. The obvious choice would have been a real analysis or differential equations class. But my past experiences with “engineering math” were dismal, and the real analysis class had consistently been offered at a very annoying time for me. (I still had high school classes to go to). With this quandary I started to browse the mathematics catalog, and I glanced at MATH 5400: the algebraic topology course.

There were several issues. For one, the class required taking real analysis first. Also, it was a graduate level course. I didn’t know a lick of analysis, and my memory of abstract algebra was quite fuzzy at this point. But also around this time, I had been meeting with a math professor at the university who worked in geometry. When I asked her if she thought I would be able to handle the algebraic topology course, she told me that if I endeavored to learn analysis over the summer then it would probably be fine.

Well, that summer I did a lot of physics instead, and then it was fall again. I didn’t really find any of the other math courses I could possibly take appealing. So I emailed the professor who would teach algebraic topology and asked him if I could take it, mentioning the other professor who could vouch for me. He gave me his cautious approval, telling me that if after the first midterm my grades were below a certain threshold then I ought to reconsider. Then I was in.

The class

I think there were seven students enrolled; by the end of the semester there were four left. (Iowa is not renowned for its mathematics programs.) Our textbook we followed was Hatcher, but we barely managed to cover the first three chapters over the term. I think that if I had taken this class at an university with a bit more teeth, I would have realized a lot quicker that the correct decision was to drop.

I remember the first lecture. Well, I don’t remember the contents, but I do remember walking out and realizing I had understood approximately nothing. This in itself wasn’t very unusual, the unusual part was that I had actually tried to comprehend. I then resorted to reading the textbook, but it didn’t clarify things. It occurred to me that I might be in over my head.

On how to read Hatcher

Up to this point, all of the math textbooks I read had been comprehensible. I learned most of my high school math from AoPS books. (They’re quite good, I would recommend.) For linear algebra and multivar, the textbooks assigned were the ones that come with their own ebook reader. In my experience this is 100% correlated with the textbook being bad, so I never read them. To be honest, I hadn’t studied from any serious math textbooks. I had fun skimming Strang along with the Napkin and various books on random topics. In all of these, I had sufficient background and the prose was pleasant enough to delude myself into thinking I was understanding the text.

A book like Hatcher is quite different. For one, it’s written in the terse style common for math textbooks, where a chapter is almost nothing but a sequence of statements and proofs followed by exercises, assuming the reader can fill in quite a lot of gaps in the explanations. If I was a grad student it would have probably been more obvious, but instead I was an overambitious high schooler. This was a book where it could easily take an hour to understand a single page.

What made it so difficult? Well, most people cannot immediately grasp all of the implications of a mathematical statement they just learned. Certainly, I couldn’t. It takes time and wrangling with examples and exercises to fit it into your mental structure of the subject. Why is this theorem important? Does it put things you previously learned in a new light? What new things does it allow us to do? Actual understanding requires you to ponder questions like these. Not having a topology background made all of this even more difficult.

To add to my troubles, you may have noticed that I am not very good at paying attention in class. My usual remedy for this is to read the textbook, and pretty frequently it’s the only thing I actually learn from. But learning from Hatcher only works well with the benefit of lectures with exposition and teachers to ask questions to. Otherwise you spend much of your time groping in the dark, struggling to make the correct connections. I suppose it’s unfair to expect textbook authors to cater their books to people learning it by themselves. Unfortunately I am mostly that person.

An ignominious end

Despite my troubles, I didn’t crash and burn immediately. The problem sets probably weren’t objectively hard, but they were grueling for me. Even though I managed to mostly solve the problems, I wasn’t actually learning much algebraic topology. I couldn’t build a coherent scheme of the subject; anything I did learn was because it was incidental on the way to solving a given exercise.

When the first midterm came around, I scored barely better than the threshold the professor had mentioned. I wasn’t happy with my performance, but I thought if I just tried to study harder, I could turn things around. Due to getting wrapped up in college applications and my general inclination to procrastinate, I did not in fact study harder. In retrospect, it’s not clear if more studying would have even helped. When the second midterm came, my score was a failing grade.

So at this point the trend was obvious. I decided the prudent thing to do was to drop the class, and emailed the professor my intention. He emailed me back asking for a chat in person. To my surprise, when we talked, he asked me to not drop the class. Although my comprehension of algebraic topology was subpar, he thought that I was showing signs of “mathematical maturity” quite advanced for my age, and that there was real benefit to me staying in the class and gaining more experience. He was willing to give me a passing grade, irrespective of how the final would go.

So, that was that, and I decided to stay in the class. Things were mostly as before, but it was at least nice to not stress about it. When the final came around, I somehow got the median score in the class.¹ My grade, at the end of all of this, was a B+.

Takeaways

So, what did I learn from all of this? Not a lot of algebraic topology!

I think that taking the class ended up being a positive experience, but I don’t think it was really the optimal thing for me to do. I do want to take another swing at algebraic topology, if I ever have the time at MIT.

During this whole ordeal I came up with the following argument in my head. If you take classes you’re unqualified for, it forces you to learn the prerequisites if you want to do well in the class. I don’t really believe this anymore. The issue is that an advanced/graduate level class often implicitly assumes you have the background of having gone through the entirety of the undergraduate core curriculum. It’s not just the raw knowledge of knowing analysis or algebra, but the accumulated experience of wrangling with these mathematical objects over a lot of hours. Without this, learning the advanced material becomes massively more difficult. There’s no easy shortcut to gaining that experience. Even if you do manage to learn some algebraic topology, that doesn’t mean that you’ve mastered all of the prerequisite knowledge.

Despite all of this, it’s still alluring to try to take some class stupidly early. Getting through it is like a status symbol, saying that even though you were unqualified, you were still intelligent enough to pass. My contention is that this is dumb. The goal of taking classes should be to learn things. If you want to seek status, you can do that in ways that don’t involve ten hours of problem sets a week.

1. Yeah, this does not mean much when there are four people. ↩