How topology helped re-ignite my love for math

Background, my attitude toward math before 2024

For a long while, I avoided thinking too much about the mandatory math courses that my computer science degree in my university includes. I had already failed the initial “introduction to university mathematics” course during the early years of my studies, and overall after my high school experience (being an “average student” in a gifted math & compsci focused high school) in math felt quite incompetent and demotivated. Then I decided to give completing my university degree my best shot. I had often felt demotivated, for reasons including: I felt my executive function was more suited for work life, I had an easier time motivating myself through applications over pure knowledge, and completing a degree felt like an insurmountable task. But I had had a difficult time looking for work during fall 2023, and ended up concluding that maybe a fresh shot at a degree, after many years of fulltime work, would be beneficial. One big reason was to try to complete something I felt was impossible for me. Thus I enrolled on a math course. I managed to “scrape by” (often completing the excercises on the last available day, with a lot of stress and support from my partner, couldn’t have done it without him) a linear algebra course. It felt insightful but also fresh. I liked being able to see the tower of knowledge that subsequent excercises built on. I liked being able to “just calculate” for many of the excercises. (Solving linear equations by hand or calculating determinants.)

Going faster

I felt empowered. Learning new math felt possible. I understood new things and also had received new tools. (And learned to read math better! To read one symbol variable names and a lot of greek alphabet sprinkled in. Something that had felt like one of the obstacles blocking me from completing my math studies many times.) So I decided to try and enroll on a difficult math course for the next period (january - march 2024), Topology IA (1/2 of a whole that introduces metric topology). The course had a non-insignificant amount of pre-knowledge I was missing, including 2/4 of the analysis courses that our university’s math majors take during their first year. The course was also aimed at those math majors. It looked like a challenge. I managed to fit some easier math courses, and another challenging one into the period, and find some friends to take with me to the topology course, and my journey was ready to go. The first two weeks of lectures were quite followable. I had many an enjoyable discussion with my friends during the lectures or after them, and learned some basics. But I did not manage to complete the excercises very well. I felt I didn’t know where to start with the heavily proof based excercises (basically 6/6 excercises were proofs for most of the weeks). I had known that I might start from a difficult spot, missing some of the prereqs, and went to the excercise solution sessions and tried to follow as best as I could. Also had to lighten my math load around halfway through the period, dropping Introduction to Logic (Ironic, it being a useful course for learning proofs, but I didn’t feel like I was learning optimally with that course, it being very self study focused), and eventually also dropping a Linear Algebra II class. At around week 5/7 of the period the Probability 1 course I was studying in also had a difficult week, and I had to miss some topology lectures. After that I didn’t manage to get my rhytm back. I was quite lost with the precise details of the newly taught, and thus I resigned to having a quite high change of failing the course. I was also lost with missing prereq info, some of which included better intuitive understandings of the following: delta-epsilon proofs, mean value theorem, non-trivial forms of triangle inequality, However not all was lost. The course exam had the possibility to bring a 3-page two-side cheatsheet, which could be written on a computer. I managed to get ahead on the probability course stuff and make some time, and determined to rehearse basically all of the topology material, and make a good cheatsheet, perhaps including all the definitions and theorems from the course material. Exam week starts. My first exam is probability 1, which should be the easier of the exams. I go there, do it, have a decently confident good feeling afterwards, if a bit tired. I rested a bit and went on to finish the topology cheatsheet. It was a painful experience of staring at mathematical notation for almost 15 hours in total, rewriting the course material in my own latex cheatsheet. But with an entire two days to go before the exam, I had managed to do it. I spent some time stressing about the exam, some staring at the cheatsheet, some watching math youtube, and in the end had a decent feeling going into the exam. I was prepared for quite a high variance when going in, but after getting the exam paper in my hands and spending maybe 30-45 mins of the allotted 4 hour timeslot getting the questions in my head properly and drafting, I managed to work on them quite well, managing to answer all 5 of the questions, if some with a couple holes. I needed 13/30 points to get a passing grade, and was more optimistic after the exam than I was beforehand. After spending some attention waiting for the results for ~two weeks, they were announced. But there was a problem. The study webpage where the results could be seen was down for mainteinance until 21 that evening. I got to watching some anime with my boyfriend and refreshed the web page the moment the clock hit 21:00. There was another problem. The scheduled mainteinance had been extended, perhaps due to technical problems. The new mainteinance end time was 21:15. And at 21:15 it was 21:30. Then 22:00. In the end I got the results maybe around 22:30 that night, after one and half hours of quite intense waiting. I had received a grade of 2/5 from the course. I was satisfied. This was not optimal understanding but it was much better than I had feared for, and it was a not-insignificant amount of understanding in university mathematics I had managed to attain. (I cannot really say I was elated, but I was relieved.)

Now

I can clearly notice my mathematical intuitions improving. It has been satisfying. I don’t know if it’s useful but also I don’t know if it would even be good to think too precisely on if it is. Maybe sometimes one can just commit to a few steps of learning some new skills and not spend all the time trying to evaluate their usefulness. I have noticed the phenomena where sometimes grokking something in math requires multiple hours or even days or weeks of being stuck with the thing in mind. The last experience of this I had when studying algorithms courses, but often when doing real world programming I have managed to avoid being stuck through various means. I’m also glad that through proper motivation (e.g. having a use for it, making math notes and writing excercise answers) I have now learned to write math in LaTeX much better. I also suffered from 4-6 days / week headaches for a month during this period of studying math. I’m actually surprised how functional I managed to stay and how “low weight” the experience has in my head. It was helpful that normal OTC pain medication handled the issue quite well. (And the situation has now become better.)

I want to learn more math. I want to understand it. I want to glance at this infinity.