Back when I went to TAMS, I took "graduate" level coursework before, which had concurrent enrollment for both undergraduates and graduate students (each having their own course number, eg Math 4500 for undegraduates and Math 5600 for graduate students). To be honest, they weren't very difficult. It felt like an accelerated undergraduate course, but still an undergraduate course. The pace was above average but not unmanageable, and the homework was very reasonable. I thought I was well prepared for graduate level mathematics, but unfortunately I was not.
People always told me I was insane for doing so much math. It started back when I was a wee child in 9th grade, who was "insane" for trying to test out of the (oh so horribly formidable) subject of Pre-Calculus over the summer. In fact, my Algebra II teacher said she was "looking forward to seeing me earn a C or possibly fail" in Calculus, which was pretty much my driving force to succeed in 10th grade AP Calculus BC. (I got a B.)
Then I was insane for trying to learn Real Analysis over a summer without having touched proofs before. Sure, I admit I was a little off my rocker when I signed up for this one. But I persevered, wrote a college essay on it, and started my senior year at TAMS in both Abstract Algebra and Real Analysis II (strongly against the recommendations of my Academic Advisor, who is a wonderful person by the way).
After acing both courses, I was out of my mind for trying to take Topology (and Abstract Algebra II). I was advised by many people, some even physicists and math majors, that Topology was the hardest undergraduate course UNT had to offer. I was warned of problem sets that would take 2-3 hours per question (sounds like Harvard's 55a), and horror stories of those who failed or dropped out of TAMS due to the class. (Galois theory was difficult too, but no one had taken it). I was told even graduate students struggled with the class!
Now the courses were non-trivial of course, but nowhere near the level of challenge that others made them out to be. It built up a sort of false confidence, a mentality that "I can do anything! (With a non-trivial but nowhere near back-breaking amount of effort)." Because to be honest, none of these courses required an insane workload: they all built upon each other, clearly defined everything from the beginning, and if one paid attention and did all the homework and readings, weren't very hard to do well in.
This marks the first week of my graduate Algebraic Topology course, and I can safely say it's the hardest thing I've ever attempted in my entire life, and BY FAR. I've spent hours and hours poring over textbook pages, going through a definiton lookup chain, grinding through homework problems, just to stay afloat. It moves at the speed of light: we defined the interval in one minute, and five minutes later, we're talking about the fundamental group of simply connected spaces.
Before class even started, we were assigned a pre-homework that asked five question on Manifolds and CW Complexes each. I've never seen a Manifold or CW Complex in my life before (at that point, I doubted whether I even learned Topology at all)! I worked on it for hours every day, looking up so many definitions and reviewing so many semesters of work, for maybe about 10 hours total, and I managed to solve ... 2.5 problems! Homework 1 was a similar experience. I worked on it for 3-4 hours every day, 6 days of the week, and managed to solve about 6 problems out of 10. It feels like I'm taking Math 55a (looking at the problem sets, they're about similar difficulty) without any of the freshman struggling with me, study groups, and tight community (thanks corona).
I don't really know what I'm trying to say anymore, maybe I'm just ranting. But let this act as a word of caution to qualified undergraduates attempting to register for graduate courses: expect those 3 credit hours to take up 30+ hours of work per week (or equivalently, the same or more as the rest of your undergraduate classes). There's a reason 9 hours is full time for grad students. Of course, I'm going to make it through. I signed up for this course to push my mental boundaries, and now I'm complaining that it's working as intended, isn't that ridiculous? It's going to be a long, very long, hard fight, but in the end I'll stumble out of the ring, face covered in blood, holding up a shaky but triumphant fist.
I was planning on adding a cheesy motivational segment here, but I'm not a good motivational speaker (it would probably have the opposite effect) so I'll leave the reader to find the motivation themselves as an exercise. Or don't, motivation is fickle anyways. Godspeed.