“How hard is Math 55?”
Math 55, for those unwilling to click on the links, is the highest first-year class offered by the second-best math department in Cambridge, MA. The class is touted as being four years' worth of undergraduate mathematics (and fairly advanced undergraduate mathematics at that) in a single year, which leaves future mathematicians (and the occasional physicist or economist) free to pursue more specialized interests. Each year, 50-100 hopefuls sign up for the class; usually, about 20 make it to the end (most simply drop back to the somewhat less work-intensive Math 25).
Why was I asked to answer this question? For starters, I'm an actual mathematician; at least, that's what my business card says (though I am beginning to suspect that mathematician is an ancient Greek word for “C programmer”). My undergraduate degree comes from the other school in Cambridge, I am addressed as “Doctor” at work, and I do still occasionally fool around with some of the nastier bits of algebra and graph theory, both at Veridian and in my spare time (though my performance in ALOTT5MAball is an indication I should probably be focusing more on probability and statistics). So, in between weaponizing pumpkins and reading old XKCD strips, I took a gander at this year's version of Math 55. So, how hard is it?
The short answer: pretty darn hard. The long answer is the same, but I get to type more.
I looked through a couple of the problem sets, and they all had one thing in common: length. It is not uncommon for students to spend almost 60 hours a week on homework for this class, and solutions tend to be 15-20 (LaTeX-formatted) pages long. Rather than slog through those, I printed out the final exams, and brought them in to work on whenever we had to install new motion sensors in the labs (which happens with surprising frequency). I also passed them around to my colleagues, except for the guy who actually took Math 55; I figured his opinion would be biased.
The consensus on the first semester final was that it would make a fine graduate-level qualifying exam, especially if your school considered linear algebra worthy of inclusion on a qual (mine did, and it wasn't nearly this hard). The first question, using Peano's axioms to prove a cancellation property, is deceptively difficult; it appears you've practically been handed the answer (pssst... it's induction!), until you realize you'll have to derive the concept of less-than in order to get the last step. Frankly, the most frustrating thing about graduate-level algebra is the beginning of each sequence, in which you have to prove things that the average third-grader has seen and most middle schoolers simply know, and your professor won't give “Come ON! I knew how to subtract before I could tie my own shoes, this is obvious!” full credit (yes, I have tried that). Question five, on field extensions, made me chuckle, because I remembered doing that exact problem on a homework assignment—during my fourth year of grad school. The last question, though, is a thing of beauty, combining complex numbers and Galois theory to prove the solvability of a specific quintic polynomial. My favorite class ever spent an entire semester building the theory necessary to show that these polynomials are not, in general, solvable. This class did it in about three weeks.
The consensus on the second final could be summed up as, “Ugh.” All good-hearted mathematicians despise analysis. For the lay-people: analysis is best described as the “theory of calculus,” and not even mathematicians like calculus. Also, it includes differential equations, which is really physics in disguise. Basically, in the meatloaf of mathematics, analysis is the onions; the whole thing really doesn't work without it, but that doesn't mean I have to like it on its own.
Anyway, the problems are actually pretty standard stuff (though I did have to look up “isoperimetric”), but none of them are anything I would classify as easy, and the sheer diversity (this covers stuff I learned over three semesters) and volume would have had the freshman version of me writing long angry missives to my graders well before finishing. My favorite is problem 9, which ends with a calculation of a specific value of the Riemann-Zeta function. I've had to do this in at least two classes (diffEq and combinatorics), and it's always fun to see another way to get these values. It's also a good question because the value is fairly well-known, so you can figure out pretty quickly if you've made a mistake.
So how do I think I would have done? I have seen nearly all of this material before, and feel like I could certainly have handled this course with just a little refresher. But I have a PhD already, and if I was doing this, I wouldn't be doing anything else. The people taking this class are 18 years old, and are taking three or four other classes (though those are usually less intense). The 18-year-old version of me couldn't have gotten near this class; I probably would have ended up in Math 23, as I liked the math, but thought at the time that I was going to do something else. Another thing to keep in mind is that in the last couple of years, there has been a pretty straightforward treatment of the class, a semester of algebra followed by a semester of analysis. Previous years included things like point-set topology and differential geometry, which can be mind-blowing to experienced students.
There may be a couple of high-school math wizards reading this; if you even think you're up for the challenge of this class, then I say go for it. (as an aside: you should apply to Harvard whether or not you care about this class, or even going to Big H; if you get in, you can say you got in, and if you don't, they write the best rejection letters.), Know what you're in for, and realize that finishing this class truly is a special achievement, so not doing so won't close any doors for you. No matter what, if you even find this interesting, then bless you, because the world needs lots of people who find math interesting. Godspeed, and I look forward to hiring you for the Jabberwocky project someday.