I visited the world's hardest math class

I visited Harvard University to check out Math 55, what some have called "the hardest undergraduate math course in the country."

But is it really as tough as it seems?

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ABOUT ME
My name is Gohar Khan, and I'm an MIT grad helping you succeed at school. I make videos about study strategies, college applications, and occasionally, my day-to-day life. I want to equip students with the tools, knowledge, and resources they need to realize their academic potential.

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[iwetmyplants]

I appreciate how much you point out how human advanced courses like these are. Hollywood likes to portray upper math courses like these as extremely cutthroat, sink-or-swim. But in reality these classes often involve a lot of collaboration, talking with professors openly and honestly, mutual curiosity in the subject, and lax grading. When I took quantum mechanics in my undergrad, everybody in my class ‘cheated’ by working together on our tests and quizzes. We did that because we all agreed that quantum mechanics isn’t something to be learned alone. We were just trying to understand the material the best we could. My second semester understood that very well, so he encouraged us to turn in assignments as a group, re-explain concepts on the whiteboard, take a stab at explaining things differently, etc.

[steelraven]

The kinda sad thing is that lots of people never liked math or like math because of a bad teacher which can make everything seem like it’s much harder than it really is
Edit: Dang I didn’t know this was that relatable

[atomicJUMP]

" as the letters turned into Hieroglyphics" got me rolling so bad 😂

[Dravignor]

Mid-way through the video, coming from someone that's learning college math right now, the lessons in Math55A are topics you would learn in year 1, 2 (Linear and Abstract Algebra respectively) and Representation Theory is something you're more likely to learn as a grad student. 55b's Real and Complex Analyses parts are something you'd learn in year 2-3, while Algebraic Topology is either 4th or in grad level

[haileychuukisu]

everyone: this is the hardest math class in the world

me: I bet the professor is using hagoromo chalk

[pekorasfuturehusband]

I’m a second year CS student in University, saying I love maths and physics is an understatement. Throughout my self study journey I taught myself Linear Algebra, Calc I-III, Differential Equations, working on Differential Geometry and Topology atm. Math 55 had always been somewhat of a dream of mine but unfortunately I’m not from the U.S nor have the budget for Harvard lol. Thanks allowing me to see for myself a glimpse of what I’ve always dreamed of 🙏🏻

[Kabcr]

Math, especially higher level math, has this beauty that occurs when you're working proofs, like a giant jigsaw puzzle that comes together in the most wonderful way and almost always interlocks with other bigger puzzles. This class seems exactly like that.
I was passionate about math in college, but also long past my time to go to college again thanks to how expensive it is, and sadly it's not that useful when building a career. I'll remember my time with math fondly.

[NeokingTech]

I went to Harvard for undergrad and studied physics, so naturally I knew a lot of kids who took 55. I myself took the intro math class just one level below it in difficulty (Math 25). This video did a better job than I expected at depicting the nature of this class and separating fact from fiction. But some of the details could use better clarification or context. For one, Math 55 is solely for freshmen. Harvard has a large set of introductory Math courses for incoming freshmen who intend to study some STEM field and have already taken calculus in high school. At my time, I think there were around 6 such classes (19, 21, 22, 23, 25, 55). All of them more or less cover linear algebra and multivariable calculus. At the start of your freshman year, you're very much encouraged to "shop" these classes for the first few weeks to find the best fit for yourself in like an open enrollment period. Most students end up enrolling in 21, which is a very standard, non-proof based linear algebra and multivariable calculus series very similar to what you'd find at other schools (though still difficult because... well it's Harvard lol).

The part Gohar gets wrong is that the mythical drop-out rate is mostly referring to the number of people who initially enroll in Math 55 first year during this open "shopping" period, then drop a level or two after a few weeks down to Math 25 or 23 once they determine they can't handle it. But of those who stay enrolled in 55a, of course they nearly all continue on to 55b. In my freshman year, I remember day 1 of 55a, there were upwards of maybe 80 people in the classroom. I went just for fun but wasn't serious about it. I think maybe around half so ~40 enrolled, and then at the end of the shopping period, ~20ish people remained in the class. They all took 55b the next semester, as far as I'm aware.

Moreover, to add some detail, people who are more serious about math (math, physics, and computer science students, mainly) are encouraged to take 22 at the minimum. 25 is where things start to get *very* difficult, and the majority of Math majors at Harvard took it. Those assignments alone usually took me 20 hours. But 55 is special in that it far surpasses the material contained in the other courses; in fact, most of the students who take it probably learned linear algebra and multi on their own in high school. I appreciate Gohar's attempt to make the class seem more inviting and less exclusive, but it truly does deserve its reputation. The student interviewed in the video who had no prior competitive math experience is very rare, and he likely has other significant technical experience, probably in physics or computer science, that has given him the technical maturity to be able to tackle 55, as well as a great level of natural talent. Otherwise, there is no level of collaboration and wishful thinking that can get one through this course unless you are very very advanced and mature in math.

[mnwtwo]

Gohar you've transformed my high school experience, keep up the grind 🔥🔥🔥🔥🔥🔥🔥🔥

[pinomarittimo]

If this comment gets 200 likes I will attend math 55

Thanks for the 200, at the start it was kind of a joke not now I want to reach this goal.
I will upload a short every day(starting tomorrow) until I get in to Harvard and pass math 55a.

[DVnc_]

Gohar is such a great listener, pls consider making a video about something related, it'll wait help alot

[ChraO_o]

Seifert-Van Kampert Theorem; Additional 5:00
Shown in an isomorphic (the part of the whole process of movement like a map) sense, though way clearer if simulated like a video/homotopy that's shown here.

Isomorphic visualization: (you can draw it on paper to help visualize)
[
Level 1:
Imagine a point in a line
Extend it until it reaches the goal
Level 2:
Now divide that line into 2
Extend those lines
Level 3:
Create 2 other points in those lines
Imagine those points as A and B
Duplicate Line A
Draw the line from A to B (assuming its a perfect drag)

And then multiple of those somewhere that can be any line
(Also this man got this theory right)
]

The algebraic expressions may seem like extremely hard unless you understand the language, you're all good.

Overall, basic maths with funny languages, "we're still lacking tbh"

[kristimotra]

Math is a language that some of us aren't taught correctly.

[bbysrec]

Gohar thank you so much for your content. You have been absolutely lifesaving this year. I look forward to starting high school next year with a prepared study schedule.

[rgamer]

One of my longtime friends got into harvard this spring, and is planning to take this course. He has no math olympiad background and took calc bc in senior year, which is wayyy behind the other kids in this course, but I know he is going to do well. To do this course, it doesnt require really strong math skills, but hard work, determination, and passion for math

Also after reading a lot of the comments, yall should understand that he is grinding over the summer. He is watching videos on all the concepts of multivariable calculus, linear algebra, and differential equations. Secondly, he is going over basic proof concepts to get a feel for the class. So he is not going into the class blind lol.

[christophvonpezold]

Hoping to take this course in September, this has made me feel like maybe it’s possible? Might eat my words though. Thanks for the great video!

[Mojojo]

Love the hard work this guys dose and he helps get better at studying thanks Gohar❤

[Mrtok]

Really enjoying this type of content…Keep it up!

[julianbruns]

8:15 my textbook at university proved something similar to problem 1, so here is a sketch of a possible solution:
By definition, f is lipschitz continuous with lipschitz constant c, so it is also continuous. (Alternatively you can use epsilon-delta, just set delta=epsilon/c and continuity will follow immediately)

Assume that there are distinct fixed points x, y. Then d(x, y)=d(f(x), f(y))≤cd(x, y) with c<1 wich is only true if d(x, y)=0 i.e. if x=y. So there is at most one fixed point.

Let X be the unit disk with the center removed. Then define f to rotate every point (x, y) by 90° and also scale it by 0.9. since the center is removed, no point stays at the same place and the contraction is given by the scaling by 0.9.

Now to the hardest part. Let X be nonempty and complete and (x_k) a sequence with x_k=f(x_k-1)
We will show that this sequence converges to the (necessarily unique) fixed point. To do that, since X is complete, it is enough to show that (x_k) is cauchy. First note that d(x_k, x_k-1)≤c^k-1 d(x_1, x_0) for all natural k. This is because you can just recursively apply the fact that f is a contraction. Now lets look at d(x_k, x_n) where n is some natural number less than k. You can split this up into k-n-1 parts using the triangle inequality such that each part is of the form d(x_j, x_j-1). Now you can use the abovementioned formula to bound each part to get d(x_1, x_0) times a factor of c^n+...+c^k-1. We can fill those facors up with c^k+... till infinity to get a geometric series and bound the factor by c^n/(1-c). Now, since c^n gets arbitrarily small for large n, for every epsilon>0 we can chose n such that c^n/(1-c) d(x_1, x_0)<epsilon and hence (x_k) is cauchy. Since X is complete, (x_k) converges to some x* wich is our fixed point since d(f(x*), x*)≤d(f(x*), f(x_k))+d(f(x_k), x*)≤cd(x*, x_k)+d(x_k+1, x*) wich goes to zero since x_k and x_k+1 converge to x*. So d(f(x*), x*)=0 and thus f(x*)=x*, i.e. we found our unique fixed point.

[srallulrich]

It seems like this course is just a mix of algebra, topology, real and complex analysis using the language of categories. Some topics must be cut out to make it all fit into 2 semesters. Seems like a lot of fun for math majors