today I'm at Harvard University to check out the hardest undergraduate Math course in the country allegedly the class is called math 55 and it condenses four years of math into two semesters apparently half the students drop out after the first semester and homework assignments can take anywhere between 24 to 60 hours Bill Gates took this class and it's usually filled with former members of international math Olympia teams in other words some of the smartest math students on the planet I want to see what this class is all about I'm here to see who is actually
taking it these are kids who were outliers in their High School how hard it actually is I say like math is really the only thing I do now and of course what actually happens inside math 55 so for the past few weeks I've been in contact with Joe Harris he's an Este mathematician working in the field of algebraic geometry and he's also the current math 55 Professor Not only was he kind enough to let me attend and even film one of his lectures but we'll be chatting with him after class my whole goal here is
to understand what this course is actually like I want to understand how hard it is who's actually taking this course and what it takes to do well because here's the thing it has no prerequisites they simply say familiarity with proof-based mathematics is a plus and as former math 55 Professor Denny aru once said and I quote if you're reasonably good at math you love it and you have lots of time to devote to it then math 55 is completely fine for you I mean I enjoy math I took some pretty hard math courses at MIT
but does that mean I'm reasonably good enough also do half the students actually drop out and do homework assignments really take up to 60 hours a week I mean that's more than a full-time job not to mention some people even say the course is a bit clicky among some math Majors taking math 55 is kind of like a status thing and apparently some students even come to Harvard just to take the class okay we're a bit early right now but the lecture is going to be inside that building right there the science center let's go
wait inside now the science center is a special place at Harvard mostly because it feels nothing like Harvard unlike the Georgian architect and beautiful brick work that spans most the campus the science center is mostly made of steel and concrete this place is Harvard's undergraduate hub for Math and Science and it was only founded in 1973 and yes I say only because that means this building has only been around for 133% of Harvard's 387e history yet it's already brimming with some of the brightest stem mins in the world including Wan Jaya a math 55 student
I met up with my first few weeks in the math math classes here it was like extremely difficult for me cuz like just simply because I've never done the proof before and I didn't really have any math background so it was like a lot of it was like self-doubt whether I could like you know succeed at math cuz like I've never been a math kid I'll be honest I'm pretty nervous right now and I'm not even taking the class I think it's because I don't know what to expect I don't know how big the classroom
is I don't know how many students there are I just hope they don't mind me walking in with this camera granted I did get approval to film but since we have a couple minutes to spare let me tell you what this class is actually about so math 55 is actually two separate classes math ma 55A and math 55b the official title of 55A is studies in algebra and group Theory and the title of 55b is studies in real and complex analysis up until recently the class didn't really have a standardized curriculum the professors simply decided
what they wanted to teach but that's been changing now bear with me I had to pull these quotes directly in 55A they teach linear and abstract algebra with a bit of representation Theory I wish I knew what that meant in 55b they teach real and complex analysis with a bit of algebraic topology now as I try to wrap my mind around these math concepts let's make our way to the lecture hall or lecture classroom I guess the first thing I noticed is that math 55 is in a pretty small classroom with around 20 to 30
students which is kind of funny because the actual enrollment is like 60 students now as the students roll in some go up to the professor to ask questions some are just chatting away but nothing feels too unusual or intense in fact Professor Harris looks like one of the friendliest guys I have ever seen but the stuff that he goes over not as friendly no there's no question as far as content goes as far as Pace goes as far as level of abstraction goes it's a tough course but it's not so bad today's topic that's not
so bad is the cyer Von compon theorem according to Wikipedia this theorem expresses the fundamental group of a topological Space X in terms of the fundamental groups of two open path connected subspaces that cover X now I don't know what that means and chances are you don't either so here's a quick 60-second explanation courtesy of chat GPT so there's a field of math called algebraic topology and it studies shapes and spaces using algebra and this field is unusually obsessed with the number of holes and shapes this dut one hole this mug also one hole so
in a sense topologist see these as the same shape because you can deform one into the other let's take a look at a couple more examples using Play-Doh here we have one hole again obviously this has two holes and here we have three holes now The Cypher Bon compon theorem explains what happens when you combine any two sh shapes it tells us how the loops and holes of the combined shape relate to those of the individual pieces of course this is a gross oversimplification the real explanation is a lot more advanced and it's almost indecipherable
even for the professor if you look at the statement of Cipher Bon compa the general form of Cy Bon compa in monre I mean it's just to me it's impenetrable I cannot understand what he's talking about but in fact it's very easy to express in the language of categories and we're not getting the language of categories doesn't add any new results here it's just a way of expressing what Cypher non-confidence telling us and that's the beautiful part about this lecture he takes this obuse almost impossible to understand concept and makes it much more digestible by
using a related concept quickly however the lecture starts ramping up B from Z to t such that V composed with v one is equal to V the diagrams on the board get more complex the symbols turn into hieroglyphics but Professor Harris's enthusiasm for the topic is infectious it makes it a fascinating circuit if we think of a star B Prime as the test object T we get an induced from a star B to that object even though the material has far surpassed what I learned and even the hardest math classes I took at MIT I
can't help but stay engaged and the same goes for the students they're not afraid to ask questions are to be judged even as they're sitting among the brightest math students at Harvard there was a nice back and forth almost an open discussion within the classroom um big part of the course um I would argue from having spoken to professors and students alike is asking questions and not being afraid to do that but then I started to wonder who even are these kids who got up nice and early on a Wednesday morning to learn about some
abstract math these are really bright kids even though a lot of the material is new to them or in some cases the way we approach the material is new to them they can handle it but why can they handle it what's the sort of background they're coming in with there's a lot of variation some have a background in competitive math like the olympiads some have been exposed to math in summer programs and some depending on geographical location are actually able to take uh university courses while they're in high school but not everyone here is coming
in with such intense experience some of the students simply wanted a challenge and discovered their love of math in the process there's also the people who might have been attracted by the notoriety of the course originally but come to find um an appeal and a beauty to it for themselves on once they've experienced it and I found that this was exactly the case with M I was like not a math major coming in here I never really did like competition math experience I I have like no no competition math experience at all it's not like
if you're like in like if you've never done a proof before like you're completely screwed it's like you just have to spend a little more time than everyone else does but like you'll get you'll get adjusted like a lot of stuff in this class is just like very elegant so this right here is a math 55 P set if you want to pause this video and try some other problems feel free now rumor has it that a single P set takes anywhere between 24 to 60 hours to complete I want you to think about that
for a second 24 to 60 hours that's a 36-hour gap it's hard to believe that there's that much volatility between pets so I asked Wan for his Insight just strictly like doing the pets probably between 15 to 20 including like lectures and like just self-studying like the material mhm upper bound like maybe 30 like probably between 20 and 25 yeah clearly the pets don't take nearly as long as people say they do and students are expected to collaborate and then best of all the grading is very lenient okay okay so the grading is very lenient
the pets don't take nearly as long as people say they do and the course has a collaborative nature so then why do half the students drop out well they don't that's just an outdated rumor the reputation might have been a little bit exaggerated over the course of the years and a lot of the sources that people were pulling from um were sources that dated back to the beginning of the course and so like any course um it evolved or the years with the students needs and with the way that pedagogy evolved itself especially mathematical pedagogy
the rumor that half the students drop out comes from 1970 back when yes half the students did drop out there's actually a quote from upen Professor David harader who took math 55 back in 1974 he said 70 students started it 20 finished it and 10 understood it but in the present day as of this Academic Year only 3.3% of math 55 students have dropped out 61 started in math 55A and 59 remain in math 55b so then what was going on a few decades ago when students were seemingly running away from this course well first
of all it used to be very different one of the problems we had with the course uh was that there wasn't really anything like a standard syllabus there were years when it was focused on one topic or another topic and there were years when it was extremely difficult ult than other years when it was normal and that's not a good thing you know if this is going to be the main introduction for a certain body of our math concentrators it should be you know more or less the same from year to year I think we're
sort of gradually converging to that also students back then didn't have nearly the level of mathematical experience students have today summer math programs didn't exist the Olympiad I don't know if it existed but I certainly wasn't aware of it at time students coming in back then were smart but they didn't have anywhere like the background and that changed a year ago I asked you all what's your favorite subject in school and math was the most popular answer and then a few months later I asked what's the hardest subject for you in school math again was
the most popular answer I think the same holds for these students math 55 is probably the hardest course they're in but it's also probably their favorite a lot of people say that the bonds that they form um through taking math 55 are bonds that last them through the rest of their time here at Harvard they come here and they discover there's actually not just one or two other kids who are like them but a whole Community MH higher education math isn't really about the individual and it's not even about how fast you can solve a
certain problem to arrive to a certain specified conclusion it's really a community based field everyone's like extremely nice people are super kind if you're confused on anything like you can just ask someone and they'll like gladly explain it to you yeah it's super rewarding and the math department accepts you with like open arms and it's very nice is it challenging no doubt is it one of the hardest undergraduate math courses in the country probably but is it this Cutthroat ruthless unforgiving chor that people on the internet make it out to be absolutely not if you're
interested in mathematics if you're motivated to learn mathematics and if you're willing to spend the time you can do this the sense of community both within the course and the math department is unparalleled these are some of the world's most Brilliant Minds working on some of the world's most pressing problems I think as a huge cohort of students who among whom are going to be the uh the leaders of the next generation of mathematicians
