Math 55 at Harvard University

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 Olympiad 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 esteemed 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 them 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 architecture and beautiful brickwork 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 13% of Harvard's 387-year history. Yet it's already brimming with some of the brightest STEM minds in the world, including Won Jae-seo, a Math 55 student I met up with. My first few weeks in the math classes here was extremely difficult for me, just simply because I've never done a proof before. 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 in math. Cause 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 an 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 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 an area of interest to me. a pretty small classroom with around 20-30 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. Now 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 Seifert Kampen 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 ChatGPT. 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 donut? One hole. This mug? Also one hole. So in a sense, topologists 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 PLATO. Here we have one hole again, obviously. This has two holes. And here we have three holes. Now the Seifert Kampen theorem explains what happens when you combine any two 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 Seifert von Kampen, the general form of Seifert von Kampen in Moncrief, 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 Seifert non-confident is telling us. And that's the beautiful part about this lecture. He takes this abstruse, almost impossible to understand concept and makes it much more digestible by using a related concept. Quickly, however, the lecture starts ramping up. phi from z to t such that phi composed with mu over one is equal to phi. 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 subject. If we think of a star b prime as the test object t, we get an induced fact from a star b to that object. Even though the material has far surpassed what I learned in 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 or 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. A big part of the course, 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 maths. Some have been exposed to math in summer programs. And some, depending on geographical location, are actually able to take 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 an appeal and a beauty to it. for themselves once they've experienced it. And I found that this was exactly the case with Juan J. I was like not a math major coming here. I never really did like competition math experience. I have like no competition math experience at all. It's not like if you're like, 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 adjusted. Like a lot of stuff in this class is just like very elegant. So this right here is a Math 55 pset. If you want to pause this video and try some of the problems, feel free. Now rumor has it that a single pset 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 psets. So I asked Wanjay for his insight. Just strictly like doing the psets, probably between 15 to 20. including like lectures and like just self-studying like the material. Upper bound, like maybe 30, probably between 20 and 25. Clearly, the p-sets 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, so the grading is very lenient, the p-sets 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 were sources that dated back to the beginning of the course and so like any course it evolved over 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 UPenn professor David Harbader, 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 3.5% of the students 53% 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 this course 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 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 the 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 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. 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 course 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, right? 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 there's a huge cohort of students who... among whom are going to be the leaders of the next generation of mathematicians.

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 esteemed 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 them 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 architecture and beautiful brickwork 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 13% of Harvard's 387-year history.

Yet it's already brimming with some of the brightest STEM minds in the world, including Won Jae-seo, a Math 55 student I met up with. My first few weeks in the math classes here was extremely difficult for me, just simply because I've never done a proof before. 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 in math.

Cause 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 an 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 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 an area of interest to me. a pretty small classroom with around 20-30 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. Now 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 Seifert Kampen 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 ChatGPT. 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 donut? One hole. This mug? Also one hole.

So in a sense, topologists 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 PLATO. Here we have one hole again, obviously.

This has two holes. And here we have three holes. Now the Seifert Kampen theorem explains what happens when you combine any two 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 Seifert von Kampen, the general form of Seifert von Kampen in Moncrief, 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 Seifert non-confident is telling us.

And that's the beautiful part about this lecture. He takes this abstruse, almost impossible to understand concept and makes it much more digestible by using a related concept. Quickly, however, the lecture starts ramping up.

phi from z to t such that phi composed with mu over one is equal to phi. 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 subject.

If we think of a star b prime as the test object t, we get an induced fact from a star b to that object. Even though the material has far surpassed what I learned in 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 or 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.

A big part of the course, 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 maths.

Some have been exposed to math in summer programs. And some, depending on geographical location, are actually able to take 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 an appeal and a beauty to it.

for themselves once they've experienced it. And I found that this was exactly the case with Juan J. I was like not a math major coming here. I never really did like competition math experience. I have like no competition math experience at all.

It's not like if you're like, 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 adjusted. Like a lot of stuff in this class is just like very elegant. So this right here is a Math 55 pset. If you want to pause this video and try some of the problems, feel free.

Now rumor has it that a single pset 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 psets.

So I asked Wanjay for his insight. Just strictly like doing the psets, probably between 15 to 20. including like lectures and like just self-studying like the material. Upper bound, like maybe 30, probably between 20 and 25. Clearly, the p-sets 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, so the grading is very lenient, the p-sets 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 were sources that dated back to the beginning of the course and so like any course it evolved over 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 UPenn professor David Harbader, 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 3.5% of the students 53% 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 this course 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 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 the 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 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.

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 course 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, right? 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 there's a huge cohort of students who...

among whom are going to be the leaders of the next generation of mathematicians.

Transcript for:Math 55 at Harvard University

Transcript for:
Math 55 at Harvard University