I visited the world's hardest math class

[@iwetmyplants2602]

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.

[@wg4112]

yeah, every single higher level math class was much more relaxed and calmer atmosphere compared to high school math or those big lectures that have 200+ students from different degrees

[@greyklopstock7155]

​@@wg4112exactly. It's like a bunch of high level musicians, or high level athletes being in a room together. Sure, one person may be "teaching" on paper, but everyone kinda recognizes that you're all more or less on the same level or capable of being on equal footing. So eventually, you get to a level of fluency where you are essentially just shooting the shit

[@Deoxys_da2]

Working as a group? Worst nightmare for introvert But there are brilliant kids who does it alone

[@wg4112]

@@Deoxys_da2 you are not getting any succsess in these type of classes without working together in some way with these homework assignments

[@geederer]

This weird internet subculture of insecure midwits vicariously fulfilling their delusions about having mathematical talent through shadowing/idolizing/gossiping about mathematicians is perhaps the most pathetic known form of existence.

[@kristimotra8179]

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

[@pumpkineer_3047]

It’s not that we aren’t taught it correctly, it’s that’s it’s such an over-natural substance that as a language it isn’t preferred or even viewed as a language that it isn’t already under its respective components towards its own inter-natural curation. We have nothing to do with math, but it has everything to do with us.

[@duckymomo7935]

If you knew anthropology, some cultures don’t even perceive math the same way that the west does

[@iMizzy777]

🧠 🪦my brain is dead

[@eglistellez9185]

I think also depends on the teachers you had, especially early on

[@steelraven117]

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

[@Oirezthefirst]

I completely agree, I used to HATE math, but in high school I had a very good and fun teacher that taught me to actually enjoy math.

[@OG_Rapid]

the professor in the vid looks so nice bro I wish he was my teacher

[@ShortsCreatorGuy]

So true. Honestly, the bigger problem I have faced is the TEXTBOOKS. Math textbooks are not digestible at all. Oftentimes they are written in a way where only those who already understand the content can accurately perceive what it's saying or trying to convey. Skipping steps, implicitly applying rules/theorems that a novice or beginner might not understand, using really high level jargon that isn't easy to understand, etc.

[@ragnarok4241]

Most teachers ever

[@steelraven117]

@@Oirezthefirst same

[@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

[@r22gamer54]

yeah, i heard most harvard students take it freshmen year though

[@bookenable6942]

​@r22gamer54 Nah, a fair few enroll at the beginning, but nowhere near most (ie ~40-80/2000) It wouldn't make any sense for humanities concentrations (unless really heavily into math or joint/doubling lol) to take the course, and honestly it is rather tough unless you have some previous experience with proof based math etc, which usually boils down to some math, physics, and cs students

[@ayushsambher920]

yep, the p set was talking about metric spaces I guess

[@wetrustinscience1825]

Both Representation Theory and Algebraic Topolgy are taught in year 2 or 3 for math students in Asia. They are rather the basic tools for solid college-level math

[@ayushsambher920]

@@wetrustinscience1825 hey! Representation Theory in form of Groups and Ring Theory is taught at the undergrad level Currently a final year Maths undergrad here from India! Lie Algebras and more is taught at the graduate level here

[@atomicJUMP]

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

[@cubecubes8288]

It is NOT that funny

[@Miguel_anitez]

Stop rolling little bro

[@MadChristoph]

Reminds me of the Futurama episode where Fry had a dream on a Final Exam of Ancient Egyptian Algebra

[@Miguel_anitez]

@@MadChristoph who asked?

[@MadChristoph]

@@Miguel_anitez you asked

[@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.

[@VGK702]

Very true then when you finally understand the concepts you're studying and it clicks in your mind it almost feels like your brain just leveled up. It's a very satisfying feeling

[@themaskedman221]

Funny, I hate proofs and always have. But then again, I am not a mathematician, but simply someone who needed to use math as a tool.

[@JohnMuhammad-mg5wp]

Nerd

[@duck-loves-tea]

I never thought I’d meet someone who felt the same way about math, you explained it perfectly. I love math when I understand it, because the things come together and make sense in my mind and feels beautiful

[@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 🙏🏻

[@Proxz]

Hello pekoras future husband, it boggles my mind when i see someone with a vtuber pfp and name type out even a single comprehensible sentence that isnt absolutely debaucherous. Thank you for letting me witness this rare sight

[@pekorasfuturehusband]

@@Proxz My pleasure! My apologies for the braindead individuals who tarnish the image of Vtuber fans…

[@firemonkey1015]

I’m a second year CS student too, made me love math because it’s pretty much all I do. Math is one of those subjects that epitomizes Stockholm syndrome.

[@jixs4v]

the notes for math 55a and 55b are out on the internet, you can self-study them.

[@pekorasfuturehusband]

@@jixs4v Oh really? I had no idea, thanks 🙏🏻

[@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.

[@michaelbodell7740]

Yeah, this is more similar to my experience nearly 30 years ago I feel like there were about 100-120 of us in 25/55 joint at the start during shopping period and then when the courses split about 25 stayed in 55 and most of the rest were in 25. But international students and IMO competitors were common, even in 25 about half the class had been past IMO participants. I did 25 and generally the problem sets took around 10-15 hours (with encouraged collaboration!). I think the 55 problem sets took about the same time for those in that class (but were harder/more material so the 55 students would have been faster if they were doing the 25 psets). If you measure difficulty by amount of work for problem sets (not really the best measure), the hardest Harvard class at my time was the OS Programming class CS 161. Those problem sets tended to take a solid at least 40+ hours of work minimum (due every 2 weeks at that time), and some few folks were probably closer to 80 hours for the problem sets (depending on debugging and preparation).

[@itskindasus9730]

Hello, kinda off topic but you got any tips on how to study physics cuz u seem to know a lot about it😅😅 (For context I am preparing for an entrance exam with MCQ type questions and I am struggling with physics ) It's not tough or anything but i am just blank when solving problems 😢

[@worsteditz7540]

How old are you

[@Sp3cterJohn]

Only for freshman?? Wtf then what do upperclassmen take????

[@noahmoore1546]

I go to a foreign university where the mathematical curriculum is very proof based and I happen to reference Math 55s course notes all the time when I'm studying

[@haileychuukisu]

everyone: this is the hardest math class in the world me: I bet the professor is using hagoromo chalk

[@deltalima6703]

Students will need magdust to keep from dropping

[@GlitchVR]

First thing I thought the video was on about in the short 😂

[@geederer]

Probably the fake Korean knockoff though.

[@JoelStudies]

@@geederer It's not really "fake" the quality is still great, writing with it is smooth and the chalk wipes off very clean. I still recommend it over any other chalk I have used in the past!

[@angel-meta]

That’s fucking exactly what I was thinking lmfao

[@eriknovak496]

My hot take is other universities have classes like this too, they just don't market the class the same way. I think UChicago has something like this. I'm an undergraduate student at Duke and took math 403 "advanced linear algebra" my freshman year, which is abstract linear algebra, point set topology, a bit of analysis and group theory, smooth manifolds, algebraic geometry, matrix perturbation theory, convexity, and a few more things I don't remember at the moment. The class was more so a tour of grad level math through the lense of linear algebra... and I took it as a freshman. Homework took around 35 hours to do for me per set, and I stressed a lot, but I made it. It was a highly collaborative and leniently graded class as well. It was a 1 semester course. So, it seems to me maybe other top unis just don't market their classes this way, even though they do offer them.

[@boomerzilean]

its jut classic harvard syndrome

[@AdrianRuiz-gp5ru]

I’m a first year at uchicago and I mostly agree with this, though I will point out that here the class that gets all the talk among students is honors analysis IBl 1-3 which does have that sense of notoriety that math 55 gets though at a much more local level. I’m not as advanced in math as to be able to speak on the topics covered, but I have heard that you must dedicate your schedule and time to the class but that the grading is very lenient so people find it satisfying to take.

[@joannabanti7590]

Who came here from the short😂👇👇💯

[@vffgaming491]

Literally everyone

[@jaseregalado2408]

Literally everyone

[@mustafahussein5849]

🤚😂😂😂

[@ivanristic5512]

Me

[@Frotdog]

No your braindead

[@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"

[@_Nibi]

k

[@GratitudeDueLife]

There may be more theories in the future once this current is explored more. I also agree(even though I’m not good at math) that we lacked at some point of the equation. Thank you for this! It really helped me understand better ^^

[@mnwtwo]

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

[@DVnc_]

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

[@SoccerKeeper]

All of Gohar’s videos are bangers

[@r22gamer54]

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.

[@goharsguide]

he's gonna crush it. 🤝

[@shakercheese5092]

Nah, he's gonna drop out.

[@ShrishGamerTV]

Bro’s a menace 😭

[@gamingbutnotreally6077]

He’s gonna drop out, he’s the initial stage of dunning Kruger rn

[@HeavyMetalShredder]

@@goharsguideimagine hacking into Mahads email and signing him up for this class without him being aware 🤣

[@byronavihai3787]

I recommend you look into the French system of the "classes préparatoires" (prep school), which condenses 4 years of undergrad math and physics into 2 years. Similarly to Math 55, all the problems we tackled were of insane difficulty, there was a strong sense of collaboration within the class, as well as competition versus other schools / classes préparatoires. These classes préparatoires prep you to apply to the "Grandes Ecoles" (the Great Schools), which are highly valued engineering schools and hard science schools (best among which the Ecole Normale Supérieure and the Ecole Polytechnique de Paris). In any case, I'm sure there's a good video or two to make about them! ;)

[@Raih-k7d]

wow! I wonder, is there any benefit to condensing the material? do people want to start grad school sooner or something?

[@joelwillis2043]

@@Raih-k7d French sure over produces brilliant mathematicians compared to their population size.

[@heeba00]

The same in Morocco

[@spinyslasher6586]

I was thinking this was gonna be some otherworldy course, but then in 3:08 when the actual content was shown, it hit me. This is literally just a combination of all the courses I took in the span of 2 semesters back in 2022 as a physics undergrad.

[@lawnside82]

The beginning of this video op makes it sound one way... further into the video its not that serious. Just over exadurated...kinda clickbate...

[@mychaelsmith6874]

Physics majors don't take most of these topics. Most likely just linear algebra and only the algorithmic parts of it. Most physics majors aren't going to care about Sylow's theorems or the Van Kampen theorem.

[@spinyslasher6586]

@@mychaelsmith6874 well I took them cause they seemed interesting. Wasn't that tough tbf.

[@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

[@samrobison6421]

Seems like the lectures would be interesting but the problem sets would drive you insane

[@srallulrich]

@@samrobison6421 yes, in the video you can find that Banach contraction theorem was given as a problem in one sheet, which is kinda crazy

[@liangyuaq-qoyunlu407]

It's trivial to make a class hard. If you teach Langlands program as a first year undergrad course, it would be much harder than this

[@srallulrich]

@@liangyuaq-qoyunlu407 from what I've seen this is far from impossible to get through for motivated 1 year students.

[@ot8479]

Not really!!! Such a non-structred mess of a 'course' just creates dumb people full of themselves without actual full understand of each topic!!! Plus, Harvard is well known like Yale for inflating grades and giving out As like fruits!

[@Spacexioms]

I’m a math major. I’ve taken abstract algebra, linear algebra, and real analysis separately in a span of three semesters. I can’t imagine doing all of these in 16 weeks.

[@Kidus5867]

not my stupid ahh pausing at 8:15 to try the questions

[@xyurem7461]

10:46 his voice is so calming.

[@goharsguide]

bro this video just dropped 60 seconds ago. how did you already get there

[@xyurem7461]

20x speed

[@SpaceBK_GD]

@@goharsguide10x speed bro

[@CLOCKROOK]

a

[@GeographyZ]

​@@goharsguide😂

[@jefferydeanmorganwife]

my toxic trait is that i think i could do this easily

[@kristophamiller4501]

😂😂😂

[@cozz124]

so me (im JUST starting calc during my first year of college)

[@PLP0321]

This brings back so many memories. I excelled very well in low level math courses such as Calculus/Multivariate Calculus, Linear, Alg, Diff. Eq, etc. I thought Math was the route for me since I loved it and understood the foundational courses very well. I entered advanced level mathematics courses in Real Analysis, Abstract Algebra, Complex Analysis, etc. and I felt like I was learning the natural numbers all over again. The scariest part was when I found out these weren't even the advance courses! It puts into perspective how smart and/or how good at mathematics some people really are.

[@MichaelPennMath]

great video. Interesting results from polling your audience. I have always said that even math majors think that math is the hardest subject!

[@Maths_3.1415]

Michael penn ❤

[@xeyalaliyev9432]

One of the greatest mathematicians -Michael Penn- Greetings from 🇦🇿

[@ReyanshTiwari-d6n]

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.

[@rinnegana.k.apurpleshariga9947]

Hey gohar, I found your channel 2 years ago. I was a sophomore who had the worst study habits, now I’m a graduating student with far better grades, thank you man

[@atomicJUMP]

keep goin' man

[@phipsy6941]

As an electrical engineering student at ETH, i could never create a deep and rigorous indirect proof of a "more complex" theorem, although i find the art and abstraction, but precision of proof based mathematics truly facinating. thank god engineering math doesnt depend on proof. we shall leave that to the mathematicians.

[@BrittanicAsianic]

Hard, but hardest in the world would be Cambridge's Maths Tripos.

[@Dreamprism]

I had a friend a couple years younger than me in this class. I was a strong math student at my university, but she was basically doing in that class what I was already doing in mine (if not harder problems). This was like 2009-2010, so I don't remember much about it. I know we talked about a few things from the class. Nice to see this video pop up in my suggestions.

[@pinomarittimo7614]

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.

[@bowlingking300]

Everyone hype this man up

[@hugyodeepwoken]

if you are begging for likes, you are not smart enough to do math 55 🙏🙏🙏🙏🙏😭😭

[@Endorcyst]

"... until I can pass the class"

[@pinomarittimo7614]

@@hugyodeepwokenI will prove you wrong

[@ConfusedCatd]

Upload a vid

[@aditya_a]

Good video! From what little I can tell from the psets and lecture material, it honestly looks difficult but doable! Seems like a mix of real analysis, advanced linear algebra, abstract algebra, etc. The psets look fun too

[@justrecorded4u984]

The only major thing is changed that number of international students increased so that is why the class became smarter than in the past.

[@sheena5006]

i’ve had terrible experiences with math teachers, a few good but mostly bad. this vid made me feel dialed in

[@Mojojo61]

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

[@Phenom0100]

I have a Master's in Computer Science, but math has never been my strongest subject. I can complete a math course and pass, but I never fully understood what I was doing in those classes. The math courses I have taken in college include calculus, discrete math, and statistics. I also throughout my life have had issues with word-based math problems.

[@misskathgandaomsim325]

isn't computer science applied math? Did you have a problem since it isn't your best subject?

[@Phenom0100]

@@misskathgandaomsim325 Not really. It is more the understanding of how computers work. You may be thinking more of computer engineering. This is understanding high to low-level programming languages, memory hierarchies, and more. Math is important, but not an integral part. Math for my bachelor's degree was mainly 3 pre-requisite courses. After those courses, there were Discrete Math 1 and 2. If you need any equations for a program it is just as easy for me to look them up get a sense of it and plug it into a program. It also depends on what you are using the degree for. For me, at least it was not about understanding all the math, but making use of the math. I am a network administrator so I don't really need any of the calculus I have gone through. Here is a description of my bachelor's degree in computer science: "This major provides a broad-based background sufficient for entry to a career in computing. It provides a systems view of computing, emphasizing the interdependence of design, object orientation, and distributed systems and networks, from basic software through systems design. This major develops core competencies in programming, database structure, computer organization, and operating systems. It allows opportunities to study a range of other aspects of computer and information science, such as information security, data communication/ networks, computer and network systems administration, theoretical computer science, and software engineering. Every student can expect to become proficient at programming in C , C++, and Assembly Language. Additionally, students will be exposed to other languages such as Java, C#, Ada, Lisp, Scheme, Prolog, Perl, and HTML. This major is available on campus or through Distance Learning." Here are some of the main courses that I have taken (Undergrad and Graduate): Advanced Operating Systems Advanced Programming with Java Artificial Intelligence C# Programming C++ Programming Computer Architecture Computer Network and System Administration Computer Organization Computer Organization 2 Concurrent, Parallel, and Distributed Programming Cybercrime Detection and Forensics

[@cozz124]

@@misskathgandaomsim325 doing math is one thing but understanding what the hell youre even doing is another

[@iamkj2008]

8:15 I wouldn't wish this on my worst enemy

[@theforeskinsnatcher373]

they arent too bad honestly. For olympiad winners this should be a piece of cake

[@aguy3119]

This is actually incredibly cool, I'm self studying ODEs and Proof-Writing and seeing this was incredibly eye opening, I definitely want to go to experience this or go to some ivy league more than ever now, man I've had 2 of these eye opening moments within 2 days lmao

[@NN-fw9il]

Huge respect to anyone taking this course😅. Quite intimidating tbh especially at an Undergrad level.

[@shirosflower]

i love lecturers and professors with my entire heart they’re so smart and so passionate

[@christophvonpezold4699]

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!

[@coochiesl4yer696]

imma need u to update this comment when ur done pls

[@zataishasims9732]

Goodluck keep us updated on how it goes.

[@thechosentaco]

How is it coming along?

[@PeterAtWork430]

Update or is your mind fried ? :)

[@pabloa2228]

8:15 the solutions are on the left and the problems are on the right for anyone wondering. The first problem looks like an epsilon-delta proof so I’m going to guess this is part of the Math 55B curriculum covering Real Analysis. As others mentioned, as a math major you take these classes separately. For example the core math curriculum at my university required Linear Algebra, Abstract Algebra, and Real Analysis. Complex Analysis and Topology were elective higher level maths you could take. Math is like a language, so it’s no different like trying to take advanced conversational Spanish when you only had Spanish 1. Once you learn the language you learn to read in math. Then it’s a matter of covering subjects. The gateway to learning how to read in math is an intro to proofs class. Math is written through rigorous proofs. At my university the class was called Fundamentals of Mathematics and was a 3-level class. It had a Calculus 1 prerequisite. Different schools make each of these subjects more or less rigorous. The fundamentals are the same. Abstract Algebra is really when you dig into the philosophy of mathematics in my opinion. I hope that helps someone

[@bobalover3736]

This video has convinced me to try taking math 55 next year

[@itstriin]

😮😲🤯

[@Dlbt]

Do it bro

[@YK-yj6gk]

i will also try meth

[@BadChess56]

How has it been going?

[@DariusOceanLife]

nice joke 🤣🤣🤣

[@lt_bigblue4934]

I’ve heard talk about Math 55 time and time again, I’m so glad to see that someone finally makes a perfect video explaining it ❤

[@SnappBacc]

Love the video and the content you got. Its so interesting to see what these people who love math so much have to share. The editing is great but I felt the excess jumpcuts were kind of messing up the flow at times

[@artichoke60045]

I went to MIT back when there was only one version of 18.100, which is now effectively the hardest of three versions. It wasn't "advanced calculus" (a redo of calculus with some rigor added; this version explicitly exists now) but point set topology, and you can derive the calculus on your own time if you want. Whether it felt good or not, I learned a hell of a lot, which has come in handy in unexpected ways in my life. I got a B but decided not to major in math. Maybe I was intimidated by a fairly unpleasant but mathematically well trained fellow student in my dorm, but mainly it was that I had other interests. Back then, though, intimidation was allowed and certainly part of the MIT culture, now it is not. No doubt in some years, those Math 55 problem sets contained problems even mathematicians would consider hard, like the 2nd semester MIT organic chem class 5..42 that asked students to show reactions that had won Nobel prizes in prior years -- with guidance but they still had to figure it out. Back then it was seen as a selection process as well as teaching, now such selection is culturally unacceptable and, also, departments are fighting for resources and want to attract more majors, not drive them away. There were well prepared students back then (having attended public high school I was not one of them), and there were things like Math Olympiad too for the few who were that skilled at such an age. At the college level the Putnam exam was already well known. AP calculus has gotten vastly decontented since then. We had many multi-step integrals and a simple proof on the Calc BC exam I took; now my kids had only easy integrals and they would not dream of asking for a proof. So there are surely more math-illiterate students at Harvard today than back then; the low end is lower. I'd be surprised if the other end of the Harvard math bell-curve is a lot higher, since the cultural influences are pervasive. Overall, since Math 55 is meant to be a survey of undergrad math topics, standardizing it and optimizing the teaching does seem like a good thing for Harvard's teaching mission.

[@julianbruns7459]

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 c0 we can chose n such that c^n/(1-c) d(x_1,x_0)

[@julianbruns7459]

@@luis_editzx6832 thanks for notifying me. I don't use instagram so i would have probably missed this otherwise.

[@AclibButLikeTheRealOne]

nerd

[@ecaorez4881]

omegalul roflkopter bin ich gerade am experiencing been 😮

[@Um-a]

y e s

[@firemonkey1015]

Nice bro

[@alex_urso]

I randomly came across this video few days before I actually have to attend one of my most important exams which can decisively change my future. I used to always suck at maths and always hate it, the 1st of July, I'll be attending an online retake of a math exam that could allow me to enter in my second year of EHL (the best management school in the world in Switzerland), and if I fail it, I'll be forever kicked out of the program, so I'm really stressed but watching this type of videos really encourages me to succeed in it. Wish me luck !

[@miguelmaravilla7646]

Hell no you will never see me taking a class like this lol. Math gives me so much anxiety , I barely survived college algebra ! But I commend those who do take it!!!

[@Messi-rw9ng]

Math anxiety usually means you had bad teachers who turned you away from math. I had the same problem and it sucked

[@Jonasz314]

Have you tried to go through Mathematiques Superieures in France? I'll bet you anything you want it completely kicks this class in the teeth, mainly on the sheer volume of math/physics and also the level of complexity. There are requirements to enroll in this type of program (mainly, you have to go through french high school program). In the first year, you will have 7~8h of math per week, and 4~6h of chemistry. If you make it to the second year, the amount of math increases to 11~14h (based on the specialization you take, either MP* or PC*, assuming you're a top student). The tests are incredibly hard, and that's to train you to the level of complexity you will face at the various competitive exams to enter top engineering schools (Polytechnique, Mines de Paris, Supaero, etc.). The main difference is that grading there is absolutely brutal. Every kid entering this program is an A+ student. Their grades will drop to D+/C at best, only geniuses will maintain anything close to B. Tests would take about 12h to complete, but you're given only 4h and have to do as much as you can - most of the time you have no clue how to tackle any of the question. And since these are graded and timed tests, of course you're not allowed to collaborate. The level of pressure is completely different, that's what makes it incredibly challenging.

[@knqsa1071]

je suis complétement d'accord (je suis en lycee pro)

[@motiv__inspirerocks]

I cant lie bro, your really helping and doing amazing keep on going!!

[@AAAAAA-n9f]

I can’t lie bro, you need help with your grammar. It’s “you’re”. Keep on going!!

[@TheRealZuia]

A class should NEVER set you up to fail. Your goal is to learn, its not a competition.

[@Qwertyuiop-p2s5f]

This course sounds super fun and interesting.

[@silkyuh]

ong

[@DaxSudo]

Those are literally my favorite concepts in math bc they are so foundational to higher level maths. Its awesome.

[@DaxSudo]

These are really good explanations as someone who’s had to self learn a lot of the higher order mathematics after basic training in nuclear physics. Just bc its interesting these concepts are super cool and its exciting to see people take an interest in them.

[@robert_raimondii]

6:02 You know the math is hard when you can’t see a single number on the blackboard lol.

[@misophobia]

this is actually a super cool video. they all seem like reslly awesome and smart minds. thanks for making this man!

[@Theseasonyard]

So yeah the moment he started writing on the board I was blown from the first question

[@bobbyologun1517]

Pause

[@rwharrington87]

8:19, I had paused and returned a little while later to find out a single pizza takes anywhere between 24 and 60 hours to complete.

[@Jude_C_Parnham]

best videos bro, not even Study Gpt understands it

[@affaan01]

first time i was actually interested on going beyond the short, you did good

[@nadish25]

Thanks a lot for your study tips in all your videos ❤❤

[@jamiec6504]

I absolutely despised math in high school and college… particularly algebra, calculus, etc. Ironically I became a bond & options trader on the Street for over 20 years which is kind of funny when you think about it. Even though I had absolutely no idea what was going on here I still found it engaging and interesting. Great work!

[@ks4willbball201]

What I would say is that Math is my favorite subject even though its hard. I love math. The things you hate you are going to love those things. I never liked math But when I got to 8th Grade- 11th grade now I love math.

[@ItzSnorez]

I love math & even decided to master it in my adult life to be fluent without needing a calculator. Particularly basic arithmetic operations; as they seemed to be the most practical to actually go through with learning for real world scenarios. For me to have knowledge stay in my head. It needs to have a relevance to my life, idk how the hell they can do these advanced mathmatics for fun.

[@bbsara0146]

What I hate is that it took the people who invented these proofs years to make them. yet we are expected to come up with the correct proof in a 2 hour long test to like 7 questions.

[@Me-uv6kc]

inventing calculus is WAY harder than repeating it...

[@loverofhumanity]

Lool replicating and essentiallly understanding + memorizing someone else's work is far far easier than having the ingenuity and creativity to invent it.

[@sixbass2588]

Are you a freshman?

[@boxmanatee]

We need more of these KZbin channels.

[@Mrtok522]

Really enjoying this type of content…Keep it up!

[@diegoaespitia]

as a fellow math major. anyone who enrolls in math heavy subjects are there for the love of math. they are NOT dropping out.

[@Rm-bf4rc]

4:20 my brother what????!???

[@Its_xEltonx]

Dude I just wanted to say how good your editing is man it’s great I wish I could edit like you ❤

[@priyanshibais1869]

Thank you so much for such an insightful video ✨✨

[@kut240]

i had a similar course like math 55 in harvard (even though its most likely not even close since im in a university thats nowhere near as good as harvard) which was called "mathematics laboratory". and i took this course when i was in my 2nd year of undergrad (im now in my 4th year). the main problem was, as the people in the video also mentioned, the lack of a syllabus. we were in all kinds of places and the prof. who gave the lecture was the same prof. who was also my topology prof. at the time. so whenever we had a lecture (90 minutes per week) she would get in the classroom and ask "so what did you covered in this weeks real analysis?" and just start off by what we said. since that year was the first time i was studying topology i was already having a hard time but this course was incredibly hard since it included both topology and real analysis. and yes i dropped out after my midterms (i got around a 40ish score out of 100 and thought "why bother when i can basically take another course instead?"). but from what i heard from other people she was teaching algebra, advanced topology and ring theory in the last week of our fall term. so basically the hardest problem after taking basic mathematics courses (im not talking about derivating hard functions etc. like real problems as in not finding the derivative of sinx rather proving that sinx has a derivative) is the lack of a syllabus.

[@HenryT2001]

And why is this the hardest undergrad math class now? Abstract algebra and some algebraic topology in a general topology class are standard compulsory courses of the second year (at least here in Europe). Any algebraic topology, commutative algebra, or functional analysis course would be harder, wouldn’t it?

[@Overlycomplicatedswede]

Any class will be better and more enjoyable to attend if the people and professor are fun to be around and easy to go up too and ask questions or ask for a quick refresher of how to do something

[@kennethvalbjoern]

Talking of difficult math courses. I remember one graduate course in differential geometry and Lie groups I took at uni. The professor was a pole (in Denmark), who could hardly speak the language. He entered class and just began writing on the chalkboard without introduction. He made no references to the course book whatsoever. Nobody knew what the man was talking about at all. You had to complete assignments to pass, and they were difficult. You had to figure out yourself what part's of the book, that contained the tools needed. Most people had to team up to figure them out. I spend 2 days on a 4-page proof of a two line exercise, that I was the only one to do. It was a great course, where I did some great proofs!

[@TheOutlier-c6o]

Kudos to the professor. He is from another dimension, it seems.

[@바둑-n8j]

발음이 너무 좋으시다!! 딕션이 귓속에 똭똭 박히게.

[@deepeshbasnet1974]

As a kid i used to feel bored of mathematics because our teachers were like robot and once i got in 9th standard and the new teacher came he just amazed me and i started taking 2 mathematics subjects (regular and additional) in which i did way lot better (used to barely pass the subject). All we need is a good teacher who makes studying fun. Since then i started having good grades in every field where maths is inserted.

[@lazyfox3649]

There is a very similar thing in Poland called MATEX, which is an experimental math program in the XIV Staszic highschool. Freshmen of that profile are often already finalists of math olympiads

[@tmjz7327]

This is not the same at all and is much harder than matex lol I just looked

[@firemonkey1015]

@@tmjz7327For highschool I’m sure it’s hard, college is another ball park though.

[@lazyfox3649]

​@@firemonkey1015yes, that's what i meant, sorry for any misconceptions

[@lazyfox3649]

​@@tmjz7327 some matex graduates i'm sure could take math 55 though, i'm assuming you looked at the entrance exams, which are much much easier

[@tmjz7327]

@@lazyfox3649 you are right, sorry I misunderstood your original comment to mean matex is equally hard. But yes I believe strong students from this program could do math55.

[@123ucr]

I took Calculus 1, 2, and 3 plus Linear Algebra and Differential Equations and had little experience with doing proofs. I would be no match for Math 55. Heck, I would even be surprised if I even lasted a week in that class.

[@plaidchuck]

Linear algebra didnt do proofs? Thats a failure of the university

[@IceGamer7810]

8:17 well: ### Problem 1 #### Part (i) We need to show that \(d_1(f,g) = 0\) implies \(f = g\). - Given: \[ d_1(f,g) = \int_0^1 |f(x) - g(x)| \, dx = 0. \] - Since the integral of a non-negative function is zero, the integrand must be zero almost everywhere. Therefore, \[ |f(x) - g(x)| = 0 \quad \text{for almost all } x \in [0, 1]. \] - Hence, \(f(x) = g(x)\) almost everywhere. But in the context of continuous functions on \([0, 1]\), this implies \(f = g\). #### Part (ii) We need to show that \(d_1(f,g)\) is symmetric, i.e., \(d_1(f,g) = d_1(g,f)\). - By definition: \[ d_1(f,g) = \int_0^1 |f(x) - g(x)| \, dx. \] - Since \(|f(x) - g(x)| = |g(x) - f(x)|\), it follows that: \[ d_1(f,g) = \int_0^1 |g(x) - f(x)| \, dx = d_1(g,f). \] #### Part (iii) We need to show the triangle inequality for \(d_1\), i.e., \(d_1(f,h) \leq d_1(f,g) + d_1(g,h)\). - By the triangle inequality for absolute values, for any \(x\): \[ |f(x) - h(x)| \leq |f(x) - g(x)| + |g(x) - h(x)|. \] - Integrating both sides over \([0, 1]\), we get: \[ \int_0^1 |f(x) - h(x)| \, dx \leq \int_0^1 |f(x) - g(x)| \, dx + \int_0^1 |g(x) - h(x)| \, dx. \] - Therefore, \[ d_1(f,h) \leq d_1(f,g) + d_1(g,h). \] ### Problem 2 #### Part (a) Show that \(f_n(x) = \frac{nx}{n^2 + x^2}\) converges uniformly to 0 on \(\mathbb{R}\). - For any \(x \in \mathbb{R}\): \[ |f_n(x) - 0| = \left| \frac{nx}{n^2 + x^2} ight|. \] - Notice that: \[ \left| \frac{nx}{n^2 + x^2} ight| \leq \frac{n|x|}{n^2} = \frac{|x|}{n}. \] - Since \(\frac{|x|}{n} \to 0\) as \(n \to \infty\) uniformly in \(x\), we have: \[ \sup_{x \in \mathbb{R}} \left| \frac{nx}{n^2 + x^2} ight| \leq \frac{1}{n} \to 0. \] - Hence, \(f_n(x)\) converges uniformly to 0. #### Part (b) Show that \(g_n(x) = \begin{cases} \frac{1}{2} & \text{if } x \geq \frac{1}{\sqrt{2}} \\ 0 & \text{if } x < \frac{1}{\sqrt{2}} \end{cases}\) does not converge uniformly to 1. - For \(x \geq \frac{1}{\sqrt{2}}\): \[ g_n(x) = \frac{1}{2}. \] - For \(x < \frac{1}{\sqrt{2}}\): \[ g_n(x) = 0. \] - Consider \(x = \frac{1}{\sqrt{2}} + \frac{1}{n}\). Then \(x \to \frac{1}{\sqrt{2}}\) as \(n \to \infty\), but: \[ g_n\left( \frac{1}{\sqrt{2}} + \frac{1}{n} ight) = \frac{1}{2}. \] - Hence, for \(x\) in this interval, \(g_n(x)\) does not converge to 1. Therefore, \(g_n(x)\) does not converge uniformly to 1. ### Problem 3 Given sets \(A\) and \(B\) with the distance function \(\delta\), define sets \(U\) and \(V\) such that they are disjoint. - Define: \[ U = \{ x : d(x, A) < d(x, B) \} \] \[ V = \{ x : d(x, B) < d(x, A) \}. \] - Clearly, \(U\) and \(V\) are disjoint because \(d(x, A)\) cannot be both less than and greater than \(d(x, B)\) at the same time. - To show that \(U\) and \(V\) are non-empty, consider points \(a \in A\) and \(b \in B\). There exists a point \(x\) such that \(d(x, A)\) is less than \(d(x, B)\), and vice versa. ### Problem 4 Claim: For \(U, V, W, C\) with \(W\) disjoint from \(C\) and closed, \(U \cup C \subset W \cup C \subset V\). - Since \(W\) is closed and disjoint from \(C\), \(U \cup C\) is a subset of \(W \cup C\). - Because \(C\) is closed, any point in \(U \cup C\) is either in \(U\) or \(C\), which means \(U \cup C\) is contained in \(W \cup C\). - Hence, \(U \cup C \subset V\). If you make a pdf file from this, you will see what's behind the '\'s.

[@Cheese_Bacon2]

Im speechless😭

[@Aravkrsingh-s4f]

Bro how

[@Vincere40]

Nice spells, wizard.

[@nope6008]

whats with the weird slashes and hashtags

[@abortive1581]

Enchantment table language

[@StephenPheenixx]

This is for people who are planning to persue Mathematics in General (Pure Math) and not applied and computational math.

[@newwaveinfantry8362]

"Undergraduate" is a detail you left out. I'm sure it's really hard, but that's an important detail to forfeit.

[@NinjaBear1993]

I did intermediate algebra when I was a freshman in high school and I finished the calculus 1-3 series by senior year.

[@jakeleo4518]

0:23 JESUSTTTTTTTTTTT bro, homie on the right locked in 💀💀😂😂😂😂 bro already premediating his life goals for the next 10 years

[@pizza8567]

Wow! Now my double accelerated course that I’m taking in a few weeks feels like a cakewalk compared to this.

[@Will_U_Read_4_U]

I adored this video so much!!

[@petem6846]

Regarding the time needed to complete problem sets. I remember my Real Analysis class from my undergrad days. At one problem, I estimated that my average time to complete a problem was 30 minutes--almost all of the time was spent understanding the problem and thinking through possible solutions or paths to the solution and just a minute or two to write down the solution once grasped. Yeah, I can believe the time range given for time needed on a weekly basis to do the homework.

[@zerotwoisreal]

4:08 be honest, who thought this was the sponsor

[@PANCAKESGGLOL]

Me

[@DupeDevice]

I

[@ShelleyWolford-r3z]

You explain even the hardest topics in such a simple way.

[@ZShadower16]

This class is like a do or do not life scenario

[@theladder06]

Very interesting class. Kudos to a great professor and students willing to accept a challenge.

[@JayJeremiahC]

Great video to have a snowball effect with students & professors ❤

[@newwaveinfantry8362]

Seifert van Kampen is not THAT bad, although other topics in Homotopy theory are quite hard.

[@Yalldone]

Gohar your amazing thnx for the amazing content

[@davidperez730]

It's all about having that urge to learn, and having enough mathematics experience to understand what is going on.

[@acropora8]

This is a very fun video, I would love to see more videos like this

[@SuperSabarinath]

I respect you for some reason

[@jay3681]

I liked that Wonjae was part of this interview. He was successful in what many would believe is an impossible course even though he had little math background. There's this fear in taking mathematics and failing for students because there's this false concept that you must love it to understand it, but I think that is far from the truth. As a practicing professional engineer for over 15+ years, people always assume that I must've loved math. My response to them is always the same: I hate it, but it was an ends to a means. My true love is engineering, but mathematics has always been my foundation and I appreciate it.

[@aryawijayaa95]

I love how indian are thinking they are the smartest race in the world

[@thekoolkat-oe9vc]

I was just thinking that this is nothing compared to jee

[@obbyperson7281]

@@thekoolkat-oe9vc JEE is more memorization based and pace, I would know as I am indian, however this course covers proofs and more complex topics that use a huge HUGE knowledge of mathematics

[@whyme3772]

Both them and East Asians have the smart stereotype, but when did this dude say that?

[@kaz9242]

U are as an idiot if you u think jee has any of the topics mentioned in that course ​@@thekoolkat-oe9vc

[@SteveVi0lence]

It doesn't seem to hard to be honest, since its collaborative. The hardest thing is getting use to the notation the professor uses. Some professors want to make it as complicated as possible, but it doesn't have to be