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.
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
@@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
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.
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
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.
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.
Correctly is very ambiguous, being part of a system that does math one way and then transferring to a system that does math another way is much more understandable.
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 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
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
@@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
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.
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).
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 😢
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
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.
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
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
@@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!
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 🙏🏻
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
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.
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.
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"
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 ^^
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.
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
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.
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.
Transitioning to analysis versus just solving math problems is such a weird transition. I had such a hard time moving into geometric proofs in high school, but that experience prepped me to be ready for analysis in college and it wasn't nearly as bad. Not having done a proof before and jumping into analysis has got to be VERY daunting. I was lucky enough to get introduced to it early, so analysis wasn't too bad. My analysis professor even said that the course will either make or break people. I personally loved it, it was so freaking cool to see the entire structure of math laid out. Because 1 + 1 = 2, therefore you prove all these other theorems and corollaries such that the new idea is proven. I remember during analysis class when math clicked with me. Beyond 1st year math you stop solving equations and mostly move into proofs. It's a hard transition but once it clicks, it's amazing!
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.
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! ;)
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.
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)
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
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!
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
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 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
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.
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.
Lool replicating and essentiallly understanding + memorizing someone else's work is far far easier than having the ingenuity and creativity to invent it.
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.
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.
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.
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.
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
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
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.
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 !
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.
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
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
@@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.
Although i'm in year seven and only started algebra and quadratics and expanding brackets in October. My goal is to make it to oxford university because i'm from England. I love your tips and I save them for when I am older and I recommend your videos to my cousin who has just left for Cambridge university. Have a great day X.
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?
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.
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.
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!
4:38 I’m not too familiar with topology but is there a clear distinction between torus-es and filled in torus-es (donuts)? I know toruses have 2 holes, the obvious visible one and the one inside the torus. Since donuts are filled in, is the hole on the inside just nullified?
Yes, we do make a distinction between a ball (filled in) and a sphere (not filled in) (they both have 0 holes though). In this case, both the donut and the coffee mug are filled in so this is no problem.
11:19 When you asked about people's favorite class and the hardest class for them, do you know if the same people answered "math" for both? I can imagine a world where a plurality of 38% picked math as their favorite course and later a majority of 52% picked math as their hardest course but yet maybe a mere 10% had actually selected both options.
I’m a thirteen yr old boy in middle school and this is very interesting to me with me wanting to understand more of this work and classes, these are the classes that I want to take in the future once I get to college. The work and math that Harvard and different colleges have are amazing, the classes and professors at these colleges are amazing. I want to go to Harvard and want to get into Astro physics. These look amazing and are very interesting to me
Yeah idk, Im heading into high school next year and have already taken Algebra 2 and am hoping to take Pre-Calc as a freshman. I cant ever imagine myself being able to understand 10% of what theyre saying @8:15 lol, absolutely ridiculous
@@niciasdorcent2652 I took Algebra 1 in 7th yes, geometry summer between 7 and 8 and then algebra 2 in 8th, though i only scored a 1310 on the psat so idk if i can even say im that smart
i did the same got a 5 on calc bc sophmore and i took linear algebra/multivariable calculus through georgia tech dual enrollment junior year and im applying for applied combinatrics and differential equations this senior year but i doubt i will ever understand any of this
@sdfsfsdf641 Jesus a 5 on calc bc in sophomore year??? That's wild... even with everything I've done, I doubt I'll get there by my sophomore year, hopefully junior tho. Just goes to show there's always someone better than you, so never get cocky
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.
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.
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!!!
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!
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.
No way in hell is a group theory class the hardest math you can take at the undergraduate level. Now, as a reference I have a PhD in stat but my masters was in math (topology) and I sat down for a few days to work through an archive of the problems. I can say, with certainty, that there are courses in physics, topology measure theory that most Ivies have which blow this out the water and it isnt even close. Hell, if the professors are particularly aggressive there are courses in optimization and linear models which would put this to shame. Group theory (and subsequently group theoretic/ analysis based calculus) is a beautiful subject which builds an almost natural intuition compared to some of these other fields Ive listed
I study in the Department of Statistics and Informatics. It is a very difficult and complex section. I completed my bachelor’s degree in 2021, and I am currently studying for a higher diploma, which is less than a master’s degree and higher than a bachelor’s degree. Greetings from the State of Kurdistan 🇹🇯
![Why you didn't learn tetration in school[Tetration]](/img/n.gif)
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.
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
@@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
Working as a group? Worst nightmare for introvert
But there are brilliant kids who does it alone
@@Deoxys_da2 you are not getting any succsess in these type of classes without working together in some way with these homework assignments
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.
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
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.
the professor in the vid looks so nice bro I wish he was my teacher
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.
Most teachers ever
@@Oirezthefirst same
Math is a language that some of us aren't taught correctly.
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.
If you knew anthropology, some cultures don’t even perceive math the same way that the west does
🧠 🪦my brain is dead
I think also depends on the teachers you had, especially early on
Correctly is very ambiguous, being part of a system that does math one way and then transferring to a system that does math another way is much more understandable.
" as the letters turned into Hieroglyphics" got me rolling so bad 😂
It is NOT that funny
Stop rolling little bro
Reminds me of the Futurama episode where Fry had a dream on a Final Exam of Ancient Egyptian Algebra
@@MadChristoph who asked?
@@Miguel_anitez you asked
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
yeah, i heard most harvard students take it freshmen year though
@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
yep, the p set was talking about metric spaces I guess
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
@@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
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.
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).
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 😢
How old are you
Only for freshman?? Wtf then what do upperclassmen take????
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
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.
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
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.
Nerd
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
Taylor series and number theory. It's always a rabbit hole!
everyone: this is the hardest math class in the world
me: I bet the professor is using hagoromo chalk
Students will need magdust to keep from dropping
First thing I thought the video was on about in the short 😂
Probably the fake Korean knockoff though.
@@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!
That’s fucking exactly what I was thinking lmfao
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 🙏🏻
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
@@Proxz My pleasure! My apologies for the braindead individuals who tarnish the image of Vtuber fans…
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.
the notes for math 55a and 55b are out on the internet, you can self-study them.
@@jixs4v Oh really? I had no idea, thanks 🙏🏻
Who came here from the short😂👇👇💯
Literally everyone
Literally everyone
🤚😂😂😂
Me
No your braindead
Gohar is such a great listener, pls consider making a video about something related, it'll wait help alot
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.
he's gonna crush it. 🤝
Nah, he's gonna drop out.
Bro’s a menace 😭
He’s gonna drop out, he’s the initial stage of dunning Kruger rn
@@goharsguideimagine hacking into Mahads email and signing him up for this class without him being aware 🤣
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"
k
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 ^^
my toxic trait is that i think i could do this easily
😂😂😂
so me (im JUST starting calc during my first year of college)
Your toxic trait is that you share it
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.
Everyone hype this man up
if you are begging for likes, you are not smart enough to do math 55 🙏🙏🙏🙏🙏😭😭
"... until I can pass the class"
@@hugyodeepwokenI will prove you wrong
Upload a vid
Excellent prep, narration and presentation. Society needs your content.
10:46 his voice is so calming.
bro this video just dropped 60 seconds ago. how did you already get there
20x speed
@@goharsguide10x speed bro
a
@@goharsguide😂
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
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.
its jut classic harvard syndrome
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.
Transitioning to analysis versus just solving math problems is such a weird transition. I had such a hard time moving into geometric proofs in high school, but that experience prepped me to be ready for analysis in college and it wasn't nearly as bad. Not having done a proof before and jumping into analysis has got to be VERY daunting. I was lucky enough to get introduced to it early, so analysis wasn't too bad. My analysis professor even said that the course will either make or break people. I personally loved it, it was so freaking cool to see the entire structure of math laid out. Because 1 + 1 = 2, therefore you prove all these other theorems and corollaries such that the new idea is proven. I remember during analysis class when math clicked with me. Beyond 1st year math you stop solving equations and mostly move into proofs. It's a hard transition but once it clicks, it's amazing!
Gohar you've transformed my high school experience, keep up the grind 🔥🔥🔥🔥🔥🔥🔥🔥
great video. Interesting results from polling your audience. I have always said that even math majors think that math is the hardest subject!
Michael penn ❤
One of the greatest mathematicians -Michael Penn- Greetings from 🇦🇿
All of Gohar’s videos are bangers
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.
The only major thing is changed that number of international students increased so that is why the class became smarter than in the past.
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! ;)
wow! I wonder, is there any benefit to condensing the material? do people want to start grad school sooner or something?
@@Raih-k7d French sure over produces brilliant mathematicians compared to their population size.
The same in Morocco
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.
not my stupid ahh pausing at 8:15 to try the questions
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)
@@luis_editzx6832 thanks for notifying me. I don't use instagram so i would have probably missed this otherwise.
nerd
omegalul roflkopter bin ich gerade am experiencing been 😮
y e s
Nice bro
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
Seems like the lectures would be interesting but the problem sets would drive you insane
@@samrobison6421 yes, in the video you can find that Banach contraction theorem was given as a problem in one sheet, which is kinda crazy
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
@@liangyuaq-qoyunlu407 from what I've seen this is far from impossible to get through for motivated 1 year students.
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!
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
keep goin' man
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.
isn't computer science applied math? Did you have a problem since it isn't your best subject?
@@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
@@misskathgandaomsim325 doing math is one thing but understanding what the hell youre even doing is another
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.
this is actually a super cool video. they all seem like reslly awesome and smart minds. thanks for making this man!
6:02 You know the math is hard when you can’t see a single number on the blackboard lol.
Look, if you’re bad at math, you’re dumb. It’s very easy, and the teachers are so kind. So, yeah, just shut up. 🙄
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.
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.
inventing calculus is WAY harder than repeating it...
Lool replicating and essentiallly understanding + memorizing someone else's work is far far easier than having the ingenuity and creativity to invent it.
Are you a freshman?
i love lecturers and professors with my entire heart they’re so smart and so passionate
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.
The beginning of this video op makes it sound one way... further into the video its not that serious. Just over exadurated...kinda clickbate...
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.
@@mychaelsmith6874 well I took them cause they seemed interesting. Wasn't that tough tbf.
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.
i’ve had terrible experiences with math teachers, a few good but mostly bad. this vid made me feel dialed in
A class should NEVER set you up to fail. Your goal is to learn, its not a competition.
8:18 Ik math is full symbols and all, but how do we get from mostly numbers to mostly words
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 ❤
first time i was actually interested on going beyond the short, you did good
Love the hard work this guys dose and he helps get better at studying thanks Gohar❤
Huge respect to anyone taking this course😅. Quite intimidating tbh especially at an Undergrad level.
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!
imma need u to update this comment when ur done pls
Goodluck keep us updated on how it goes.
How is it coming along?
Update or is your mind fried ? :)
Dude I just wanted to say how good your editing is man it’s great I wish I could edit like you ❤
8:15 I wouldn't wish this on my worst enemy
they arent too bad honestly. For olympiad winners this should be a piece of cake
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.
je suis complétement d'accord (je suis en lycee pro)
I cant lie bro, your really helping and doing amazing keep on going!!
I can’t lie bro, you need help with your grammar. It’s “you’re”. Keep on going!!
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
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
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.
Really enjoying this type of content…Keep it up!
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 !
Thanks a lot for your study tips in all your videos ❤❤
This video has convinced me to try taking math 55 next year
😮😲🤯
Do it bro
i will also try meth
How has it been going?
nice joke
🤣🤣🤣
Hard, but hardest in the world would be Cambridge's Maths Tripos.
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.
"Undergraduate" is a detail you left out. I'm sure it's really hard, but that's an important detail to forfeit.
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
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
This is not the same at all and is much harder than matex lol I just looked
@@tmjz7327For highschool I’m sure it’s hard, college is another ball park though.
@@firemonkey1015yes, that's what i meant, sorry for any misconceptions
@@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
@@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.
Although i'm in year seven and only started algebra and quadratics and expanding brackets in October. My goal is to make it to oxford university because i'm from England. I love your tips and I save them for when I am older and I recommend your videos to my cousin who has just left for Cambridge university. Have a great day X.
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?
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.
0:26 is the image ai generated? Look at the text on the wall
Might be an AI upscale but when an AI makes text it's never coherent
We need more of these UA-cam channels.
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.
Im speechless😭
Bro how
Nice spells, wizard.
whats with the weird slashes and hashtags
Enchantment table language
Very interesting class. Kudos to a great professor and students willing to accept a challenge.
Thank you so much for such an insightful video ✨✨
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!
So yeah the moment he started writing on the board I was blown from the first question
Pause
I see those go stone bowls at 8:52. Do you play?
4:20 my brother what????!???
The left page from the problem set at 8:19 isn't even a problem set.. it's an answer key. Not even an answer key for the questions on the right
4:38 I’m not too familiar with topology but is there a clear distinction between torus-es and filled in torus-es (donuts)? I know toruses have 2 holes, the obvious visible one and the one inside the torus. Since donuts are filled in, is the hole on the inside just nullified?
I think the biggest distinction is that one has inside points and other doesnt
Yes, we do make a distinction between a ball (filled in) and a sphere (not filled in) (they both have 0 holes though). In this case, both the donut and the coffee mug are filled in so this is no problem.
@@julianbruns7459a torus is empty on the inside tho, right?
11:19 When you asked about people's favorite class and the hardest class for them, do you know if the same people answered "math" for both?
I can imagine a world where a plurality of 38% picked math as their favorite course and later a majority of 52% picked math as their hardest course but yet maybe a mere 10% had actually selected both options.
best videos bro, not even Study Gpt understands it
I’m a thirteen yr old boy in middle school and this is very interesting to me with me wanting to understand more of this work and classes, these are the classes that I want to take in the future once I get to college. The work and math that Harvard and different colleges have are amazing, the classes and professors at these colleges are amazing. I want to go to Harvard and want to get into Astro physics. These look amazing and are very interesting to me
I adored this video so much!!
Wow! Now my double accelerated course that I’m taking in a few weeks feels like a cakewalk compared to this.
4:08 be honest, who thought this was the sponsor
Me
I
DUDE THIS EDITING IS INSANE!!!
Yeah idk, Im heading into high school next year and have already taken Algebra 2 and am hoping to take Pre-Calc as a freshman. I cant ever imagine myself being able to understand 10% of what theyre saying @8:15 lol, absolutely ridiculous
Wait, have you taken algebra 1 in 7th grade?
@@niciasdorcent2652 yeah some schools like mine offer that for some students
@@niciasdorcent2652 I took Algebra 1 in 7th yes, geometry summer between 7 and 8 and then algebra 2 in 8th, though i only scored a 1310 on the psat so idk if i can even say im that smart
i did the same got a 5 on calc bc sophmore and i took linear algebra/multivariable calculus through georgia tech dual enrollment junior year and im applying for applied combinatrics and differential equations this senior year but i doubt i will ever understand any of this
@sdfsfsdf641 Jesus a 5 on calc bc in sophomore year??? That's wild... even with everything I've done, I doubt I'll get there by my sophomore year, hopefully junior tho. Just goes to show there's always someone better than you, so never get cocky
Those are literally my favorite concepts in math bc they are so foundational to higher level maths. Its awesome.
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.
This course sounds super fun and interesting.
ong
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.
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!!!
Math anxiety usually means you had bad teachers who turned you away from math. I had the same problem and it sucked
These kids are brilliant but the level of difficulty really makes you think about what level the people who makes these theorems are
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!
I need more videos like these
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.
No way in hell is a group theory class the hardest math you can take at the undergraduate level. Now, as a reference I have a PhD in stat but my masters was in math (topology) and I sat down for a few days to work through an archive of the problems. I can say, with certainty, that there are courses in physics, topology measure theory that most Ivies have which blow this out the water and it isnt even close. Hell, if the professors are particularly aggressive there are courses in optimization and linear models which would put this to shame. Group theory (and subsequently group theoretic/ analysis based calculus) is a beautiful subject which builds an almost natural intuition compared to some of these other fields Ive listed
0:23 JESUSTTTTTTTTTTT bro, homie on the right locked in 💀💀😂😂😂😂 bro already premediating his life goals for the next 10 years
This was fascinating. Great job, thanks.
This is a very fun video, I would love to see more videos like this
I respect you for some reason
I study in the Department of Statistics and Informatics. It is a very difficult and complex section. I completed my bachelor’s degree in 2021, and I am currently studying for a higher diploma, which is less than a master’s degree and higher than a bachelor’s degree. Greetings from the State of Kurdistan 🇹🇯
발음이 너무 좋으시다!! 딕션이 귓속에 똭똭 박히게.