I had a math teacher that once joked, that in the highest level math classes maybe one person in the entire class would understood what was going on, and some times that person wasn't even the teacher.

Glad to hear even math professors sometimes just had to wing it in a tough class xD

if a professor tells you its a take home test, you know damn well it will be a HARD AF test!

Agreed. I'll skip the long story but I was working full-time at Sony Electronics, in the warehouse taking inventory and operating a fork truck. After work, I'd drive straight to University of Pittsburgh for several graduate level classes that I paid out of my own pocket. Of course I had a degree in math at that time. And for an undergrad, I had a good understanding of how to prove a statement. In the Fall semester, I took complex variables and regular point set topology. The Spring semester I took "Topology II" which was algebraic topology. There were about 6 other students. All of them were full-time grad students. Then there was me, who came after work in my dirty pants and steel toed boots. The professor was old school and passionate about the subject. He made it clear that all questions were to be asked during office hours, not during class. I took at least 10 pages of notes per class. There was no assigned textbook, but the professor did tell us what he used for reference. I bought the two most used texts. After each lecture, I went home and feverishly filled in the gaps (which were plentiful) and used the books to make as much sense of it as possible. Our homework was to fill in the gaps of the lecture in addition to a problem or two or three that he'd write on the board, seemingly as a side note. But he never collected them for grading. I think I was the only one who actually did those problems and fill in the gaps. The prof told me privately that by far, I seemed to be the hardest worker in that class. About the 2nd week before the end of the semester he said (almost in the same words that you used) that he was required to give a final (always blame it on the department!!) On exam day one student turned to me and said the only words that any of the other students have spoken to me for the entire term: "I'm surprised that you're even bothering with this." They all knew each other, but they didn't know me. In spite of keeping up with the class, my confidence at that moment was on the low side so I just smiled it off. Five minutes later we got the exam. Seven handwritten problems (prove this, prove that) on one sheet of paper and a dozen or so blank sheets of paper to do the problems. I didn't say much about the time I spent in his office during office hours, but that professor knew that I could do the proofs on that exam. In fact, I swear that he hand picked those problems with me in mind. I was the first to complete the exam. I was thrilled that he said ("you shouldn't have had any difficulties with this") for the entire class to hear. The guy who said that he was surprised that I bothered had a puzzled look on his face. I responded, "yeah I got them all!" But that was definitely the hardest class that I ever had!! I needed that professor to write me a letter which was why I worked so hard. He did. Two years later I was awarded a TA and became a full-time grad student.

Saw a tiktok edit on 3rd quarter algebraic topology lmao

I could not even write, a, b or c to take an algebra class in high school, was more interested in racing motorcycles and girls. My high school guidance counselor told me to go to trade school and be a brick layer...a very well respecting job. Everyone laughed when I decided to study mechanical engineering in college. They thought I was nuts...well some struggling with math occurred until I took my first introduction to calculus course. My professor had a very severe stutter and that slowed the class down and then a very large light bulb came on in my head and it all CLICKED. I smoked all of the rest of my math classes, but Advanced Engineering Mathematics was a rough class, made a B. I retired as an Engineering Advisor for ExxonMobil :)

I love how we go from 1+1 to this

I can completely relate as someone who is working in algebraic topology and doing research that heavily relies on spectral sequence computations (computing homotopy groups of various things). I had my first master's student a while ago, and at that time I started realizing that it is very difficult to teach spectral sequence in a good way. For me, it is like a craft I gradually learned from my PhD advisor during my years working with him. In the end, spectral sequence computation is more like painting (both in the sense of a craft, and literally, painting) than logical reasoning for me, but that makes it very difficult to convey to other people.

The last class i attended was a pure maths class called topological quantum field theory. There were only 4 people in the class. One post doc with master degrees in physics and mathematic and two physic professors attending the class...and me a regular graduate Student. I went there two times and the professor was always very impressed by my reapearance ^^

I finished my engineering degree in Robotics and Mechatronics at 48 years old, high school was a distant memory and I dropped out in grade 11 anyway. Every maths class was torture for me and it was a real grind to finish everything. I found the physics much easier as I could picture what was happening, control systems was difficult as I couldn't picture what was happening as easily. This subject seems difficult to understand as you can't see what the picture should look like to understand the question. Smarter and better people than me dropped out along the way, I just fell across the line at the end. I will never be a great engineer but I am an engineer.

Of the whole spectrum of different math classes I've had (here's some Group Theory, Algebra, Differential Equations, Calculus, Discrete math, Probability theory & Statistics, e t.c.), the hardest one was Biostatistics. The problem with this course was that it's not like we can open R and call some functions for some data and get results. It was not about biology but a PURE theory about longitudinal data analysis, Mixed models, GLMM, and a bunch of other statistical methods used in biology. He used his notations which I've never seen. Notational hell if put simply. His presentations were filled with errors, and I mean lots of errors. I've had to write him about 12 emails almost for each presentation to list all kinds of errors starting with simple typos and ending with fixing formulas used in theorems, finding out what had to be proven, showing counter to the teacher's formula, and then proving the correctness of other formula. 80 errors in total were found. Got me a month to scour through them. Saved me on finals, which turned out to be a great preparation for them.

Engineer here - I feel soooo much math education is wasted because of teachers/professors who dont have an ounce of didactics. I had courses in the Kreyszig book during my bachelors and both our professors did exactly what you describe in the video; rolled up in the class room and started filling blackboards while talking to themselves. The choice was to either frantically take notes or try to understand what the hell was going on. And the courses famously had a 80% fail rate - good times 🙄

My master thesis was mainly about spectral sequences. It turned out to be the roughest time of my life so far. The amount of indices just drives you crazy. But at the end I came up with a good analogy in my mind of a building of infinite height and every floor has infinite spaces that are ordered in a 2D grid. When you look on top of the 0th floor the differentials just go up. When you go up one floor you take the kernel of a differential from the floor below and create a new differential using the underlying morphism by extending it to the homology group. This causes a change of direction of the new differential (which took me some time to figure out but if you write it down it makes sense). One of my main takeaways was that the floors can "stabilize". For example in the case of chain complexes every floor above the first floor is just the homology of the complex. This is in deed for trivial reason since the only differential on the homology group is equal to zero. I guess the professor showed a similar conclusion during the lecture. But as I said before: It took me a lot of pain and suffering to arrive at that conclusion ... At least I got a good grade at the end but after that it was really time for me to leave university ;-)

I think I recall hearing about that course. Fortunately, my professor followed the book you were holding up, so it was a pretty reasonable class. I actually needed spectral sequences to solve one of the harder parts of my thesis problem. I learned what I need from the first 40 pages or so (if I recall correctly) of A User's Guide to Spectral Sequences by McCleary. Impressive that you still remember enough to even write down what you did on the board.

What a lovely video to "meet you" through! Honesty, humour, and maths -- I already think you're a wonderful communicator! I look forward to watching a lot more of your content! All the very best to you from the UK! Cheers! Patrick.

See my Star-Wars themed edit of the beginning of this video on TikTok: www.tiktok.com/@billkinneymath/video/7272921845878263086

I love math stories like this! Fun to hear. Probably not fun to experience.

My most difficult class (when I took it) was called "differentiation". Sounds like just calculus, but was basically a setup course for manifolds. At the time it was notorious!

Great video, great pacing, thanks for sharing a gem of your learning experience sir!

I liked abstract algebra a lot and I want to know more about topology from a written form so this might be cool to look into one day. Though my academic years are behind me and most of my math is used on programing. It's always fun to expand the mind however.

I took a homological algebra course as an undergrad and the last month or 2 was on spectral sequences. A few years later now I'm finishing my PhD I'm still kinda scared of spectral sequences.

I think my hardest math class dealt with differential forms on manifolds. We started that year with Spivak's Calculus on Manifolds, then moved on to tensor analysis, Christoffel symbols, Lie groups, etc. And a lot of head scratching.

i agree. spectral sequences, exact sequences, snake lemma ... are difficult to grasp. this all is because of moduli spaces

In my Algebra II class the professor decided to spend a month on spectral sequences, and their dark magic is still completely impenetrable to me. I recall a distinct sense of despair after trying to learn bigraded complexes and then being told I was only looking at the first page...

You are amazing Professor ❤️🇲🇨

Love it!!!!!!!!!!!!

I've never heard of spectral sequences, but i have run into algebraic topology via the circuitous route that follows from investigating the curry-howard isomorphism. As I began to really dig into things I found that just as programs can correspond with logic, there's also a correspondence between type theory and category theory, and thus also with algebraic topology. That all 3 of those are effectively just different ways of looking at the same fundamental thing really blew my mind. Not that I actually understand much of the abstract algebra, I didn't spend the kind of time to actually learn these concepts (especially since I don't even have formal education on calculus. Best I've got formally is precalc and some basic stats), but just becoming familiar with some of the concepts at a super high level has certainly opened my eyes to what some parts of pure math looks like.

This was very funny, I always enjoy your videos, spectral sequences sound interesting

That was actually a pretty cool story. Thanks for sharing.

I agree on the difficulty of Algebraic Topology. I have a Bachelors in Math, where I got mostly As and some Bs in math classes, except for a D in Algebraic Topology. That was in the early 90s. It was conceptually difficult to grasp and I fell behind.

Thanks for sharing your story. Makes me feel a bit better about those lecture where I leave having only understood about 10% of the material. Lol as students we at times feel bad or even embarrassed leaving a lecture confused

The edit is fire

This is one of the most relatable videos I've ever seen on YT. I had a similar experience in my ODE class. The class was way beyond anything I was good at. Advanced calculus mixed in with some Matlab. It was so hard. I was trying my best minimum. I say best minimum cause all I did was pay attention in class. Listen with the hardest focus a student could muster. But I couldn't for the life of me get better at it.

Currently self teaching myself, all the way from algebra to calc 3, and then I'll delve into the not terrifying world of topology and advanced statistics, set theory group theory, I'm also starting university in a different subject, so wish me luck.

What a great story… so glad I scrolled across this video!

Fun fact: Kids mathematics deals with numbers Adult mathematics deals with letters.

I had a similar experience trying to self-study algebraic geometry from Hartshorne with barely any knowledge of commutative algebra. I pushed through it but I let it go after the second chapter.

Real analysis was actually really pleasant for me to read about, because it brings insight into what was otherwise just accepted at face value by everyone except me. I'm not claiming I could become an expert on it, nor do I know a lot of pure maths, but to me real analysis was the first real(pun intended) math class. I first read about RA in a Terence Tao's books, and it was so remarkable to me that I printed the books. Most people I've talked to would say the most demanding area of maths is algebraic geometry. I know nothing about that.

Great stuff, had to learn some differential top. for my recent paper 📜

Absolutely awesome pitch-perfect video man!!!! 😂😂😂………loved it ❤❤❤ I know and feel to the core of my bones EXACTLY WHAT you are describing being a math major myself the bane of my (course work)existence having been Chaos and Fractals theory that jumped off from Mandelbrot and took me to some dark corners (pun fully intended) that I still have nightmares about, the other such course being Advanced Measure theory that I also suffered through …… loved to see how B is the average of F and A for attendance ….. like some famous guy said ….just show up for work everyday and plug away at it ….and have faith that the rest will slowly and surely fall into place ….. Awesome video …..👏👏👏👏👏🙏🙏🙏🙏

the hardest course i've taken (so far) was an advanced probability and stochastic processes class. there were only 5 other people doing it and the only assessment was 4 assignments that were just questions from the book the professor wrote himself. i did well on the first 2 assignments and got cocky, so i stopped turning up to the class. i ended up with 52/100, 2 marks above a minimum pass because the final 2 assignments on martingales and ito calculus absolutely stumped me

Definitely Advanced Algebraic Geometry. We were 10 students, which was exactly enough to make it into a seminar; the professor kicked off the first two lectures and after that each of us delivered one lecture. No further assignments, no final exam. Just the lecture and overall participation; there would be plenty of discussion during each lecture. I was given the lecture on the Grothendieck-(Hirzebruch-)Riemann-Roch theorem, after we had covered a bunch of scheme theory, category theory, homological algebra and whatnot. I think the stress of that semester has shaved at least a year off my life. I'm proud of having understood the material (to an extent; I recall zoning out on a regular basis and just blindly copying notes in hopes of understanding them later. Never figured out what the deal is with Koszul complexes). But that course also where I drew the line and decided not to pursue a PhD.

That's such a funny story. Good delivery too.

This brought back painful memories of my hardest math class, rheology the study of flow in materials. This was in the late 60's and the professor was from India and he had such a strong accent no one could understand a word he spoke. I have no idea how I passed the class, although I think everyone did, but it was probably just for turning up.

One or more of these four: Introduction to Mathematical Analysis I (I think Analysis II was a bit easier, got a better grade in it too), Geometry (Euclidean and non-Euclidean), Intro to Modern Algebra & Number Theory, or Introduction to Complex Analysis (my worst math class in terms of grading: the only math class I got a D in). Unfortunately I wasn't able to take classes like Introduction to Topology (mostly due to schedule conflicts), and algebraic topology isn't even offered at my university, not even at graduate level.

Mathematics as a whole is pretty broad - while some people might struggle with the formulas, others might struggle with the visuals. Algebraic topology is full of tough dense topics. Geometry as a whole is probably one of the tougher areas of mathematics, because it encapsulates a fair portion of complex number analysis, calculus, and algebraic topology. Outside of working with geometries and manifolds - there are areas of mathematics that need have challenged us for forever. Prime numbers, we still don’t know very much about (despite so much research). Randomness is not well defined in mathematics as a whole - chaos theory I think is the implied idea, and there isn’t a lot of development there. Transcendental mathematical operations are often unsolved - think of all of the functions we don’t know how to integrate yet. Dimensionality: ie (Fractals: Non-integer dimensions) and optimization of space for spheres in R^N (is really specific) - just demonstrates we don’t know very much about anormal dimensions. Algebraic topology, though, dense and difficult- is pretty well studied. I think it’s the direction a lot of pure mathematicians go into. There’s a lot more out there though.. Don’t let the symbols or the proofs discourage you (the reader) from learning it- it’s pretty neat.

That story about the final exam reminds me of my Software Engineering midterm, where we have to do a Proof By Induction. This professor did a pretty awesome thing where he made an appointment with every student to go over their midterms (sadly he didn't do that for the finals, for obvious reasons). I got full credit on the induction question, even though I'd struggled to understand the concept. He said: "This is incorrect, but the answer shows that you at least understood the concept, most people didn't even have that." 15 years later, I still don't know how to do a proof by induction.

This reminds me of a senior numerical analysis class that was taught by a visiting professor who thought his job was to make us all feel dumb. It was completely off the rails wrt the course description. I think a couple of the PhD students did OK, but my top tier math undergrad friend took one look at the final and stormed out.

This is relatable on such a deep level but I did analysis. My class on wavelet theory was also a really small group of students, the teacher said things like "as you learned in functional analysis", or "was this clear for everyone". During the breaks I asked the other students what was going on but everyone was as confused.

Functional analysis in a master program. My first F grade. Not only was I unaware there were prerequisites I was supposed to take first (measure theory , L_p spaces, etc.), but the course was also taught as a 3rd or 4th semester functional analysis where the professor just assumed we had learned how to deal with infinite dimensional linear spaces and basic functional analysis results such as open mapping thm. and even harmonic analysis. When I showed the materials to another instructor, they agreed this was really not meant for master students.

I KNEW it was going to be a pure math topic before even starting the video. Omg that sounded like a horrible math class!! I had a bit of a similar experience in a graduate level course on applied analysis. No idea what was going on until professor out us in groups for a final project in which we would be presenting a chapter of the book we were using. Our group happened to get a chapter on Spectral Theorem!! I mostly didn't know what I was talking about but I'm guessing I played it off pretty well as our group got an A for the project, and thus for the whole class. 😂

damn, spectral sequences in a first year course is insane. my uni doesnt have a proper algebraic topology course, so I was first introduced to spectral sequences in Bott & Tu's "Differential Forms in Algebraic Topology" and a little bit later read "A User's Guide to Spectral Sequences" by John Mccleary. i love and appreciate them now, (in a weird masochistic way i guess) but if i had to relearn it again from scratch, i think i'd quit and become a farmer or something.

The fact that the only numbers used in that math equation are 0s, 1s and 2s and they are all subscripts or exponents has to be the most intimidating math problem I've ever seen in my life. It doesn't even look like math anymore, it's more like an alien language.

It had long been a set of running jokes when I was in grad school to say that everything in math, ever, is either trivial (and therefore no explanation is needed) or vacuously true (and therefore true because nothing). Also, I took a course on dynamical systems that was pretty mind-bending, but spectral sequences definitely has it beat. Our professor also simply gave us all B's for absolutely no other reason except we were there and we stayed there, justifying the class's existence.

Finally, a math video with notation I don't understand. Now I have to read some new wiki pages, or go to the math library.

I'm currently enrolled in a graduate course on differential geometry (jan 2024). I had since time immemorial pondered about what on earth the sexy sounding exotic objects like manifolds, tensor fields, riemannian surfaces, lie derivatives etc with their ubiquitous use in physics, were all about; so as soon as I heard that the course was being run, I jumped into the chance of learning about them. The beginning classes were a recap of advanced calculus, definition of a derivative, inverse function theorem etc, which I didn't have a hard time following, so I thought I'll glide over this course fairly easily. But soon enough, came the reality check: the definition of a differentiable manifold was a mouthful, hard enough to decipher and visualize, so when the professor first wrote it down and proceeded to scribble an atlas for the n-dimensional sphere and asked us to verify its satisfaction of manifold axioms as home task, it felt as if a bullet train hit and ran over me. After that, the classes seemed to get exponentially harder and harder. Next he gave us the definition of real projective space and a hand waving proof that it forms a differentiable manifold, and tasked us with writing a differentiable structure for an n dimensional projective space over the complex field! The classes now go on exactly like the way you mentioned: prof comes in and writes esoteric algebraic objects and their morphisms on the board, which I silently stare at and jot down, hoping I will understand all that one day. Apparently, differential geometry is supposed to be an advanced undergrad course, so being a graduate student, it feels way harder to me than it really is, mainly because the other classes are much easier in comparison and partly because the focus of mathematics education has shifted more towards the applied side in recent years, depriving the modern undergraduate curricula of the (once normal) formalism and rigor which this course expects. Nevertheless, I love the thrill of challenge it gives me, and the satisfaction of learning a truly abstract part of mathematics which I had always wanted to learn. I plan on taking algebraic topology next semester (if I survive this semester).

The toughest class I had in college was Latin; the final exam involved translating a passage that had grammatical formations I had never seen before. As it turned out, not one person in the class had, either (there were six of us), and the professor admitted that she had not prepared us for that. She also said that if even one of us had succeeded, she would have given the rest of us Fs.

Very interesting story!

Great Story. Thanks.

I did algebraic topology this semester, it was very interesting, however, it does take some time to wrap your mind around it, and when i came out the other end of the rabbit hole, i barely compleated the class

Yeah, we also had that kind of lecturer. No one had an idea what he was talking about. And we had a test - a task. Fortunately some of us developed a kind of intuition to write correct symbols w/o a tiny understanding of their meaning, and helped others. All passed. Nothing as learnt. It was something about abstract theory of databases. Fortunately we had just one such a teacher. Most others were great.

To be honest, I don't think I've ever encountered a math class that was harder than my Analytic Geometry class, and that wasn't even that bad. When I did algebraic topology I definitely did not do... whatever you were talking about lol

Awesome show. After Calculous, you should have enough mathematical knowledge of the mathematic fork stuck in road. What road do you want to keep following.

Can confirm. Walked out of Algebraic Topology course after two classes in grad school - incomprehensible.

Ugh take home finals were the worst. In my first quarter algebra class the professor gave us this 12 question take home final where every problem took me at least 2-3 pages and I spent the whole finals week just doing this exam and then when we turned it in he said he'd grade it like a qual exam and only cared if we nailed 3 or 4 solutions. Worst part was I'd tell my friends how much my algebra take home final was kicking my ass and they'd look at me like I was an idiot because to non math majors algebra was the crap they learned freshman year high school with quadratic formula and FOIL and such. But then I'd grab my copy of Lang and ask them for help with a problem and they'd see those commutative diagrams and think WTF kind of new math is this school teaching? haha

Reminds me of when I took Linear Algebra 2 for my math minor. Although I probably would understand it a lot more now, I had never taken on a class with any major proof writing and the whole class was proofs. I, along with many others probably got Bs. _shrug_ I am still seeking to be better and better at math whenever I get the chances to really go hard on a subject.

Manifolds was definitely the hardest for me! From smooth atlases to De Rham cohomology still don't get a lot of it!

Right on!

I had a similar situation. I took homological algebra and commutative algebra courses with the same prof who happened to be the dept head and two of his grad students, both foreign exchange students. He started with the snake lemma and went briskly over our heads from there. it was misery throughout. 😂 none of us could quit for various reasons, so we sat there trying and failing sometimes to complete a single item on those exercise sheets wading through an impenetrable thicket of alien ideas. I hated feeling stupid but loved gazing at those fancy diagrams diagrams. ❤

Your math prof must've been highly inspired by Lovecraft.

The most advanced course I took was differential geometry which was the only four semester course (not even counting seminars and senior seminars) in the entire curriculum. Ended up getting my master’s degree with a thesis in that field so it was doable, but I never had to work so hard to understand something. The hardest subjectively was probability theory because it went against my intuition so often.

What you're describing with not understanding anything and having a final the last week which is take-home is *exactly* what happened to me this semester when I was taking differential geometry. Literally also had three questions on the final exam. Hopefully I pass tho

Our undergraduate Topology professor skipped metric spaces, "Because we live in one". Enrico Fermi when teaching Quantum Mechanics at the University of Chicago failed every student who took the course.

You are amazing! I want to ask is there going to be any Algebraic Topology lecture? Thank you for your work!

It looks like some solid skills with category theory are needed in order to learn spectral sequences well. I’ve studied some category theory profitably on my own, and I will do more. Most math departments at universities never have actual courses on category theory, but I think they should.

amazing video 👍👍

Calculus 1: I got an A. Calculus 2: I dropped out after the first week. A bunch of my friends dropped out and took it at a community college and all passed with As and Bs. I asked what helped them get through it and they all said the teacher took a lot of time to explain every single step. And most importantly, he said it was the algebra that would hurt them- not the calculus.

What's your opinion on graduate level real analysis? (Measure theory). I don't know any but I'm definitely hoping to learn some eventually. Rudin has another book on the subject too, I think what makes his book principles book harder than some other texts is a lot of stuff he straight up leaves you to figure out on your own so if you haven't seen some of this stuff before it'll take a while for you to get what you're even reading, it definitely shouldn't be someone's sole text unless they have a lot of time

Advanced physics J numbers and finite element analysis i never will go beyond that ever, never ever evarr just passed that module on the 3rd try, Find it remarkable how people like yourself have the capacity to crunch expressions and hold em like thoughts, world is fortunate to have mathematicians like you

Like wow, I was an undergrad in the late 80s and I used that exact same textbook Advanced Engineering Mathematics. Really shocked to see it still around.

I took abstract algebra as an undergraduate and it was very hard. Definitely the most abstract course I took in college

I can only imagine how hard it must have been studying without the internet! We should definitely appreciate it more. My hardest class was probably my last masters class. A very specialized topic in numerical analysis. There were 5 of us at the beginning - 3 very smart people fled in the middle of the semester. We also did not understand at all what was happening. We were past the point where it was appropriate to ask him, so we went with it. Smiling and nodding. Also: no homework, pretty loose schedule. Our main problem was that there were no other sources, every single paper related to that topic has been written by him. At the end there was an oral exam. To this day, I have no idea how we passed it. Probably he felt so sorry for us. And very, very disappointed. Only two kind of people can make me feel a guilt that deep: mums and professors who are so passionate about their research topic.

Loved this video! I hit subscribe so fast I saw Spacetime fold in on itself. Fun fact I had a math professor drink coffee out of a cup that read “Not a Donut.” I’d always look at the cup and say yes it is, so is a paper towel roll and a toilet roll. Fun times indeed❤

i have always said that there is not hardest math but people just make them hard by the way they explain them

2:40 had me weak with laughter 😭

Mine was differential geometry for general relativity- I took both in the same semester as an undergrad. I got the basic concepts- gravity is the warped geometry of 4D space-time, differential geometry (DG) tells you how to calculate this warping in a way where the math respects the fact the geometry must be the same for all coordinate choices (tensor calculus, covariance). What I found really hard to follow was the overly arcane terminology and notation of formal DG, which was made worse by the fact that we never seemed to explicitly calculate anything. It was part of what convinced me that much of super “high” level mathematics is far more esoteric and difficult than it needs to be, because many mathematicians refuse to use plain English or provide clear examples. Which was especially ironic for DG given that I needed to use this math for GR problems.

Toughest class I took was during my Masters. We had this lecturer who looked like a vampire and taught us Ergodic Theory, something we had to take because we wanted to focus on Dynamical Systems. 10 people started the class, and in the end only 3 people took it. No one had any idea what was going on and he would often skip 2-3 pages of work between sentences on the board. We still remember him turning around and saying things like 'That is obvious'. To this day, when I don't understand something, I say 'it's obvious' haha

Algebraic Geometry with schemes is pretty evil too. Not to mention topos theory.

I’m sure math majors do much worse but general relativity (tensors) was ridiculously complicated.

i always find it funny when watching US college math content, hearing them talk about real analysis and calculus. Here in Germany we don't have calculus, we start with real analysis in first semester. yea it's about as fun as it sounds haha

Good to know that analysis isn't the worst it can get

How do you feel about number theory? Sounds relatively harmless until you actually open the book…then it’s terrifying

Solve the hardest maths Problem no more maths

Been there, done that. MA Berkeley, ‘78. PhD, CUNY, 2015. Still learning at 71. Thx!

We had the luck of our professor sticking entirely to Kosniowski and Armstrong's books and only sometimes going into Hatcher. The consequence was that most of us got either an A or an A+ (I got an A+). This was a lifesaver as I had taken Functional Analysis, Probability Theory, PDE and Representation Theory along with Algebraic Topology. I managed to barely pass PDE but did average on the other three. The hardest sem of my graduate life.

Omg I literally have the same advanced engineering mathematics book it was given to me when I was 9 by an engineer haha

Me thinking 6th grade math is hard until…..: 👁👄👁

My last grad level class 20 yrs ago!!! Still shudder at that thought!!!

I think any math class where the main tool of study is abstract algebra is going to be a difficult class. Just studying abstract algebra on its own is challenging enough, but when you study a topic which assumes mastery of it, you really have to know what you're doing in order to learn anything. I'm in the middle of my master's research in an area of algebraic geometry, and I don't know what I'm doing half the time lol.

I wonder if our math teacher can solve this, since they know what they’re doing but not really good at explaining/teaching