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.
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.
@@billkinneymath , It wasn't a class. Your teacher taught nothing. The goal of teaching is to communicate. He made no effort. He did what amused him, with no interest in conveying useful information to the students. That's not a teacher. If there is such a thing as the opposite of a teacher, he was it.
@@maskedmarvyl4774 Lots of bored teachers do it. They teach what they should 1, 2 or 3 years, then they realize how boring it is, and they rarely take satisfaction in actual teaching and then they steer to what INTERESTS them, not students. To be a good teacher one needs to be less selfish and have observation skills and undurstanding that perception of others is different from their.
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 have a similar story. In Years 9-12 at High School. I had the same great maths teacher. In Years 9-10, out of 2 classes that included re 60 students, position-wise, I'd come like 57-58th.....I was hopeless. Didn't even know how to balance a simple equation like x+3 = 2x -1. Then one of my best friends left school and in Year 11, all I did was do my homework (what a concept lol) and read the textbook for re 20 minutes every night. Then something clicked, and from there it was smooth-ish sailing. When I sat the HSC (Higher School Certificate, NSW, Australia) in 1986, out of re 30 000 students, the 1st exam of 2, was for 3 hours. I did it in 50 minutes including going back and checking my answers, and came in the top 3% in the State (NSW). For the 2nd exam, I took an hour instead of 2 and came in the top 10% of the State (unknown number of students). Strangely enough, even though it has been explained to me several times, I still suck at HSC physics, (Mechanics), yet electricity and magnetism light, and waves, I excel at. Like, go figure. Never did complete my Undergraduate Pure Maths degree as my life changed direction, yet when I move overseas in the next few months to be with my Mrs and settle, since I am in my mid-50s now, I may go back and study some maths. I actually think I have a way to approach the 3n+1 problem (Collatz Conjecture) yet haven't looked into who to speak with.
When a professor tells u its a take home test, u know it is gonna be a hard test. When a professor tells u its a take home test, u have a week and u can do anything in your power to answer the questions including working together, then u know this test is just the test maker flexing on the difficulty.
@@orang1921 It wasn't that bad. The professor just didn't believe in giving out any tests. He wanted to test our thought process. Most of our grade was from homework with the rest coming from the final.
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.
Me and some others in my dept. are going through Huybrechts' book on "Fourier-Mukai transforms," and one of the sections was devoted to spectral sequences. It was a difficulty spike I was not expecting lol. I hope one day I'll be able to look back on that section and think it was all trivial 😅
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 🙄
Yes, probably 50% of my teachers in grad school were like that. Another 40% were okay but not inspiring. Only about 10% were both clear and inspiring. But I made it through anyway.
I find that odd - if the department knows that the courses have an 80% fail rate, why not change the way that they're taught? They're clearly not accomplishing their objective the way that they're currently setup.
@@aiberengi I got some insight into the management of a university department when I did my ph.d - in all the meetings during those three years the only aspect of coursework and teaching that was discussed was how to reduce it to an absolute minimum while still meeting all the legal obligations. Add to this that "academic performance" (at least back then - this was 2007-2010 in Denmark) only includes publications/citations and funding and you end up with teaching simply not getting prioritized. At least not from the vast majority of professors or the departments as a whole. That is my experience anyways.
I had a class that used that Kreyszig book! Now that I'm older and have learned more math on my own, that book and class was garbage. The problem is it crams a ton of math from various parts of math into one course. You just don't get the foundation you need to understand any of it. And it's a huge book.
@@Ninja-wc4un I made a TikTok account just to reply to it with a link to this video but so far nobody seems to have noticed my reply (after about 2 hours). When videos go viral it seems to be just for a very short time.
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.
Wow. I think maybe we talked about this at a Joint Meeting. If you've still got a copy of your thesis, send me a few pictures of how you used spectral sequences over facebook. Did Rick have to review spectral sequences himself to understand what you did, or did he already know it well enough? I didn't remember enough when I wrote it on the board. I cribbed it off of Wikipedia.
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.
I had a "basic" PDE class that focused more on technique (Weinberger's text) than more advanced PDE classes, so it wasn't too bad. It was also taught by a young postdoc who enjoyed and was good at teaching. In my ODE class, the professor delved into what I thought were more interesting topics about dynamical systems rather than "classical" ODE stuff that you might more typically get before 1960.
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!
@@billkinneymath The course description for the upper-level Combinatorics course at my uni is "counting techniques and coloring problems". I found that really funny, given that it's graph + partition theory but sounds like an elementary school class.
@@elliottwalker8624 That is so funny! I actually made a "map" drawing of Snoopy (making it like a puzzle) and brought it to my daughter's 3rd grade class to hand out to the kids and teach them about the 4 color theorem.
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.
@@antronixful and do you think that all of this math is really correct, consistent and describing something in the real world? feynman said that reality doesnt exist, only math does, and this idealism is enough for me to throw in the garbage this stuff; how can you imagine that einstein could really grasp the enormity of millions of terms in his ten gr equations, if written extensively, and instead contrapting them to a simple equivalence by the introduction of a zero term? i dont buy this, as i dont buy the quantum mechanics practice of introducing some wild virtual particle, entanglement, action at distance, superposition and other logical absurdities when the calculus doesnt resolve as desired, only to match the math to the experiment. Modern physics is a big scam.
My differential geometry class was taught by a Russian physicist who conned me into taking the class by saying [insert Russian accent], "This is very good class for physicist." I get halfway through the class, past the drop date, when he says one day, "We are going to skip chapter 5. What this is is applications in physics. You should read it. It's very good." My jaw dropped. My pencil dropped. I thought, "Then what am I doing here?" Just like the guy in this video, we had a final project. I rewrote some fluid dynamics equations I learned in one of my atmospheric physics classes using differential forms and got a B. I still don't think I learned anything.
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.
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.
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
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...
@@RigelNV Oh, probably should have worded that differently. This was a course in graduate school and completely different from Algebra II in high school, even though they have the same name. I would have failed so badly if it were high school…
That's a very funny way of putting it! There have been other textbooks that I have also been stuck on the first page of too. I think one of them was on hyperreal numbers (non-standard analysis). I never took a course on it, but I wanted to learn it. I picked up the book, looked at the first page, and then put it back on the shelf. When you think about infinitesimals intuitively and not rigorously however, it becomes kind of a fun subject.
@@GabeKhan Yeah. A friend of mine told me highschool algebra was like learnnignto count your toes as a toddler. It teaches you what they are, where they are and some idea of how many there should be. its up to you to figure out how to make sure they are all there!
@@billkinneymath not really a math guy myself, more of an engineer, but the nonsense supposedly required in "rigorous" probability theory with Borel sets got me to take a course in measure theory. So much work and boilerplate notation for no worthwhile intuition at all. Actually it should be needed anywhere calculus is applied, anywhere real numbers are used -- the answer is meaningless because it's all unmeasurable, and you have to use something like Borel sets instead! Kolmogorov made a name for himself by, I guess, rewriting existing probability theory in that much more complicated notation, so it all became associated with probability theory. Then I noticed the thin book "Radically Elementary Probability Theory" by Edward Nelson in the library probably because of its orange paper cover and read most of it in an afternoon. It's apparently a correct way to define real numbers. The definition is a bit weird, but not nearly as weird as a Borel set. The original guess for real numbers, the set of limit points of sequences of rational numbers, was a reasonable try in the 1600's or 1700's, but it turns out to have been a wrong choice, because what results is essentially meaningless numbers that cannot be used to denote quantities.
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.
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.
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.
@@spiderjerusalem4009It's probably best to just do a bit of real analysis beforehand if you really want to learn complex analysis. You don't need that much so it shouldn't take too long if you're committed
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.
@@billkinneymath What's more was the professor even said "this is not easy stuff, this is very serious mathematics. And I expect you to spend at least 10 hr per week studying for this". Then I thought: this had to be some kind of misunderstanding or miscommunication between him and the department. Because it was offered as a 3-credit free-elective course, just like any other. If it takes that much effort, why isn't it set up as some standalone research project? We were also not grade on homework or participation. Our entire grade consisted of just a midterm and a final. Maybe it is expected of a master student to be matured enough to know what courses they should take and I made a wrong judgement call. But even if there was someone who knew all the courses in the department AND knew my background well, they couldn't have predicted that this professor personally wanted to teach this course in such an unordinary way. I don't know if it was because it was a private institute or because he's very famous, he just had a lot of freedom in how he teaches a class. And now my academic future is hindered if not buried by this blackspot.
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.
I'm always impressed when people try to learn advanced math on their own. Fortunately there are many good RUclips channels (3Blue1Brown is my favorite) that help with the intuition. The full details have to be worked out in your own mind, however.
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.
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.
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.
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.
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.
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.
Currently failing calc 2. I need a 77 in the test I’ll be taking in about 45 minutes, a 77 in a retake this Thursday and a 77 in my final next Tuesday just to get to a 70%. 😭
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
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.
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
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
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.
I agree. I think if the professor had actually assigned some homework so that we were motivated to study it, that would have helped. I appreciate everything more and more as time goes by.
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 felt that way to us too. Even though the first two quarters were okay (the professor was better too, but they switched us to a different professor for the third quarter).
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.
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. ❤
That is a subject that can be "fun" without knowing what it means. I remember learning about tensor products and thinking it was just a "game". But I didn't know what it was for, or how it was related to tensors in physics. I still don't know. Maybe someday I'll be able to study it further.
@@billkinneymath Tensor algebras and more distinct algebras like quasi-triangular Hopf algebras and Hecke algebras arise in the quantization of gauge theories and string theories. Here we are often dealing with modules. For example the Verma modules as the highest weight representation of Kac-Moody algebras. These are quantum symmetry algebras of two-dimensonal conformal field theories living on the string worldsheet or the boundary of the topological membrane. If you'd like to delve into some very interesting new math, which was predicted by phyicists in 2009 I would emphesize the subject called AGT-relation/correspondence/conjecture. Its core statement is that the generating function of equivariant Euler characteristic classes of the moduli space of self-dual solutions (instantons) to the SU(N)-Yang-Mills equations on R^4 is equal to the conformal blocks of an affine W_N-algebra. This statement has profaound and whide-reaching consequences, which we are yet to understand.
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).
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.
😅 That is so funny! Maybe I should tell that to my multivariable calculus students so we don't need to learn about Euclidean space. (just kidding). But it's sad to hear that about Fermi. He's so famous these days because of the Fermi paradox too.
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.
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.
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.
We did category theory at the start of my grad school abstract algebra class. I had no idea what it was about but thought it was kind of fun. Later on I've appreciated how unifying it is.
I think all of the different 'Algebras' beyond Master's level are difficult primarily because of the abstract component /reasoning in the sense that it's Pure Math & not Applied akin to Kreysig's massive tome, which I still swear by as one of the best AEM texts ever written for self-study & revision. As an ongoing Master's level wannabe, I'd single out a few beauties but 'tuffies' such as: *Differential Geometry *Tensor Analysis *Complex Analysis *Chaotic Dynamics As tough as it is, I'm still in awe of the sheer beauty, depth & truth in all Mathematics! It's a journey of discovery & a quest in pursuit of determination to succeed & unlock the human potential...
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.
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
Yes. I wish we had stayed with the standard content of algebraic topology. That was challenging enough, but at least I understood it better than spectral sequences. The professor I had for the first two quarters was a better teacher too.
@@nestorv7627calculus is too broad a term. I'm ignorant about this field, but analytic geometry could maybe be the same thing as geometric analysis? Or maybe analytic means holomorphic, so maybe it means complex geometry? But I'm sure it is something unique.
@berserker8884 in the US we call calculus 1,2, and 3 "analytic geometry." In some rare instances professors may call that differential geometry, but it's usually referred to by diff geo itself
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
no, undergrad algebra is easy, very easy. galois, representation theory, group theory, all the standard topics are kindergarten stuff. like this guy said, the hard stuff is topology, topology is fking nightmarish. if u think spectral sequences are hard, look up A1 homotopy theory, algebraic K theory. spectral sequences are just a computational tool, you just need to get used to it, but those subjects are so hard you'll be crying trying to understand them. if you want to learn something really abstract, try to learn modern homotopy theory, non commutative geometry. topology and algebraic geometry are the hardest subjects, you'll cry learning them trust me.
Calc 3 prof did something analogous. It was supposed to be a standard multivariable calc class at an average university but he lectured stuff not in the book, had exam questions about things we never discussed or only touched on briefly and borrowed exams from harder classes at MIT etc, with stuff we didn't cover. I'm not a wiz so I was pretty lost. Gave everybody in the class a C that made it through to the end out of guilt. Sucked because it broke my all A record.
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.
You can kind of think of it as a "game" where you just play by certain rules. But that doesn't necessarily help you understand why it "works". There are some easier examples out there though. I would try those.
I remember when I finished middle school and got to highschool, the math teacher said "So far you haven't done real Math, now we start wit real Math". Then I go to university and the Math lecturer says the exact same thing. Then a year later, just for fun, I sit in a graduate level math class to see if I can follow (spoiler, I couldn't follow a thing), the math professor says "now it's time that you learn some real mathematics". And of yeah, everybody at Uni loved to say "trivial", to avoid having to explain things.
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.
In my real analysis (measure theory) course the professor was too easy on us during the course. I got an A, but I couldn't pass the written prelim, which was much harder!
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.
Wow! A grad school math class with discussion. In most of mine we just listened to the professor (again, mostly because we were afraid to reveal our ignorance). I did have one professor who really encouraged questions though, even if they seemed really "basic".
@@billkinneymath I'd like to be a patroen of this leture, if you're gonna record this course, I'll be willing to pay 100$ to support your work through www.patreon.com/.
I think if you are interested you can always start reading gtm 119 (rotman), or Munkres (his algebraic topology book, not his general topology or differential topology). I personally recommend these two, and I especially like rotman, it’s very easy to read and it’s very detailed in my view.
Mathematicians are usually some of the most humble and wise people. Little to none of the ego affliction of many of the other disciplines. Only artists can possibly claim something similar and this is why they are brother and sister.
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
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
That was definitely a tough one for me too. A funny story there is that my professor was actually too easy on us. Like the other class from the video, he hardly gave us any homework (he was a slightly better teacher though). So the class didn't prepare me well enough for the written preliminary exam for the Ph.D. program. Fortunately I passed the Complex Analysis prelim so I didn't need to pass the Real Analysis one on measure theory.
For me, the second grad course in real analysis was the second hardest course I took in undergrad, the hardest being a course in algebraic geometry. We covered L^p spaces, the interpolation theorems, Fourier analysis, distribution theory, and applications to PDEs. I've never used Baby Rudin, but I think it would not be used in a graduate level class. More likely, such a class would use something like Rudin's other texts, Folland, Stein and Shakarchi, etc.
Rudin is great, his approach to measurable functions is by a different (equivalent to the definition I often see in other books of course) definition , I like it very much.
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. 😂
They are symbols from abstract algebra. The plus circles are "direct sums": en.wikipedia.org/wiki/Direct_sum. The kers and ims are "kernels" and "images": en.wikipedia.org/wiki/Kernel_(algebra) and en.wikipedia.org/wiki/Image_(mathematics)
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.
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.
Mine is not very hard overall compared to others but for me it was calculus in college last year. But the reason why it was tough is I went back to college 11 years after high school and my highest math up to that point was 10th grade algebra. I failed out of pre-algebra at my first go around at community college 10 years prior and dropped out completely. Coming back to get my degree I had to take this class and they allowed me to take it. (I did pass stats before this so that may have been the reason they let me) Well, I studied very hard, did my best learning the concepts of algebra and passed my calculus course. It was a wild ride cramming everything into my brain for that semester.
Fortunately I never had to take that class. I knew someone who was doing his Ph.D. thesis in Algebraic Geometry but I heard he never finished because his advisor never thought what he did was good enough.
@@billkinneymath Yes. As the saying goes, there are two kinds of mathematicians: those who don't understand Grothendieck, and those who pretend to understand Grothendieck.
Yeah I feel the same. I feel Hartshorne is difficult, I read algebraic geometry and arithmetic curves by Q. Liu instead. It’s supposed to be easier to read, I still read it very slowly 😢. I read about topos theory in some categorical logic books. I feel it’s very interesting , but I lack motivation to read further about it. I guess that’s because I know too little about logic, I read some basic about it purely because I like categories.
@@billkinneymath It’s one of those things that I think would get a lot easier once I got used to the notation, but I didn’t have enough time for that in the class I took and definitely need to go back and reread the textbook. It was Sydney Coleman’s Lectures on General Relativity for anyone interested; I’m sure a math major who already has experience with tensors would pick it up much more quickly than I am.
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 actually haven't had the opportunity to study it beyond the basics that are learned in abstract algebra. I know that what Andrew Wiles did to prove Fermat's Last Theorem was incredibly hard though.
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.
It would have made much more sense to name the class Homological Algebra. It is in the last chapter of the book on Homological Algebra that I am studying and they look terrifying indeed.
Bill, I really think the title of this video should have been "How not to teach" and you could add "mathematics" if you like: The students not knowing what's going on, teacher talking to the board, little to no engagment or interactivity, not knowing how you're going to be assessed or what on, etc; all too common yet so wrong! When are these "teachers" going to be held accountable! A proposition to consider: a course is only as hard as the teacher is bad
Yes. Unfortunately, especially at research universities, there's not much effort to hold poor teachers accountable as long as they keep the grant money flowing and bring research prestige. I think it is somewhat better now than 30 years ago with some teaching specialists added to the mix (though they are paid less), but there's still a lot of institutional inertia. It's the way the system is designed.
@@billkinneymaththe problem is that there's no incentive to structure things in any other way. If every tenured prof doubles as a researcher, then they maximize the amount of grant money they can bring in, and the prestige. The moment you introduce someone who specializes in teaching rather than research, your wasting an opportunity to bring in another researcher. And money. It's a terribly broken system which prioritizes the university over the students in a hideously capitalistic fashion.
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.
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
Actually I looked back at my high school math notes from when I was 17 years old and realized that my math teacher was doing real analysis. Of course, I didn't know it at the time. I thought that was what most 12th graders were doing. He didn't go super in-depth, but he was doing the basics!
That was the funniest story!!! I am delighted how you tell it without a pinch of resentment, God forbid in my country for a professor to do that, students would riot, the professor would be lynched, fired or at least reprimanded.
I love it! I am a professor of classes related to math (calculus, physics, statistics) and think a lot about this learning process, things like how much effort is expected from the students and from the teacher, and where I work I think it is expected almost all effort must come from the teacher and almost nothing from the student, that is why I was in awe how you seems to have no hard feelings towards your teacher@@billkinneymath
I was a math minor but placed higher from high school credits. So the last course I had to take to get my minor was Differential Equations. The beginning of the course with predator/prey models was fun and I was able to understand that fairly well. However, when we moved to Laplace Transformations I was lost. The only thing I can remember is you have a real function and it's transformed into an imaginary function. There were a handful of function definitions that you had to use in order to perform the transform, and I could never identify which one was needed. I love math but Laplace Transforms was way over my head.
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.
Very funny!
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.
This was truly an inspiring story. Thanks for sharing 🙏
@@brozors -- Thanks for reading!!
Awesome story! Congrats!
@@billkinneymath -- Thank you thank you!! I don't remember jack squat from that class. I ended up specializing in applied analysis and ODEs.
@@artichoke60045 Thank you!
Glad to hear even math professors sometimes just had to wing it in a tough class xD
I sure did. And that wasn't the only one, just the hardest one.
@@billkinneymath , It wasn't a class. Your teacher taught nothing. The goal of teaching is to communicate. He made no effort. He did what amused him, with no interest in conveying useful information to the students. That's not a teacher. If there is such a thing as the opposite of a teacher, he was it.
to be honest professors and tutors in universities aren't "teachers" @@maskedmarvyl4774
@@maskedmarvyl4774 Lots of bored teachers do it. They teach what they should 1, 2 or 3 years, then they realize how boring it is, and they rarely take satisfaction in actual teaching and then they steer to what INTERESTS them, not students. To be a good teacher one needs to be less selfish and have observation skills and undurstanding that perception of others is different from their.
"Trivial right"? 💀💀💀💀💀
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 have a similar story. In Years 9-12 at High School. I had the same great maths teacher. In Years 9-10, out of 2 classes that included re 60 students, position-wise, I'd come like 57-58th.....I was hopeless. Didn't even know how to balance a simple equation like x+3 = 2x -1.
Then one of my best friends left school and in Year 11, all I did was do my homework (what a concept lol) and read the textbook for re 20 minutes every night. Then something clicked, and from there it was smooth-ish sailing.
When I sat the HSC (Higher School Certificate, NSW, Australia) in 1986, out of re 30 000 students, the 1st exam of 2, was for 3 hours. I did it in 50 minutes including going back and checking my answers, and came in the top 3% in the State (NSW). For the 2nd exam, I took an hour instead of 2 and came in the top 10% of the State (unknown number of students).
Strangely enough, even though it has been explained to me several times, I still suck at HSC physics, (Mechanics), yet electricity and magnetism light, and waves, I excel at. Like, go figure.
Never did complete my Undergraduate Pure Maths degree as my life changed direction, yet when I move overseas in the next few months to be with my Mrs and settle, since I am in my mid-50s now, I may go back and study some maths. I actually think I have a way to approach the 3n+1 problem (Collatz Conjecture) yet haven't looked into who to speak with.
if a professor tells you its a take home test, you know damn well it will be a HARD AF test!
I had a linear algebra 24 hour take home final.
No. Just no.
When a professor tells u its a take home test, u know it is gonna be a hard test.
When a professor tells u its a take home test, u have a week and u can do anything in your power to answer the questions including working together, then u know this test is just the test maker flexing on the difficulty.
😂😂😂😂
@@robertlunderwood taking a linear algebra class this year... u scare me
@@orang1921 It wasn't that bad. The professor just didn't believe in giving out any tests. He wanted to test our thought process. Most of our grade was from homework with the rest coming from the final.
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.
Me and some others in my dept. are going through Huybrechts' book on "Fourier-Mukai transforms," and one of the sections was devoted to spectral sequences. It was a difficulty spike I was not expecting lol. I hope one day I'll be able to look back on that section and think it was all trivial 😅
no idea wtf you are saying, but thumbing up your comment!
One of my colleagues used to call a spectral sequence - a procession of ghosts.
Can you explain for us mortals what it means and/or what it is used for? Actually curious but I’m on like…. calculus 2 level.
@@incizor1273 It was meant as a joke. Sorry for not mentioning it explicitly.
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.
Wow amazing, can I ask when you started?
you don't need to be exceptional in everything you do, the fact you finished is amazing.
@@yerb. I was working full time so could only study part time, took 9 years.
Congrats on persisting and becoming an engineer!
Congratulations!
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.
Sounds like you learned a lot correcting him!
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 🙄
Yes, probably 50% of my teachers in grad school were like that. Another 40% were okay but not inspiring. Only about 10% were both clear and inspiring. But I made it through anyway.
I find that odd - if the department knows that the courses have an 80% fail rate, why not change the way that they're taught? They're clearly not accomplishing their objective the way that they're currently setup.
@@aiberengi I got some insight into the management of a university department when I did my ph.d - in all the meetings during those three years the only aspect of coursework and teaching that was discussed was how to reduce it to an absolute minimum while still meeting all the legal obligations. Add to this that "academic performance" (at least back then - this was 2007-2010 in Denmark) only includes publications/citations and funding and you end up with teaching simply not getting prioritized. At least not from the vast majority of professors or the departments as a whole. That is my experience anyways.
I had a class that used that Kreyszig book! Now that I'm older and have learned more math on my own, that book and class was garbage. The problem is it crams a ton of math from various parts of math into one course. You just don't get the foundation you need to understand any of it. And it's a huge book.
@@GizmoMaltese My sentiments exactly - and yeah, you can beat a horse to death with that book 😀
Saw a tiktok edit on 3rd quarter algebraic topology lmao
Cool! Do you have a link?
One of my students just shared a link with me: www.tiktok.com/t/ZTRs6psCD/
@@billkinneymath hahaha yeah that's the one, it's really cool to see you go viral on TikTok!
@@Ninja-wc4un I made a TikTok account just to reply to it with a link to this video but so far nobody seems to have noticed my reply (after about 2 hours). When videos go viral it seems to be just for a very short time.
@@billkinneymath damn that sucks no one saw your comment. I tried looking for your comment but it’s nowhere to be found in the abyss.
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 ;-)
Glad to hear you were able to figure it out!
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.
Wow. I think maybe we talked about this at a Joint Meeting. If you've still got a copy of your thesis, send me a few pictures of how you used spectral sequences over facebook. Did Rick have to review spectral sequences himself to understand what you did, or did he already know it well enough? I didn't remember enough when I wrote it on the board. I cribbed it off of Wikipedia.
@@billkinneymath The stuff on the board looks to me like the spectral sequence arising from a filtered cochain complex - might be mistaken though
@@gregwong-dg5jq I don't remember it well enough. I just found it on Wikipedia. 🤣
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.
ODE wasn't too bad. PDE kicked my butt.
@@robertlunderwood sorry I meant pde. I said ode cause the coheres name was ode 2
I had a "basic" PDE class that focused more on technique (Weinberger's text) than more advanced PDE classes, so it wasn't too bad. It was also taught by a young postdoc who enjoyed and was good at teaching. In my ODE class, the professor delved into what I thought were more interesting topics about dynamical systems rather than "classical" ODE stuff that you might more typically get before 1960.
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!
That's funny! Our graduate level abstract algebra course was just called "Algebra"
Riemannian Manifolds? You had fun. It's like Banach's algebra😮.
@@billkinneymath The course description for the upper-level Combinatorics course at my uni is "counting techniques and coloring problems". I found that really funny, given that it's graph + partition theory but sounds like an elementary school class.
@@elliottwalker8624 That is so funny! I actually made a "map" drawing of Snoopy (making it like a puzzle) and brought it to my daughter's 3rd grade class to hand out to the kids and teach them about the 4 color theorem.
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.
yeah that stuff was a requirement for GR a d was wild... differential geometry is absolute madness
The very first sentence in the Spivak's book is "This little book." That's all I can remember from that book
@@antronixful and do you think that all of this math is really correct, consistent and describing something in the real world? feynman said that reality doesnt exist, only math does, and this idealism is enough for me to throw in the garbage this stuff; how can you imagine that einstein could really grasp the enormity of millions of terms in his ten gr equations, if written extensively, and instead contrapting them to a simple equivalence by the introduction of a zero term? i dont buy this, as i dont buy the quantum mechanics practice of introducing some wild virtual particle, entanglement, action at distance, superposition and other logical absurdities when the calculus doesnt resolve as desired, only to match the math to the experiment. Modern physics is a big scam.
@@zdenekburian1366 ok
My differential geometry class was taught by a Russian physicist who conned me into taking the class by saying [insert Russian accent], "This is very good class for physicist." I get halfway through the class, past the drop date, when he says one day, "We are going to skip chapter 5. What this is is applications in physics. You should read it. It's very good." My jaw dropped. My pencil dropped. I thought, "Then what am I doing here?" Just like the guy in this video, we had a final project. I rewrote some fluid dynamics equations I learned in one of my atmospheric physics classes using differential forms and got a B. I still don't think I learned anything.
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.
Thanks so much for the encouraging message. So happy you to have you watch!
This was one of the most humorous videos on RUclips, Well done on that and the course!
Thanks!
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.
Me too...30 years later! 😅
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
I just started learning about that subject this summer!
I love math stories like this! Fun to hear. Probably not fun to experience.
Yes, I can look back on it now with fondness, but not so much when it was happening.
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 have proffesors in high schools? I don't get it.
@@RigelNV Oh, probably should have worded that differently. This was a course in graduate school and completely different from Algebra II in high school, even though they have the same name. I would have failed so badly if it were high school…
That's a very funny way of putting it! There have been other textbooks that I have also been stuck on the first page of too. I think one of them was on hyperreal numbers (non-standard analysis). I never took a course on it, but I wanted to learn it. I picked up the book, looked at the first page, and then put it back on the shelf. When you think about infinitesimals intuitively and not rigorously however, it becomes kind of a fun subject.
@@GabeKhan Yeah. A friend of mine told me highschool algebra was like learnnignto count your toes as a toddler. It teaches you what they are, where they are and some idea of how many there should be. its up to you to figure out how to make sure they are all there!
@@billkinneymath not really a math guy myself, more of an engineer, but the nonsense supposedly required in "rigorous" probability theory with Borel sets got me to take a course in measure theory. So much work and boilerplate notation for no worthwhile intuition at all. Actually it should be needed anywhere calculus is applied, anywhere real numbers are used -- the answer is meaningless because it's all unmeasurable, and you have to use something like Borel sets instead! Kolmogorov made a name for himself by, I guess, rewriting existing probability theory in that much more complicated notation, so it all became associated with probability theory. Then I noticed the thin book "Radically Elementary Probability Theory" by Edward Nelson in the library probably because of its orange paper cover and read most of it in an afternoon. It's apparently a correct way to define real numbers. The definition is a bit weird, but not nearly as weird as a Borel set. The original guess for real numbers, the set of limit points of sequences of rational numbers, was a reasonable try in the 1600's or 1700's, but it turns out to have been a wrong choice, because what results is essentially meaningless numbers that cannot be used to denote quantities.
i agree. spectral sequences, exact sequences, snake lemma ... are difficult to grasp. this all is because of moduli spaces
Yeah...all that category theory stuff...it's so not fun(ctorial)! Such abstract nonsense! 😂
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.
Hope it goes well for you!
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.
Hope you get a chance to do it someday!
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.
I'm in the same boat with you on both real analysis and algebraic geometry.
I feel like complex analysis is just so much more beautiful and I almost wish it was taught before real analysis.
@@Seeker265729 I agree. I think it is the most beautiful subject in math, though it is not my field of expertise.
@@Seeker265729Any complex analysis books that do not require real analysis as prerequisite? Needham's? Any other as suggestions 🤔
@@spiderjerusalem4009It's probably best to just do a bit of real analysis beforehand if you really want to learn complex analysis. You don't need that much so it shouldn't take too long if you're committed
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.
Wow! It sure would have been helpful if someone had advised you about what to take before you joined that course.
@@billkinneymath What's more was the professor even said "this is not easy stuff, this is very serious mathematics. And I expect you to spend at least 10 hr per week studying for this". Then I thought: this had to be some kind of misunderstanding or miscommunication between him and the department. Because it was offered as a 3-credit free-elective course, just like any other. If it takes that much effort, why isn't it set up as some standalone research project? We were also not grade on homework or participation. Our entire grade consisted of just a midterm and a final.
Maybe it is expected of a master student to be matured enough to know what courses they should take and I made a wrong judgement call. But even if there was someone who knew all the courses in the department AND knew my background well, they couldn't have predicted that this professor personally wanted to teach this course in such an unordinary way. I don't know if it was because it was a private institute or because he's very famous, he just had a lot of freedom in how he teaches a class.
And now my academic future is hindered if not buried by this blackspot.
What did you study there?
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.
I'm always impressed when people try to learn advanced math on their own. Fortunately there are many good RUclips channels (3Blue1Brown is my favorite) that help with the intuition. The full details have to be worked out in your own mind, however.
I am self-studying too. Hobby only.
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.
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.
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.
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.
She should have been fired.
@@JB52520 Nah, she was a tough teacher, but she taught us well.
@@jeffreysommer3292 What even was the point of a test where you're rigged to fail?
awful teacher
Latin ezez lol
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.
Glad to hear the others were good. Was that for a computer science degree?
@@billkinneymath Yep
That was actually a pretty cool story. Thanks for sharing.
You bet! Glad you liked it!
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.
Currently failing calc 2. I need a 77 in the test I’ll be taking in about 45 minutes, a 77 in a retake this Thursday and a 77 in my final next Tuesday just to get to a 70%. 😭
@@Capitan_Chaos I believe in you!!! You can do it!!!
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
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.
The typical graduate student experience. But it was good you talked to others so you knew you were not alone!
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
Did you pass?
Great video, great pacing, thanks for sharing a gem of your learning experience sir!
Thanks for watching! I can look at it back with fondness now, even though it wasn't fun at the time.
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
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.
I agree. I think if the professor had actually assigned some homework so that we were motivated to study it, that would have helped. I appreciate everything more and more as time goes by.
Can confirm. Walked out of Algebraic Topology course after two classes in grad school - incomprehensible.
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 felt that way to us too. Even though the first two quarters were okay (the professor was better too, but they switched us to a different professor for the third quarter).
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.
I had an abstract algebra class that did category theory right away. I had no idea what it was about at first, but I came to appreciate it later.
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. ❤
I think the proof of the snake lemma in gtm 196 is the most simple and elegant one I ever know. Others I saw always use brutal diagram chase.
That is a subject that can be "fun" without knowing what it means. I remember learning about tensor products and thinking it was just a "game". But I didn't know what it was for, or how it was related to tensors in physics. I still don't know. Maybe someday I'll be able to study it further.
@@billkinneymath Tensor algebras and more distinct algebras like quasi-triangular Hopf algebras and Hecke algebras arise in the quantization of gauge theories and string theories. Here we are often dealing with modules. For example the Verma modules as the highest weight representation of Kac-Moody algebras. These are quantum symmetry algebras of two-dimensonal conformal field theories living on the string worldsheet or the boundary of the topological membrane. If you'd like to delve into some very interesting new math, which was predicted by phyicists in 2009 I would emphesize the subject called AGT-relation/correspondence/conjecture. Its core statement is that the generating function of equivariant Euler characteristic classes of the moduli space of self-dual solutions (instantons) to the SU(N)-Yang-Mills equations on R^4 is equal to the conformal blocks of an affine W_N-algebra. This statement has profaound and whide-reaching consequences, which we are yet to understand.
@@youtubesucks1885 I think all that will take me the rest of my life to understand. But maybe I can get there someday!
@@billkinneymath I am 29! You will get there faster than you think.
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).
Hope it goes well! Have you found any good resources to help?
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.
😅 That is so funny! Maybe I should tell that to my multivariable calculus students so we don't need to learn about Euclidean space. (just kidding). But it's sad to hear that about Fermi. He's so famous these days because of the Fermi paradox too.
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.
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.
Hope you have the time to do that and hope it goes well!
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.
We did category theory at the start of my grad school abstract algebra class. I had no idea what it was about but thought it was kind of fun. Later on I've appreciated how unifying it is.
The edit is fire
I think all of the different 'Algebras' beyond Master's level are difficult primarily because of the abstract component /reasoning in the sense that it's Pure Math & not Applied akin to Kreysig's massive tome, which I still swear by as one of the best AEM texts ever written for self-study & revision.
As an ongoing Master's level wannabe, I'd single out a few beauties but 'tuffies' such as:
*Differential Geometry
*Tensor Analysis
*Complex Analysis
*Chaotic Dynamics
As tough as it is, I'm still in awe of the sheer beauty, depth & truth in all Mathematics!
It's a journey of discovery & a quest in pursuit of determination to succeed & unlock the human potential...
Thanks for sharing! I love math too, even spectral sequences now that I don't have the pressure of the class anymore!
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.
Sir i think, elliptic equation analysis, complex descretization, advanced vector calculus, abstract harmonic analysis are also way harder.
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
Yes. I wish we had stayed with the standard content of algebraic topology. That was challenging enough, but at least I understood it better than spectral sequences. The professor I had for the first two quarters was a better teacher too.
@@billkinneymath Oof
Isnt analytic geometry just calculus?
@@nestorv7627calculus is too broad a term. I'm ignorant about this field, but analytic geometry could maybe be the same thing as geometric analysis? Or maybe analytic means holomorphic, so maybe it means complex geometry? But I'm sure it is something unique.
@berserker8884 in the US we call calculus 1,2, and 3 "analytic geometry." In some rare instances professors may call that differential geometry, but it's usually referred to by diff geo itself
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.
Hope you are able to learn it. Then you can help me understand it.
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
That's a perfect picture of advanced math classes right there!
Manifolds was definitely the hardest for me! From smooth atlases to De Rham cohomology still don't get a lot of it!
I took abstract algebra as an undergraduate and it was very hard. Definitely the most abstract course I took in college
no, undergrad algebra is easy, very easy. galois, representation theory, group theory, all the standard topics are kindergarten stuff.
like this guy said, the hard stuff is topology, topology is fking nightmarish. if u think spectral sequences are hard, look up A1 homotopy theory, algebraic K theory. spectral sequences are just a computational tool, you just need to get used to it, but those subjects are so hard you'll be crying trying to understand them.
if you want to learn something really abstract, try to learn modern homotopy theory, non commutative geometry. topology and algebraic geometry are the hardest subjects, you'll cry learning them trust me.
the money i got after taking these classes was abstract as well.
Calc 3 prof did something analogous. It was supposed to be a standard multivariable calc class at an average university but he lectured stuff not in the book, had exam questions about things we never discussed or only touched on briefly and borrowed exams from harder classes at MIT etc, with stuff we didn't cover. I'm not a wiz so I was pretty lost. Gave everybody in the class a C that made it through to the end out of guilt. Sucked because it broke my all A record.
You are amazing Professor ❤️🇲🇨
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.
You can kind of think of it as a "game" where you just play by certain rules. But that doesn't necessarily help you understand why it "works". There are some easier examples out there though. I would try those.
Fun fact: Kids mathematics deals with numbers
Adult mathematics deals with letters.
I remember when I finished middle school and got to highschool, the math teacher said "So far you haven't done real Math, now we start wit real Math". Then I go to university and the Math lecturer says the exact same thing. Then a year later, just for fun, I sit in a graduate level math class to see if I can follow (spoiler, I couldn't follow a thing), the math professor says "now it's time that you learn some real mathematics". And of yeah, everybody at Uni loved to say "trivial", to avoid having to explain things.
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.
In my real analysis (measure theory) course the professor was too easy on us during the course. I got an A, but I couldn't pass the written prelim, which was much harder!
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.
Wow! A grad school math class with discussion. In most of mine we just listened to the professor (again, mostly because we were afraid to reveal our ignorance). I did have one professor who really encouraged questions though, even if they seemed really "basic".
Been there, done that. MA Berkeley, ‘78. PhD, CUNY, 2015. Still learning at 71. Thx!
Nice😊
Awesome! I love life-long learning of advanced topics!
You are amazing! I want to ask is there going to be any Algebraic Topology lecture? Thank you for your work!
Thanks! I'd like to do Algebraic Topology in the future, but I don't have time to do it at the moment. Blessings!
@@billkinneymath I'd like to be a patroen of this leture, if you're gonna record this course, I'll be willing to pay 100$ to support your work through www.patreon.com/.
@@bzhkl Thanks for the offer! Maybe I can do it in a couple years.
I think if you are interested you can always start reading gtm 119 (rotman), or Munkres (his algebraic topology book, not his general topology or differential topology). I personally recommend these two, and I especially like rotman, it’s very easy to read and it’s very detailed in my view.
Mathematicians are usually some of the most humble and wise people. Little to none of the ego affliction of many of the other disciplines. Only artists can possibly claim something similar and this is why they are brother and sister.
Love it!!!!!!!!!!!!
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
Unfortunately, it's a very common experience.
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
That was definitely a tough one for me too. A funny story there is that my professor was actually too easy on us. Like the other class from the video, he hardly gave us any homework (he was a slightly better teacher though). So the class didn't prepare me well enough for the written preliminary exam for the Ph.D. program. Fortunately I passed the Complex Analysis prelim so I didn't need to pass the Real Analysis one on measure theory.
Haha, you should make a video on that one aswell one of these days, glad it turned out okay
For me, the second grad course in real analysis was the second hardest course I took in undergrad, the hardest being a course in algebraic geometry. We covered L^p spaces, the interpolation theorems, Fourier analysis, distribution theory, and applications to PDEs. I've never used Baby Rudin, but I think it would not be used in a graduate level class. More likely, such a class would use something like Rudin's other texts, Folland, Stein and Shakarchi, etc.
Rudin is great, his approach to measurable functions is by a different (equivalent to the definition I often see in other books of course) definition , I like it very much.
Your math prof must've been highly inspired by Lovecraft.
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. 😂
Solve the hardest maths Problem no more maths
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Interesting though.
Mind to elaborate what the big plus circles are? The ker and ims?
They are symbols from abstract algebra. The plus circles are "direct sums": en.wikipedia.org/wiki/Direct_sum. The kers and ims are "kernels" and "images": en.wikipedia.org/wiki/Kernel_(algebra) and en.wikipedia.org/wiki/Image_(mathematics)
i have always said that there is not hardest math but people just make them hard by the way they explain them
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.
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.
We got to know almost all the books in the library that were relevant for our courses!
Mine is not very hard overall compared to others but for me it was calculus in college last year. But the reason why it was tough is I went back to college 11 years after high school and my highest math up to that point was 10th grade algebra. I failed out of pre-algebra at my first go around at community college 10 years prior and dropped out completely. Coming back to get my degree I had to take this class and they allowed me to take it. (I did pass stats before this so that may have been the reason they let me) Well, I studied very hard, did my best learning the concepts of algebra and passed my calculus course. It was a wild ride cramming everything into my brain for that semester.
I hope you succeeded at understanding it!
Algebraic Geometry with schemes is pretty evil too. Not to mention topos theory.
Fortunately I never had to take that class. I knew someone who was doing his Ph.D. thesis in Algebraic Geometry but I heard he never finished because his advisor never thought what he did was good enough.
@@billkinneymath Yes. As the saying goes, there are two kinds of mathematicians: those who don't understand Grothendieck, and those who pretend to understand Grothendieck.
Yeah I feel the same. I feel Hartshorne is difficult, I read algebraic geometry and arithmetic curves by Q. Liu instead. It’s supposed to be easier to read, I still read it very slowly 😢. I read about topos theory in some categorical logic books. I feel it’s very interesting , but I lack motivation to read further about it. I guess that’s because I know too little about logic, I read some basic about it purely because I like categories.
My last grad level class 20 yrs ago!!! Still shudder at that thought!!!
Did you have a similar experience in a course?
2:40 had me weak with laughter 😭
I’m sure math majors do much worse but general relativity (tensors) was ridiculously complicated.
I came across a really thick book in the library once whose title was just "Gravitation". Yes, it looked really hard!
@@billkinneymath It’s one of those things that I think would get a lot easier once I got used to the notation, but I didn’t have enough time for that in the class I took and definitely need to go back and reread the textbook.
It was Sydney Coleman’s Lectures on General Relativity for anyone interested; I’m sure a math major who already has experience with tensors would pick it up much more quickly than I am.
@@billkinneymathI'm sure it's misner Thorne wheeler. It's a classic
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.
That was my old book from the 80's too. I still keep it in my messy office!
I wonder if our math teacher can solve this, since they know what they’re doing but not really good at explaining/teaching
You'll have to ask your teacher! It's not a commonly taught subject.
@@billkinneymath I know it’s not taught from school
Great stuff, had to learn some differential top. for my recent paper 📜
Glad to hear you liked it!
How do you feel about number theory? Sounds relatively harmless until you actually open the book…then it’s terrifying
I actually haven't had the opportunity to study it beyond the basics that are learned in abstract algebra. I know that what Andrew Wiles did to prove Fermat's Last Theorem was incredibly hard though.
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.
Hope you can finish!
It would have made much more sense to name the class Homological Algebra. It is in the last chapter of the book on Homological Algebra that I am studying and they look terrifying indeed.
I still get scared by it!
It's my favorite topic, once you understand what it's talking about you cant unsee it, but if you dont get it looks like gibberish
@@CrucialFlowResearch That's really cool. Did you just learn it recently?
This was very funny, I always enjoy your videos, spectral sequences sound interesting
Glad you enjoyed it! Let me know if you study the subject and understand it!
Bill, I really think the title of this video should have been "How not to teach" and you could add "mathematics" if you like: The students not knowing what's going on, teacher talking to the board, little to no engagment or interactivity, not knowing how you're going to be assessed or what on, etc; all too common yet so wrong! When are these "teachers" going to be held accountable! A proposition to consider: a course is only as hard as the teacher is bad
Yes. Unfortunately, especially at research universities, there's not much effort to hold poor teachers accountable as long as they keep the grant money flowing and bring research prestige. I think it is somewhat better now than 30 years ago with some teaching specialists added to the mix (though they are paid less), but there's still a lot of institutional inertia. It's the way the system is designed.
@@billkinneymaththe problem is that there's no incentive to structure things in any other way. If every tenured prof doubles as a researcher, then they maximize the amount of grant money they can bring in, and the prestige.
The moment you introduce someone who specializes in teaching rather than research, your wasting an opportunity to bring in another researcher. And money.
It's a terribly broken system which prioritizes the university over the students in a hideously capitalistic fashion.
@@Bobbias I agree it is broken, though there is value in some of the research. I just wish teaching would be valued as much too.
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.
Very interesting story!
It was a strange experience. I was very fortunate that it worked out the way it did!
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
Actually I looked back at my high school math notes from when I was 17 years old and realized that my math teacher was doing real analysis. Of course, I didn't know it at the time. I thought that was what most 12th graders were doing. He didn't go super in-depth, but he was doing the basics!
Shouldn't the bottom left E_2 be p+2, q-1? Loved the video though!
Thanks! You're probably right because I don't remember. I just copied what I found on Wikipedia, but maybe it was wrong there too! 😂
That was the funniest story!!! I am delighted how you tell it without a pinch of resentment, God forbid in my country for a professor to do that, students would riot, the professor would be lynched, fired or at least reprimanded.
I'm so glad you liked it! Yes, the professor had tenure, so there were no penalties. We were thankful that we could move on to other things!
I love it! I am a professor of classes related to math (calculus, physics, statistics) and think a lot about this learning process, things like how much effort is expected from the students and from the teacher, and where I work I think it is expected almost all effort must come from the teacher and almost nothing from the student, that is why I was in awe how you seems to have no hard feelings towards your teacher@@billkinneymath
Omg I literally have the same advanced engineering mathematics book it was given to me when I was 9 by an engineer haha
I was a math minor but placed higher from high school credits. So the last course I had to take to get my minor was Differential Equations. The beginning of the course with predator/prey models was fun and I was able to understand that fairly well. However, when we moved to Laplace Transformations I was lost. The only thing I can remember is you have a real function and it's transformed into an imaginary function. There were a handful of function definitions that you had to use in order to perform the transform, and I could never identify which one was needed. I love math but Laplace Transforms was way over my head.
Hope you get a chance to review Laplace transforms in the future. They are kind of fun when you get used to them.
So far my hardest was surface area
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For me according to high school syllabus it is permutation and combination and complex number