I'm convinced it's almost impossible to learn grad level math outside of a university environment.I graduated with a math degree about 7 years ago and did the graduate sequences in analysis and algebra that used Lang and Folland in my senior year, and worked in tech since then with zero need for advanced proof based math. I've been relearning math again naturally having forgotten a lot. Undergraduate level analysis and algebra was easy to get back up to speed. But I just hit a wall with the graduate level material, which is pretty much inaccessible without lectures, lecture notes, classmates to discuss the problems (and the natural feeling of healthy competition with them) to create the sort of environment that I had while taking the real life classes. And it's not because the material is too difficult although the difficulty is definitely a large part of it. But there is a lack of resources for it that the undergraduate material doesn't have, where there are a lot of books, solved problems, even videos.I could do all this stuff before, but I'm finding it impossible now. Unless you're a legitimate off the charts genius with a photographic memory I don't think it's really possible to learn advanced mathematics totally on your own without taking the classes.
>>16930437>the natural feeling of healthy competitionlead poisoning
>>16930437Kind of. You need to be exposed to the general culture around math, and that might take some time, but after that, you can tackle pretty much anything that's sub-research-level on your own.
>>16930437I would like to emphasize that this a very case-by-case issue. I have had a mixture of academic and individual study, and it has mostly depended on the specific field of "grad" mathematics. The existence of the internet has eliminated a good deal (not all) roadblocks to grad material via the free accessibility of the supplemental media online. I suspect your main issue was one of lack of synergy with the material more than any inherent incapabilities.That being said, no matter how easy it is these days to get supplemental materials without access to a university, it's still a pain in the ass unless you really like individual studies.I am also speaking from more of a pure mathematics specialist's perspective so I'm not sure of your specific context.
>>16930442This is true; mathematics has just of a dynamic culture as any other field, and if you go in blind you can get shell-shocked.
>folland as grad analysis Do americans really?
>>16930456shhhDon't remind the American's how much of shit hand they got dealt with analysis materials.
just read a book jesus christ if you're relying on lectures to teach you then you're learning a dumbed down version just intended to convey the general concepts so you're not totally lost when reading the book
>>16930437I'd argue you likely only ever had a very superficial understanding unsuitable for anything beyond solving homework problems.
>>16930473just attend the lectured jesus christ if you're relying on books to teach you then you're learning a over specified version just intended to convey the computable difference as example so you're not totally filtered when attending the lecture
>>16930437This is completely wrong. Math is the easiest subject to self-learn because no expensive lab equipment is needed.
>>16930437It is possible. You'll just have to accept the reality of being a crank.
>>16930437The problem is the insane amount of gatekeeping/obfuscation which is pervasive throughout all mathematics. The notions aren't inaccessible if you lack some superhuman abstraction ability (math is in a sort of programming language that compile into mental processes), but you do need to learn notions in order and not only a logical order but one that allows intuition to build accordingly if you don't want to be overloaded with technical details that remain meaningless to you.But in order to have that, the road has to be mapped before you get started and someone have to do it for you. For your picture with the snake lemma for instance: homological algebra is next to impossible to learn alone (it is up to 120 years of technical developments over technical developments starting from Poincaré). Recently I've come across one of the most wonderful books I've read: "algebraic topology" from Wiliam Fulton. In the last chapters he presents homological groups in an extremely efficient manner, linking them to triangulated spaces (which homological algebra is really about). This was one of the most enlightening moment of my life and where (in my opinion) you should get started. How do you do if no one has told you that and you try to get through thousands of pages of functors, spectal sequences and derived categories without that?
>>16931204I actually think AI can solve this for you now. Unless what you want to learn is genuinely only in working mathematicians' heads, it should be able to find you a path through the literature.
>>16931213>muh AIkyskyskyskyskys
>>16931213It actually can't.
>>16931234>>16931440If you didn't comprehend my post, then you genuinely never stood a chance in the first place.
>>16931581I did comprehend it. It was complete gibberish that's not even worthy of contempt.
>>16931234>>16931440If I asked AI about books that could help me understand Snake Lemma, it would never recommend me Fulton's?
Anyways let R be a ring, a differential complex is a pair (M,d_M) with M an R-module and d_M: M -> M an homomorphism such that d_M o d_M = 0 (*). We define H(M) to be the quotient of Ker(d_M) by Im(d_M) (the latter being included in the former thanks to (*)). If (A, d_A) and (B, d_B) are differential complexes, a morphism f: A -> B of modules is said to be a morphism of differential complexes if d_B o f = f o d_A (**); such a morphism induces a morphism between quotients H(f): H(A) -> H(B) thanks to (**). Let (A,d_A), (B,d_B) and (C, d_C) be differential complexes, f: (A,d_A) -> (B, d_B) and g: (B, d_B) -> (C, d_C) two morphisms of differential complexes such that Im f = Ker(g), g is surjective and f is injective (we say that "0 -> A -> B -> C -> 0 is an exact sequence"). Then you can prove construtively that there is a morphism D ("boundary") from H(C) to H(A) such that Im(H(f)) =(Ker(H(g))), Im (H(g)) = Ker(D) and Im(D) = Ker(f) (we say that "H(A) -> H(B) -> H(C) -> H(A) is a long exact sequence").This lemma actually is what matters (must know!!) and its superseds the "snake's lemma". In most case s A,B,C will comewith a graduation, inducing a graduation into H(A), H(B) and H(C) such tthat D shift the indices but this would obfuscate the result and the proof (which is a rather simple calulation where every next step is rather contrived).
>>16931204I actually understand the snake lemma and its proof just fine. It's just tedious diagram chasing. That's not one of the things I am finding abstruse.
>>16930437>without taking the classesAs a spontaneous savant genius, once you finish the books you attend lectures on record while reading papers (if helps if it in a different field).Ive attended maaany lectures, too many. People teach in different ways and some use rarer approaches to things.
>>16930437Imho, you basically lack a lot of basic skills a math student should have developed during their study. Not even talking about problem solving and proof-writing skills. Although you definitely suck at those too.“How do you check your answer without a solution manual?”“How do you approach problems that are too difficult for you?”“How do you massage a problem into something more familiar?”“How do you study a book that has no problems?”“How do you come up with your own questions?”“When to ask for help?”That sort of thing. Call it “mathematical maturity” if you want. They are just stuff that you learn naturally, subconciously, and not taught anywhere. Definitely not written on any textbook lol. This is something you need to work on by yourself. You relied on crutches too much during your previous study. Googling shit, solved problems, forums, videos, etc. And didn’t use your brain enough instead when you were faced with anything slightly difficult. Basically what people warned about modern use of AI.
>>16933213that's a whole lot of assumptions
>>16930441this
>>16930437>snake lemmaTerrible pic to prove your point, I learned diagram chasing on my own and it's easy as fuck
>>16930437>But I just hit a wall with the graduate level material, which is pretty much inaccessible without lectures, lecture notes, classmates to discuss the problems (and the natural feeling of healthy competition with them) to create the sort of environment that I had while taking the real life classes.Don't some colleges give free access to older lectures online? Like Harvard & MIT, if i remember correctly.
>>16936256try reading all the posts first >>16931735
are most graduate level math textbooks free or paid?
>>16930442>>16930448Wtf does this even mean? Sounds like bait
>>16930437"Methods of homological algebra" by Yuri Manin and Israel Gelfand, is an excellent book to get into this fantastic topic and to be introduced tp the modern tools of it, namely derived categories and sheaves over a site.The book contains typos and is demanding but you'll be rewarded.
>>16930456Is folland bad?
>>16936296Free if you're lucky, paid if you can't find it anywhere else.
