Advice for which "advanced math" topic for a self-taught student to start with? (number theory, group theory, probability, etc) Since I have been doing contest-math for a very long time, I want to try out more advanced math. Now the term "Advanced math" is a bit vague.
I want to learn

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*Analytic number theory

*group theory

*probabilistic methods

*Topology

Being a self-taught student, I don't know which area I should start. I don't have any preferences too, all of them looks nice to me. So, assuming I have learnt all the high school maths, which topic would be a bit easier to grasp? Also, I don't know what their prerequisites are? I know basic calculus and modular arithmetic ( Theory in David burton's book). Also, I will be doing it as a hobby, so I would prefer YT videos and handouts more than books, at least for the starting.
Can anyone please guide me on the prerequisites, which area to start learning first and what later? Which books, handouts or YT videos I should refer to?
Thanks in advance!
 A: Disclaimer: Most of what I suggest is informal and applied mathematics but I think is still good mathematics because it sets up as a primer for when you will actually do those topics in university (?).

Hi! I too found myself in this position many times. There are a few routes you can take once you are done with the school mathematics (I am assuming you have done the NCERT syllabus). I'll try to mention all of them:

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*Finish vector calculus and multivariable calculus with this animated lecture series by professor Greist. I suggest doing it with a standard book for solving problems. This will open up a lot of door into 'applied' mathematics books.


*Visual Complex Analysis by Tristan Needham. Again an informal book but it's fully based on geometry and it exposes you to a lot of beautiful parts of CA. Highly recommend.


*Tensor Calculus (?) : This maybe controversial but I think it is really helpful to know tensor calculus as presented in this video series by Pavel Grinfeld. He also has a book which he links in his videos, which I recommend. The reason I suggest this is because with some of the methods/ techniques he discusses here, a lot of the derivations for vector calculus identities becomes much simpler.


*Generating functionology : This book explains the links between counting problems and calculus. If I recall correctly, around the fourth chapter or so, the main theme is solving a graph theory problem and some abstract algebra concepts are brought up. I found it a nice primer for those ideas.


*Geometric anatomy of physics : A somewhat 'rigorous' build up of physics but from the basics of set theory and analysis. It goes really deep and the pacing is fast. I gave up on this one after the third lecture, but I plan on continuing when I get more time again.


*There is actually some 'deeper' maths you could explore while sticking to stuff you know with Olympiad  math. Check out the Olympiad geometry book by Evan chen, near the end, he explores ideas in projective geometry ( I haven't reached here yet) but it's really a cool topic. Primer


*Topology :  Lectures by Dr Tadashi Tokieda, here I've only seen about the first two videos but he explains in a very interesting way and I think someone determined for learning topology will find it easy to follow his lectures to till the end. There are also these notes:Snoopy Topology Notes which I heard are high quality.

Final comment: There are detailed notes made for Schuller Lectures already (The geometry anatomy of physics one) , see here
A: Of the four things you mentioned, the prerequisites for analytic number theory can be brutal. Setting up the $\zeta$-function, for example, isn't particularly difficult, but you'd probably need a background in complex analysis to cover standard results like the prime number theorem and Dirichlet's theorem. (The basic idea is that the poles of the $\zeta$ function control the growth of primes, but you need some sort of Tauberian theorem and some familiarity with complex analysis to extract an actual proof from that idea.) Number theory is a lot of fun, but there's a lot of machinery (in algebraic or analytic number theory) required to get to interesting results.
General probability can involve pretty deep measure theory, but the probabilistic method may not be so bad if you're in the finite category. Most of my experience with it is in contexts like hyperbolic groups, so I can't say much about it from the perspective of someone coming in from a contest-math-style background. If it interests you, though, go ahead and give it a try. The worst-case scenario is that you decide it isn't for you and move on to something else.
Group theory is pretty broad. There are standard texts like Artin and Hungerford, though you might be more interested in combinatorial group theory. If you're familiar with the basics already, you might want to look at representation theory. It's a great subject, and there are lots of sources (I'd recommend Fulton and Harris) that are well-written and accessible.
Topology's another great choice, though the problem there is that point-set topology is tedious but necessary before getting to the good stuff. Munkres is the standard text for it, but it's a pain to slog through. Books like Bott and Tu or Hirsch are great for the differential case, and Hatcher is beautifully written (and available for free online) for the algebraic case, but both have prerequisites that you might not be familiar with already.
