I am a self-learner who loves pure mathematics. I study mathematics alone without university and I don't have professor or advisor to help me (So I am not sure what I can learn next). I mostly study math just for curiosity and I have learned calculus, real analysis, and linear algebra. I have read these books:
- Thomas’ Calculus (and Stewart's book).
- Half of Advanced Calculus by Patrick Fitzpatrick.
- Introduction to Real Analysis by Robert G. Bartle.
- Principles of Mathematical Analysis by Walter Rudin.
- Linear Algebra by Kenneth Hoffman and Ray Kunze.
I want to know what I can learn next or in other words What topics that I have their pre requisite? I also don’t like real-world applications in mathematics; I prefer to learn the abstract concepts.
I know this question might have different answers depending on the school or professors, but I am pretty sure that these differences are not that big and many will agree on what I can learn or I can’t learn now. For example, I am sure that most people (or all) will agree that I can’t learn algebraic topology right now. I was thinking of these topics:
- Elementary Number Theory: I don’t know if I should learn number theory now or later as I have heard that it uses many concepts from complex analysis and many special functions like $li(x)$.
- Complex Analysis: I think this will be a good complement to my real analysis books, but should I use Rudin’s Real and Complex Analysis or not? (By the way, I don’t know any other books on complex analysis).
- ODE: I don’t know if I should study this after or before complex analysis as I heard they use complex variables and I heard that most of this topic has to do with applications. As I mentioned before, I don’t like learning applications with new concepts; I like to learn the mathematical concepts in abstract.
- General Topology: I liked chapter 2 of Rudin’s book the most and it was about basic topology. I think real analysis is the only prerequisite, but I heard some people like Zach Star recommend a course in abstract algebra.
- Abstract Algebra : I heard that number theory is a prerequisite for this topic as many concepts involve (somehow) prime numbers.
There are other subjects I don’t know much (or anything) about: Combinatorics, Probability Theory, Graph Theory, Naive Set Theory, Measure Theory, Functional Analysis, and Fourier Analysis. But I think Combinatorics and Graph Theory are some of the options that I can study now.
I would suggest taking a look at I.N Herstein's Topics in Algebra, specifically to learn Group Theory. If you like the exposition to Group Theory, you can go through the rest of the book.
In my opinion, the subject is beautifully presented accompanied by an excellent set of problems. Also, the elementary number theory required for your study is present in the book or can be studied from other sources (not a hard task as not much depth is required).