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Hello! I am a fourth year undergraduate in pure mathematics, and have taken many of the classes in your list (especially in the first and third categories), so I'll try to give some advice. First of all, what you're about to do is an very large endeavor - mathematics is a difficult subject, and learning math will take great persistence and self-motivation, especially if you are self-learning. However, it is also extremely rewarding - mathematics is a beautiful subject, and learning math has easily been one of the most enjoyable things I have ever done. For the next point, if you want to go deep into math, then you will have learn how to prove things. The heart of math is not at all computation, but ideas, and to know that ideas are true, we need proofs. All of pure mathematics is based on rigorous formal reasoning and proofs, and sadly, most high schools and even universities never touch this part of math. If you have never seen proofs before, I would first recommend reading the book How to Prove It: A Structured Approach by Daniel J. Velleman, which goes through basic set theory, logic, and various proof techniques. Most importantly, it will give exercises for you to practice. Let me say this now: it is impossible to learn math without doing exercises. Again, this will take some work, and the beginning may be a bit slow, but as I said above, it is extremely rewarding - there are few things so satisfying as finding a beautiful, clean, or elegant proof. I hope you will enjoy this as much as I have. Now then, let's dive into the courses and textbooks. I'm going to model this after what I did in my degree. Many of these topics require earlier ones as prerequisites, so I'm going to organize them into several layers. Some of the textbook recommendations may be a bit difficult, since in many of my classes the professors taught out of their own notes and left textbooks only as references, but I'll do my best. In your "first year", so to speak, there are three main things to learn: - Single variable calculus, differential and integral. You likely know calculus already, but again, we are now taking the proof based road! The canonical text for this topic is Calculus by Michael Spivak. It's what I used in my first year, and most importantly, comes with a solution manual :) - Linear algebra. As others have noted, linear algebra is absolutely crucial for many other subjects. I personally learned from Algebra by Michael Artin, but have heard very good things about Linear Algebra Done Right by Sheldon Axler, so I'd probably start there. - Graph Theory and Combinatorics. These are I think are somewhat more accessible than the others (perhaps at least more intuitive), so I might actually recommend trying these first. For the basics, try A Walk Through Combinatorics by Miklos Bona. By the way, whenever I need to find a textbook on a subject, I just Google "best (subject) textbook", and try to find the Math Stack Exchange post where someone has asked this question. (Eg. here's [0] the one for graph theory, which is where I got the combinatorics book.) Now, this post is already getting long enough, so I'll post this for now and follow up the rest in another comment. [0] https://math.stackexchange.com/questions/27480/what-are-good... |