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Background: I am currently studying physics and except for high school mathematics I have experience with linear algebra, single- and multivariable calculus, differential equations (mostly ODE's, but I have some limited experience with PDE's), and some transform theory. I also know some basic set theory, graph theory and really simple enumerative combinatorics. I am interested in higher mathematics such as abstract algebra, the more rigorous (than calculus) analysis and topology, according to what I know about them.

I saw a recommendation on Hacker News, for three books about exactly these areas of mathematics and according to the post, the books are "hardcore" in the sense that they are hard to go through, but provide many rewards for doing it. This is exactly what I want. What do you think, are the books to be recommended?

Also, which area would you recommend that I start with?

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  • $\begingroup$ I'm also a physicist (student), and Munkres' Topology is the book I used to learn about it. I don't know the other two, but that one is, in my opinion, great. Very understandable, and everything is greatly explained. I really recommend it. $\endgroup$ Jun 14 '14 at 21:49
  • $\begingroup$ What did you as a physics student study first of the three areas I listed? (if you have studied all three, of course) $\endgroup$
    – user157111
    Jun 14 '14 at 21:49
  • $\begingroup$ I first learnt group theory (only finite groups), then topology, and then more group theory (more about representation theory and continous groups). As for analysis, I've never learnt more than what a regular physicist studies, and I notice a lack of knowledge I would like to get rid off :) $\endgroup$ Jun 14 '14 at 21:53
  • $\begingroup$ Similar questions have been asked on this site several times, try looking them up. I think the recommendation you got on Hacker News is a good starting point. $\endgroup$ Jun 14 '14 at 22:04
  • $\begingroup$ I bought spivak's Calculus for analysis and found it useful. Also I like Apostol's books on calculus and his book 'Mathematical Analysis' which I find really good. I like Gertrude Ehrlich's 'Fundamental Concepts of Abstract Algebra'. And then you've also Dummit and Foote's book. Kuratowski has an alright topology book, from the bits I've read, and also Dugundji. The order I came across them was 1. Real Analysis 2. Groups 3. Topology. As pandabear says, many references are already on this site, although they'll prob all be done separately, not too much hassle. $\endgroup$
    – snulty
    Jun 14 '14 at 22:31
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Read The Princeton Companion to Mathematics and then, in a few years, read it again. If you want more recommendations I have a list of recommendations for approximately your background level on my blog here.

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