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I am a Physics student but I finally found that I've entered the wrong department that I am in fact much more interested in mathematics. I want to self-learn mathematics.

I am now reading Artin (Algebra) and Rudin (Prinicple of Mathematically Analysis). Both books are terrific.

Could anyone suggest a study sequence of subjects after that and some classic textbooks in each subject?

I am more interested in pure (I don't mind they being abstract) mathematics (especially those can be applied in quantum information theory, QFT, GR, quantum gravity, String theory, etc.).

Thanks in advance!

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    $\begingroup$ @user109256 No, that post doesn't help me a lot. Please please don't delete my post here. $\endgroup$ – velut luna May 11 '16 at 16:44
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    $\begingroup$ Duplicates will be removed, aren't they? $\endgroup$ – velut luna May 11 '16 at 16:48
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    $\begingroup$ @user109256 OK. Thanks! $\endgroup$ – velut luna May 11 '16 at 16:59
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    $\begingroup$ I am personally of the opinion that anyone who calls Rudin "terrific" is either a masochist or overly charitable, but all the power to you! You might want to look into Munkres' Topology as a next step. $\endgroup$ – Omnomnomnom May 11 '16 at 17:18
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    $\begingroup$ If you're particularly interested in the things that tie into quantum information theory, you should also look into functional analysis! I recommend either Kreyszig (because it's readable) or Pedersen's Analysis Now (because it gets to the point). $\endgroup$ – Omnomnomnom May 11 '16 at 17:20
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Here's a list of possible topics to look into after surviving Rudin:

  • Topology (Munkres is one of the canonical undergrad texts here). In a nutshell: "what can we say about closeness without a direct notion of distance (i.e. a metric)? What can we say about 'continuous functions'?"

  • Functional analysis, to be taken after some topology (Kreyszig and Pedersen are my go-tos here). This topic is key to understanding quantum information theory. In a nutshell: linear algebra, but on infinite-dimensional vector-spaces. Note: infinity is weird.

  • Algebraic geometry (reference 1, reference 2)

  • Representation theory

  • Lie Groups/Lie Algebras, together with some differential geometry.

(See also my comments above)

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See the suggestions given for this mathoverflow question.

Also, for functional analysis, consider volume 1 of the 4-volume series Methods of Modern Mathematical Physics by Reed/Simon, and keep Kreyszig's book handy for when you don't understand something -- note that Kreyszig's book has applications to quantum mechanics at the end.

Finally, for an nice overview of most areas of modern mathematics by a physicist, see Paul Roman's 2-volume Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis (Volume 1 here with views to table of contents of both volumes, an amazon review of both volumes here).

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