Self-study mathematics subject sequence and recommended books 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!
 A: 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)
A: 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).
