Self-study mathematics subject sequence and recommended books

Question

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!

Answer 1

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)

Answer 2

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).