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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!

velut luna
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  • I think this post may help you http://math.stackexchange.com/q/1447558/317580 and if you really find that post helpful then kindly flag it to remove it from hold. Also let me know if it helped you by leaving a comment here. – Heisenberg May 11 '16 at 16:35
  • It is often advised to read several books simultaneously, as they can be complementary (as lectures are). I do not particularly know the english literature, but in general, if you are familiar with mathematical proofs (I mean, the way mathematics is done at university), I'd recommend to learn analysis jointly with topology and linear algebra. It is good to learn some statistics and probability but not too deeply since they require measure theory (taught in advanced analysis, in general). These are a basis, of course; there are plenty of subjects. – MoebiusCorzer May 11 '16 at 16:39
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    @user109256 No, that post doesn't help me a lot. Please please don't delete my post here. – velut luna May 11 '16 at 16:44
  • @Mathaholic What made you think that I'll delete you post? – Heisenberg May 11 '16 at 16:46
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    Duplicates will be removed, aren't they? – velut luna May 11 '16 at 16:48
  • @Mathaholic I don't think it's a duplicate but I just redirected you to that post because I thought that may be helpful to you. – Heisenberg May 11 '16 at 16:50
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    @user109256 OK. Thanks! – velut luna May 11 '16 at 16:59
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    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. – Ben Grossmann May 11 '16 at 17:18
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    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). – Ben Grossmann May 11 '16 at 17:20
  • @Omnomnomnom +1 for the recommendation. I have followed both of them on similar lines. – Heisenberg May 11 '16 at 17:21
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    If you're more of an algebraist than an analyst (that is, if you prefer homomorphisms to epsilons and deltas), you might also want to consider looking into algebraic geometry. An Invitation to Algebraic Geometry is good for a first look, Hartshorne is good if you're looking for a real project. – Ben Grossmann May 11 '16 at 17:24
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    @Omnomnomnom Thank you for the recommendations! – velut luna May 11 '16 at 17:26
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    Finally, if you feel like you want a good mix of linear algebra and abstract algebra, you should consider looking into either "representation theory" or "Lie Groups and Lie Algebras". I don't think I have helpful recommendations for references on those subjects, though. – Ben Grossmann May 11 '16 at 17:27
  • You're welcome! Hope you enjoy it, whichever way you go. – Ben Grossmann May 11 '16 at 17:28
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    @Omnomnomnom I request you to convert those comments as an answer providing a few more similar details. It'll be helpful to other users seeking answers to similar questions. – Heisenberg May 11 '16 at 17:32
  • @Mathaholic May you please briefly explain why Artin (Algebra) and Rudin (Prinicple of Mathematically Analysis) are terrific? I once considered to use those books for self-learning purposes. – Mc Cheng May 16 '16 at 11:41
  • @McCheng Very systematic, proofs are very rigorous and yet detailed and not difficult to follow, plenty of examples etc. I have no much background in rigorous mathematics (I am a physics student and you know the way physicists apply mathematics) can follow with no much difficulties. – velut luna May 16 '16 at 11:57
  • Does this answer your question? How do I self-learn undergraduate math? – user1147844 May 28 '23 at 00:25

2 Answers2

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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)

Ben Grossmann
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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).

Community
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Dave L. Renfro
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