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I am currently studying mathematics in the 5th semester and at my university, we are offered something called a "reading course", where a student will have to read a/several book/s about a topic he is interested in and afterwards he will be examined by a professor. As a lot of subjects of lectures at my university are not very interesting for me, I'd really like to do such a reading course.

However, I don't know a lot about the things that I have not yet been officially taught. So far, I've attended and passed the following courses:

  • Calculus 1 & 2
  • Linear Algebra 1 & 2
  • Geometry
  • Computer Science
  • Numerical Mathematics 1 & 2
  • Physics 1 & 2 & 3
  • Complex Analysis
  • Methods of Mathematical Physics 1
  • Algorithms and Complexity
  • Measure Theory
  • Algebra 1 & 2
  • Topology
  • Probability and Statistics

In Physics 3 and Methods of Mathematical Physics 1, we have been introduced in quantum mechanics, which I loved. I'd really like to learn more about quantum mechanics, which is why I'm currently attending a lecture in Functional Analysis 1 (& 2 next semester) and General Mechanics. I am also attending a seminar about "Renormalization" (as far as I understood, it is a mathematical "tool" which has broad applications in QFT and QSM) and I intend to attend a lecture called "Theoretical Physics" next semester, a lecture for mathematicians about elecrodynamics, quantum mechanics and general relativity.

Question: So far, all I've seen about "quantum" was a shallow introduction about quantum mechanics. I'd really like to learn more about quantum mechanics, quantum field theory and also quantum information. Are there any other "quantum" subjects I might be interested in? Can you recommend any books about topics in these subjects which will not exceed my previous knowledge by too much?

Thank you very much in advance for any help.

EDIT: I almost forgot: My favourite subject so far was Complex Analysis. I don't know if there is any "quantum" field which makes use of Complex Analysis but it would be awesome to learn about that if there is anything.

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It sounds like you'd be more inclined to study the mathematical aspects of quantum mechanics rather than its physical applications. Check out this link: mathoverflow.net/questions/2917/… –  Alex R. Sep 24 '13 at 19:59
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