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I heard, however, that the grad level physics courses here uses a lot of math. So, in order to prepare for those courses, which of the following math subjects should I strongly consider taking? (among those grad level courses, I'm considering GR, advanced E&M, QFT, and Atomic/optical physics, solid state physics).

1. Analysis (proof-based course, uses Rudin)

2. Functional analysis (requires analysis)

3. Topology (requires analysis)

4. Complex variables (fairly basic course) , different from complex analysis (which is more advanced)

5. Abstract Algebra (requires linear Algebra)

6. Advanced calculus (more like calculus techniques&PDE's for engineers type of class)

A. Would you be able to rank by the importance of the following subjects, as it is relevant to my physics studies?

B. Would you advise me to entrench myself deeply into the math, even though I'm not a math major? There's an alternative to taking 3-4 extra math classes, which is to pick up a book like Mathematical methods for Quantum and Classical Physics by Byron, and learn it over the summer. I started it , and I like the style. It is mathematically rigorous, but still very relevant to my physics studies.

Thank you.