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I'll be precise: how much mathematics (in terms of courses in a university mathematics department or mathematics textbooks) can a very good theoretical physicist be expected to know? If this depends on the field, please specify. I am particularly interested in those working in areas of physics amenable to geometrical methods, e.g., general relativity, gauge theory, and quantum gravity.

I've seen similar questions but no concrete answers from physicists.

Edit: By "mathematics" I mean what physicists probably call "pure mathematics". Of course any physicist can solve all kinds of ordinary and partial differential equations, is competent in vector calculus and complex variables, etc. Perhaps other ways to phrase it are "rigorous mathematics" or "mathematics that isn't in Arfken".

Another Edit: To be more specific, how much differential geometry do physicists working in general relativity, gauge theory, etc. know?

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closed as too broad by ACuriousMind, DanielSank, user81619, Kyle Kanos, jinawee Oct 1 '15 at 21:28

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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It can vary widely. In the US, undergraduate physicists are expected to take math classes including:

  • Calculus up to Multivariable Calculus
  • Differential Equations
  • Linear Algebra

They also often take a class in mathematical physics that covers things like advanced integrals, contour integrals, gamma functions, special functions (Bessels, spherical harmonics, etc.).

As a graduate student (I'm a graduate student in Physics, but an experimentalist), you aren't required to take any additional pure mathematical classes (at least not in my department) other than maybe another mathematical methods course. However, you will be expected to learn some more differential equations, potentially some group theory, tensor calculus, basic differential geometry, and things like that.

Of course, this depends heavily on your specialty. I'm in the field of fundamental particles, so these are useful to me. Basically, you pick up what you need for your discipline.

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