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I am a second year math major, and I intend to do my PhD from a top math school. Until now, I haven't taken any university level physics course. Somewhere in this site, I saw someone writing that courses related to theoretical physics (heavily loaded with mathematics) are also very important besides the regular math courses. So, my question is: How important are the physics courses? Or any other courses?

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Math grad programs do not look at physics courses on the transcript, or think in those terms.

The point is that higher-level (not formulaic) physics courses can be beneficial to math people by providing other inputs for intuition. A common obstacle is that the higher-level physics courses do speak in terms of the lower-level ones, which are often quite alien to/from any sensible mathematical world-view.

But if one skips over those "immediate" things, one can find that there are "physcial imperatives" mandating mathematical "facts"... which might not be obvious on "purely mathematical" grounds.

The grandest example is "Green's functions" ... about which volumes can be written... An immediate point is that the idea is wonderful, is necessary, even if one cannot justify it. Green got the idea pre-1850, and it was completely understood in "rigorous" terms by L. Schwartz in 1950. Not easy, ...

That is, understanding other (very serious) inputs to mathematics is obviously helpful.

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    They might if you say you're planning to do research in one of a very few specific fields where physics is relevant. But in general, they don't. – Peter Shor Oct 7 '13 at 19:54
  • Indeed, as @PeterShor notes, if a student's proposed specialization explicitly claims to bear on specific issues in physics... well, ignorance of physics wouldn't be sensible. – paul garrett Oct 7 '13 at 20:02
  • Indeed, one of my tutors accidentally became involved in theoretical physics once, or at least his name did. He's a topologist/geometer and he constructed the first examples of manifolds with certain properties that it turned out some string theorists had been looking for. I don't think his prior level of exposure to physics really mattered, he said the fact his work had application came as a surprise. Had he intended to do something significant in physics then knowing some physics might have made that more likely! – Steve Jessop Dec 18 '14 at 10:37
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A possible benefit of non-math course(s) in the resume of a PhD applicant in math, is the fact that it shows that he has a wide research perspective and is eager to study different areas. That is definitely a huge plus for a grad student as one of the major sources of creativity, is bringing in ideas from areas that are sometimes totally irrelevant to the area under study. There are a lot of instances of innovations in for example Agile software engineering that came from manufacturing.

Though, as others mentioned, most universities should not care much. But if you target top universities, then you must know that they do receive a lot of good applicants. And this might be something that make your application stand out!

just my 2 cents..

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Make sure to have a decent overall GPA. Personally, even though physics uses a lot of mathematics, I would recommend that you take a more basic conceptual physics course, to understand the physics itself and learn it for its own sake. That way you will be better to appreciate the physics. If you are really passionate about math, then take math-related electives. That would show your commitment to the major. Maybe even do math-related research if that is your bent. Treat the physics courses as important, but do not worry if you are not doing as well in them as in your math courses. Because you are a math major, they will pay more attention to your major courses, but don't let yourself be discouraged.

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In addition to "math" courses, a good math major should take courses in "related" areas; i.e. subjects that either use a lot of math, or contribute a lot of applications to math. Two of those subjects are physics and computer science.

Someone studying advanced calculus will do well to learn physics concepts such as gravity, charge and flux, as used in say, Newton's or Maxwell's equations. These offer the basis of gradients, divergences Gauss and Stoke's Theorems, and others. Likewise, a good computer science course might use mathematical topics such as recursion, graph theory, or various forms of logic. You might also consider Economics (specifically econometrics) courses that cover optimization and systems of equations as well as more advanced applications using partial differential equations.

Just avoid the kinds of courses sometimes referred to as "physics for poets" (algebraic applications only), or "programming for data processors" (elementary programming devoid of advanced mathematical concepts).

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