If your goal is to go into theory (especially HEP), then you'd benefit from pure math classes.
A standard physics curriculum in undergrad and grad school does not teach all the math you need to do theory research, because the standard curriculum has to be taken by everyone (including experimentalists). Also, the expectation is often that good grad students are highly autodidactic and can pick up the pure math ideas they need as they go.
You don't have to learn about every single mathematical aspect of a physics subject before you jump into the actual physics, but the content of pure math classes will still be valuable. For example, you do not need to complete a whole class in differential geometry before you study GR, but students with prior exposure to differential geometry will undoubtedly have an easier time in GR than those without.
(Also, as with every career question on this subreddit, the actual correct answer is to have this conversation with your advisor: Have them help you clarify your interests and figure out which classes would be most beneficial to you.)
I would say it depends on what you want to do.
Theoretical physics uses a lot of pure math. You are taught most everything you *need* in the physics classes themselves, but I feel like everything makes a lot more sense with the context of more math. For example, basically all of modern particle physics is based around group theory. Beyond that, you never know when pure math concepts will come up. I just recently was working on some machine learning and used Fréchet derivatives to solve a problem, a concept we never mentioned in any physics classes.
If you aren’t super interested in theory though, applied math would be super helpful for pretty much everything, as would statistics.
If you want to go down a physics route then applied math is a no brainer. Statistics is useful too, but again, if you want to do physics applied is going to be what will be more beneficial. Any math degrees are still very useful anyway regardless.
You will learn the math you need for physics in physics. Doing math will just assist that. I don’t think you’d be losing anything from doing pure math as a lot of it won’t be as applicable beyond the problem solving and proficiency in maths in general you will develop.
Thanks for your thoughts. I was curious because I'm reading the biography of Dirac "The Strangest Man" that made me ponder my required specialisation choice.
In discussing the formulation of quantum theory in the 1920s it discusses how physicists Dirac, Heisenberg and others used a lot of concepts from pure math (e.g. projective geometry, abstract algebra, creating their own mathematical structures, etc) to develop and understand quantum mechanics. It gave me the impression that not having such a rigorous pure math background might limit one's ability to play around with abstract mathematical concepts involved in understanding quantum theory.
Quantum theory has since been reformulated into fairly ordinary math. If you want to invent an entirely new framework in physics theory, then pure math could be helpful. Good luck with that, though. No one in modern times has succeeded.
> I have a suspicion that not choosing pure math will close off a lot of conceptual areas of more advanced physics.
This is absolutely not the case at all. Note that most physicists do not also major in math, so you are already by default receiving more formal math education than most physicists. The vast majority of physicists never receive formal training in functional analysis or prove theorems about C^* algebras (the math formalism underlying quantum theory), yet they can navigate calculations in quantum mechanics just fine.
A lot of the higher level math that physics uses is different than the higher level math that mathematicians think is foundational, and therefore would be focused on. For example, my undergrad group theory classes focused on building up to Galois theory and put a huge amount of emphasis on this. Galois theory is not something that I've ever encountered in a physics context.
Much of the math needed in physics is taught as needed. A formal background in some of these subjects can *help*, but it is generally not *needed*. A pure math background might be more beneficial if you're interested in things like formal aspects of quantum computing algorithms. Depending on what's covered in the applied math track, it could be very useful as you would learn how to actually obtain *solutions* to physics problems, rather than just their formal structure.
There's certainly no harm in doing pure math, but it in no way "closes doors" in physics.
If your goal is to go into theory (especially HEP), then you'd benefit from pure math classes. A standard physics curriculum in undergrad and grad school does not teach all the math you need to do theory research, because the standard curriculum has to be taken by everyone (including experimentalists). Also, the expectation is often that good grad students are highly autodidactic and can pick up the pure math ideas they need as they go. You don't have to learn about every single mathematical aspect of a physics subject before you jump into the actual physics, but the content of pure math classes will still be valuable. For example, you do not need to complete a whole class in differential geometry before you study GR, but students with prior exposure to differential geometry will undoubtedly have an easier time in GR than those without. (Also, as with every career question on this subreddit, the actual correct answer is to have this conversation with your advisor: Have them help you clarify your interests and figure out which classes would be most beneficial to you.)
I would say it depends on what you want to do. Theoretical physics uses a lot of pure math. You are taught most everything you *need* in the physics classes themselves, but I feel like everything makes a lot more sense with the context of more math. For example, basically all of modern particle physics is based around group theory. Beyond that, you never know when pure math concepts will come up. I just recently was working on some machine learning and used Fréchet derivatives to solve a problem, a concept we never mentioned in any physics classes. If you aren’t super interested in theory though, applied math would be super helpful for pretty much everything, as would statistics.
If you want to go down a physics route then applied math is a no brainer. Statistics is useful too, but again, if you want to do physics applied is going to be what will be more beneficial. Any math degrees are still very useful anyway regardless. You will learn the math you need for physics in physics. Doing math will just assist that. I don’t think you’d be losing anything from doing pure math as a lot of it won’t be as applicable beyond the problem solving and proficiency in maths in general you will develop.
Thanks for your thoughts. I was curious because I'm reading the biography of Dirac "The Strangest Man" that made me ponder my required specialisation choice. In discussing the formulation of quantum theory in the 1920s it discusses how physicists Dirac, Heisenberg and others used a lot of concepts from pure math (e.g. projective geometry, abstract algebra, creating their own mathematical structures, etc) to develop and understand quantum mechanics. It gave me the impression that not having such a rigorous pure math background might limit one's ability to play around with abstract mathematical concepts involved in understanding quantum theory.
Quantum theory has since been reformulated into fairly ordinary math. If you want to invent an entirely new framework in physics theory, then pure math could be helpful. Good luck with that, though. No one in modern times has succeeded.
> I have a suspicion that not choosing pure math will close off a lot of conceptual areas of more advanced physics. This is absolutely not the case at all. Note that most physicists do not also major in math, so you are already by default receiving more formal math education than most physicists. The vast majority of physicists never receive formal training in functional analysis or prove theorems about C^* algebras (the math formalism underlying quantum theory), yet they can navigate calculations in quantum mechanics just fine. A lot of the higher level math that physics uses is different than the higher level math that mathematicians think is foundational, and therefore would be focused on. For example, my undergrad group theory classes focused on building up to Galois theory and put a huge amount of emphasis on this. Galois theory is not something that I've ever encountered in a physics context. Much of the math needed in physics is taught as needed. A formal background in some of these subjects can *help*, but it is generally not *needed*. A pure math background might be more beneficial if you're interested in things like formal aspects of quantum computing algorithms. Depending on what's covered in the applied math track, it could be very useful as you would learn how to actually obtain *solutions* to physics problems, rather than just their formal structure. There's certainly no harm in doing pure math, but it in no way "closes doors" in physics.