What mathematics do physics phd students learn?

[playoff]

Hello again PF! I think I ask excessively often here so I tried my best to search in google. The result I got was this: http://superstringtheory.com/math/math1.html

Well, that appears as all the math there is (no offense to mathematicians)! Likely the field of super string theory is the most math-intensive, but I would like to know what math courses a phd student would take in various fields of physics (particularly particle/nuclear physics, but I would also like to know others' as well). Would they compare to the maths an undergraduate math major learns?

And few other questions regarding the same topic:
1.) Although likely an extreme example, the amount of math needed as a physicist stated in the aforementioned website was astonishing. Of what classes is a phd student's semester typically consisted of?
2.) If a student in his undergraduate years took/is educated in a math course one would take as a phd student, do you relearn it during your phd program?

Thank you in advance and happy Tuesday

 

[Tosh5457]

That list seems ok for an undergraduate, but looking forward you may need to learn more mathematics depending on the field you choose. I'd add differential geometry, topology, functional analysis and abstract algebra (group theory and Lie algebra). That's what Roger Penrose puts in his book as requirements to learn theoretical physics. Particularly, learning differential manifolds will change your physical perspective in many ways, while giving you the background for general relativity.

 

[playoff]

@Tosh5457: I think you forgot the next two pages which I was talking about. The list not only includes differential geometry, real and complex analysis, but also homology, fiber bundles, index theorems and bunch of other topics that seems rather overwhelming to even read their descriptions. Abstract algebra and topology are actually not in the list. May I ask their applications in physics?

 

[micromass]

Abstract algebra and topology are actually not in the list. May I ask their applications in physics?

They are in the list, just not under that name. Abstract algebra is in the list under the guise of group theory. Although the group theory you need as a physicist is very different from what mathematicians see as abstract algebra.
Topology is in the list under multiple items. For example, if you want to learn manifolds, homotopy, homology, then you need to know topology to some extent.

 

[WannabeNewton]

@Tosh5457: I think you forgot the next two pages which I was talking about. The list not only includes differential geometry, real and complex analysis, but also homology, fiber bundles, index theorems and bunch of other topics that seems rather overwhelming to even read their descriptions. Abstract algebra and topology are actually not in the list. May I ask their applications in physics?

If you want to do actual physics then focus on the physics. Youll pick up the math along the way. Time has to be spent wisely. Without stating your field of interest in physics one cannot make a remark as to how much math you would actually need to learn as theoretical physics can mean a lot of different things. The link you posted certainly does not apply to all areas of theoretical physics and one would be wasting their time trying to learn all that for any and all such areas.

 

[playoff]

@WannabeNewton: Thank you for the reply, and as you said I will focus on physics. However as I'm still curious some of my questions have not been answered. If it helps, my primary field of interest is theoretical particle/nuclear physics. But I would like to know a general list of courses a phd student would take in a semester. For example, a phd student in his very first semester may take mechanics/electromagnetism, quantum field theory, abstract algebra, and spend other hours in research or teaching (then again this is a cluelessly hypothetical case).

 

[Matterwave]

@WannabeNewton: Thank you for the reply, and as you said I will focus on physics. However as I'm still curious some of my questions have not been answered. If it helps, my primary field of interest is theoretical particle/nuclear physics. But I would like to know a general list of courses a phd student would take in a semester. For example, a phd student in his very first semester may take mechanics/electromagnetism, quantum field theory, abstract algebra, and spend other hours in research or teaching (then again this is a cluelessly hypothetical case).

A physics grad student would take whatever math classes he wished to take. But generally speaking, the only required one is probably a mathematical methods for physicists course that covers most of the basic necessary subjects of mathematics.

For example, in my math methods course, we covered complex analysis (including complex integration, the Cauchy formula, method of residues, method of steepest descent, etc.), special functions and their generators (E.g. Hermite polynomials), Sturm-Liouville type differential equations, and Green's functions. More complex math can always be taken if the student so requires. And often a course that requires more advanced math will start off by teaching a little math first.

 

[AlephZero]

1.) Although likely an extreme example, the amount of math needed as a physicist stated in the aforementioned website was astonishing.

Really? If I was astonished by that list, it would be because it was so little, not so much.

There is nothing "advanced" on that web page. I work as an engineer, developing computational methods, and I use everything on that page (and a lot more) all the time.

The answer to the question "what math does a PhD student need to learn?" is "whatever it takes to do the research". That's the basic difference between "research" and "solving the problems in a textbook".

 

[WannabeNewton]

Really? If I was astonished by that list, it would be because it was so little, not so much.

There is nothing "advanced" on that web page. I work as an engineer, developing computational methods, and I use everything on that page (and a lot more) all the time.

You might want to take a look at the next page of the list.

 

[AlephZero]

You might want to take a look at the next page of the list.

Fair comment. I didn't check that the OP had linked to the wrong page!

 

[Hercuflea]

My e&m professor is a nuclear physicist and he said he had never taken real or functional analysis.

 

[Dr Transport]

Other than a course in wavelet theory, my only math course during my graduate career was a two semester mathematical physics course out of Courant and Hilbert along with my teachers group theory notes (point and Lie groups)

 

[playoff]

@Matterwave: Ok I'm starting to get a clear picture of how it works. Very useful information, thanks!

@AlephZero: Well, it necessarily isn't the wrong page. But I owe you apologies for the confusion.

@Hercuflea: Wow, I suppose nuclear physics isn't as math intensive as I thought. Thank you for the info!

@Dr Transport: Then I suppose the need of math varies vastly upon different fields. May I ask what was your specialization?

I'm starting to come to a conclusion that sophisticated math isn't required but useful if you know it. By sophisticated I mean anything above perhaps partial/ordinary differential equations and linear algebra. Any comment to this?

 

[Jorriss]

I'm starting to come to a conclusion that sophisticated math isn't required but useful if you know it. By sophisticated I mean anything above perhaps partial/ordinary differential equations and linear algebra. Any comment to this?

It just depends on what you want to do, and even saying you need PDE's is ambiguous. You need to know separation of variables and a few other techniques, sure, but do you need to know the cauchy-kovalevskaya theorem? Most certainly don't.

The bare minimum of what a PhD student needs to learn is likely to be covered in the departments mathematical methods course (or EM/mechanics if methods are taught there). Otherwise, it is too highly dependent on specialization to comment in meaningful detail.

Crudely though... PDEs, ODEs, diff geo, group theory, asymptotic analysis but probably not anything that wouldn't be found in a math methods, GR or quantum text.

 

[PhysicsGente]

@Matterwave: Ok I'm starting to get a clear picture of how it works. Very useful information, thanks!

@AlephZero: Well, it necessarily isn't the wrong page. But I owe you apologies for the confusion.

@Hercuflea: Wow, I suppose nuclear physics isn't as math intensive as I thought. Thank you for the info!

@Dr Transport: Then I suppose the need of math varies vastly upon different fields. May I ask what was your specialization?

I'm starting to come to a conclusion that sophisticated math isn't required but useful if you know it. By sophisticated I mean anything above perhaps partial/ordinary differential equations and linear algebra. Any comment to this?

That depends what you want to do. If you want to do experimental nuclear/particle physics, you will have to take QFT and particle physics classes, but that's it. It really won't matter if you understand the math behind them. But if you want to do theory, you will have to learn a lot of math to even master QFT, i.e, algebra, topology, etc. From personal experience, the amount of math you would need to understand those topics deeply would take you a couple of years (or maybe even more) to learn at the graduate level.

 

[WannabeNewton]

@Matterwave: Ok I'm starting to get a clear picture of how it works. Very useful information, thanks!

@AlephZero: Well, it necessarily isn't the wrong page. But I owe you apologies for the confusion.

@Hercuflea: Wow, I suppose nuclear physics isn't as math intensive as I thought. Thank you for the info!

@Dr Transport: Then I suppose the need of math varies vastly upon different fields. May I ask what was your specialization?

I'm starting to come to a conclusion that sophisticated math isn't required but useful if you know it. By sophisticated I mean anything above perhaps partial/ordinary differential equations and linear algebra. Any comment to this?

What makes you think nuclear physics isn't math intensive? Just because it doesn't make use of marginalized esoteric math doesn't mean it isn't math intensive. You have a very wrong idea about how physics research or even physics grad school works. Seriously very few people have the time or motivation to learn all of that math. Not to mention unkess youre doing string theory or related subjects like topological field theory that math will be absolutely useless.

The above poster is also mistaken. You certainly do not need algebra or topology to master qft and knowing these subjects won't help your understanding of qft at all beyond the basics of lie groups if what you care about is physics as opposed to mathematical physics. And if youre like me youll find math outside of a physics context very dry which makes the job of learning all that math even more unmotivated. There is a big difference between physics and mathematical physics. I can assure you that my past EM professor, who works in particle phenomenology, makes little use of pure math in his work and for good reason as it is often a hindrance.

 

[playoff]

@WannabeNewton: Sorry if it sounded offensive in any way. I should have put the emphasis on "as," as I obviously had the wrong idea before that every physics discipline needed all the math there exists. I understand that physics and math are although to an extent related, very different matter. Although I may take a math course or two, my focus will always be physics.

Thank you everyone for the reply!

 

[WannabeNewton]

@WannabeNewton: Sorry if it sounded offensive in any way.

I didn't take any offense, sorry if it came off that way.

 

[PhysicsGente]

The above poster is also mistaken. You certainly do not need algebra or topology to master qft and knowing these subjects won't help your understanding of qft at all beyond the basics of lie groups if what you care about is physics as opposed to mathematical physics. And if youre like me youll find math outside of a physics context very dry which makes the job of learning all that math even more unmotivated. There is a big difference between physics and mathematical physics. I can assure you that my past EM professor, who works in particle phenomenology, makes little use of pure math in his work and for good reason as it is often a hindrance.

There is a difference between theory and phenomenology. I stand by my argument that if you want to master QFT you MUST know topology and algebra beyond the usual group theory stuff, otherwise, you'd just be taking many things for granted, unlike quantum mechanics for example. To do the physics, you have to implement the math one way or another, and if you do not know the math behind QFT, then there is no way you are going to be able to do that.

 

[Dr Transport]

@Matterwave: Ok I'm starting to get a clear picture of how it works. Very useful information, thanks!

@AlephZero: Well, it necessarily isn't the wrong page. But I owe you apologies for the confusion.

@Hercuflea: Wow, I suppose nuclear physics isn't as math intensive as I thought. Thank you for the info!

@Dr Transport: Then I suppose the need of math varies vastly upon different fields. May I ask what was your specialization?

I'm starting to come to a conclusion that sophisticated math isn't required but useful if you know it. By sophisticated I mean anything above perhaps partial/ordinary differential equations and linear algebra. Any comment to this?

Semiconductor Theory, specifically Boltzmann Transport Theory and $\vec{k} * \vec{p}$ theory

 

[Jheavner724]

This is entirely dependent of your field, but that website seems to exclude a decent bit of the necessary mathematics if you want to do string theory research (it may be good for an undergraduate though, just definitely not active research, and most likely not for a Masters or PhD student either). Of course, concepts and qualitative aspects are still important in physics, but, as has been said by many a physicist, mathematics is the language of physics — especially at the higher levels of theoretical physics.

If you're a graduate student in a highly experimental area, then you might get by with just calculus (single and multivariable — usually called Calculus I-III), differential equations (ODEs and PDEs), a decent course in probability and statistics (e.g. Stat110 at Harvard), linear algebra, and one or two others for your speciality specifically.

In more theoretical physics, more mathematics is common, especially in quantum gravity theories (i.e. string theory variants and the like). It's quite difficult still to give a list of necessary mathematics because there are so many different types and areas of research in string theory too. Also, I don't think it's all that helpful to just list, considering I will likely include too much or too little. An added layer of complexity comes from the mathematical physics involved. Things like Grassmann intergration are unlikely to be found in a mathematics class but are common place on physics.

There are, however, some mathematical topics you can almost definitely rule out (e.g. more advanced combinatorics) unless you're in a peculiar field (unlike string theory). Anyway, here's what I recommend: go to a good physics university's website (MIT may be best due to OCW) and check prerequisites to courses as well as program requirements and the sort.

You can also try to clarify some specifics and I can reevaluate.

Oh, and I think you may be confused about graduate chill. Grad students don't take that many classes and most of the one's they do are more like seminars (in fact many are named as such). The big thing for PhD candidates is their thesis or dissertation (at least in most countries I know of such as the UK and US). So, you're doing a lot of research and reading papers. This is also where you learn enough. Sure, you can watch lectures and read textbooks (there are especially some good graduate texts), but you gain most of your graduate student knowledge from keeping up with current events and reading papers in your field.