What Math Courses Should a Physics Major Take?

[stormyweathers]

Every physics major knows he needs to take a bunch of math courses. But there are so many offered at my university its making my head spin! I've taken (aside from lower division linear algebra/calculus/DEQ) real analysis and abstract algebra so far, and I've tentatively decided to aim for differentiable manifolds and dynamical systems as well. I know that I'll be able to take more than those two, so I'm looking for some advice.

Whenever I ask this question people answer with "Well, what kind of physics do you want to study?", and I don't have an answer to that yet. It would be pretty useful to know which areas of math are most useful for some common/major areas of physics, both theoretical and experimental.

 

[micromass]

If your aim is to do differentiable manifolds, then a topology courses will be very useful. If they offer calc on manifolds, then that's good too.

For dynamical systems, you might want to take a class on PDE's.

A proof-based linear algebra course wouldn't hurt as well.

 

[nonequilibrium]

Will you be taking a math methods class down the road that covers topics like complex analysis? If not, a course in complex analysis in the math dept. sounds like a good idea too! Complex analysis isn't specifically tied to any branch of physics, but it is commonly used in modern branches, e.g. path integral approach to quantum (field) theory, or if you would consider doing string theory down the road, complex analysis is simply some mathematical baggage that is seen as elementary. Besides, for its own sake complex analysis is simply beautiful. Also it will give you a much better understanding of, say, power series, such that other mathematical physics-related branches will become accessible, e.g. asymptotics, or making sense of divergent sums that turn up in physics. BTW complex analysis is related to the "zeta function" (which itself is related to the infamous Riemann hypothesis, one of the greatest problems in math) which pops up surprisingly much in physics (for being an object from number theory), i.e. when analysing Bose-Einstein condensation or quantizing the string in string theory!

 

[ahsanxr]

For experimentalists and computational physicists, I would say you could be doing things that would be more useful than taking pure math courses such as improving your lab skills through advanced lab courses and taking more scientific computing courses, and doing more research.

For theory however, regardless of whether you want to go into particle physics, condensed matter or mathematical physics, you're going to have to know a ton of math.

Classes that'll be useful regardless of the theory subfield:

I second micromass' suggestions on the first 3 classes:

- Point-set topology. This stuff isn't directly useful, but a lot of math is based on concepts from here such as algebraic topology and differentiable manifolds.
- Calculus on manifolds (usually offered as "Analysis II"). So I second micromass' suggestions on the first two classes.
- Proof-based linear algebra. This depends on how good the class you took was. A lot of the courses that are just concerned with crunching matrices don't really give you an understanding of the stuff you should take from a class on LA. If you're confident in your understanding of general vector spaces, linear functionals, transformations, diagonalization, inner products then you don't need this. If not, certainly take it.

- Complex Analysis.
- Representation theory and Lie algebras/groups.
- "Math Methods". While this kind of class usually isn't rigorous, it's still a good way of getting familiar with doing computations with green's functions, PDEs and what not which are skills every theorist should master.

Mathematical Physics:

Functional analysis is the bread and butter of mathematical physics, from what I can tell at least, since everything in rigorous quantum mechanics and quantum field theory is based on that.

Hardcore particle theory (strings, loops etc):

Everything: All of the above (maybe not functional analysis) along with Algebraic and Differential Topology, Algebraic and Differential Geometry and a whole bunch of other topics.

Another thing I found useful to get some direction as far as what math courses to take, is to browse through some advanced mathematical physics texts (Hassani, Stone, Nakahara etc) and see what they cover.

Disclaimer: I'm an undergrad who hasn't studied a lot of these topics himself, but this is the general impression I've gotten from browsing through books, papers and talking to professors.