What additional math course should I take?

[Shackleford]

I would choose between partial differential equations or differential geometry. I would keep away from real anay. (just because there isn't much use in physics)

Would just one semester be all right even though they have two semesters of PDEs?

 

[Nolen Ryba]

My suggestion is MATH 4355: Mathematics of Signal Representation. Why? Because I think it will pack the most punch for the money. The idea of a signal is incredibly useful and unifying, and you don't really have go into full rigor to understand things conceptually. I was lucky to learn these ideas outside of class early on and I can hardly think of any more useful in understanding physics in general. Think of just a few examples of physical things that are signals:

position of particle(s) as function of time
string
membrane
sound field
electromagnetic field
voltage as function of time at a point
temperature field
any time series data

A few places where these ideas pop up:

Heisenberg uncertainty principle (a property of the Fourier transform which applies to any signal)
decomposition of string, membrane, sound field, quantum wavefunction, or EM field into modes
condensed matter (phonons)
electronics/communications
spectroscopy
imaging (CT scans, holography, etc)
system identification and control
general interpretation and storage of data

As far as I can tell, every branch of physics uses these ideas in a nontrivial way. In math, this ties together linear algebra, functional analysis, and ODEs/PDEs.

I don't see how the other courses could be this useful. Things like real analysis and (point-set) topology are good to know, but you'll get very little working knowledge out of the entire semester- mainly you'll learn how to make what you already know rigorous. If you'll immediately be doing research in GR you should probably take differential geometry but otherwise it'll give you little milage. The linear algebra and differential equations courses are probably useful, but I think the signals course would be better- for example, it will make those courses easier to learn on your own later. The nonlinear dynamics course would also be useful, but I still think the signals would be better- for NLD you could pick up a copy of Strogatz and learn it on your own. The abstract algebra and number theory courses might be fun (if you like that sort of thing) but are more or less useless for physics.

I don't know much about the financial stuff or how you might use it, so I can't comment there. Maybe one of those courses would be worth more to you than my suggestion- I don't know.

 

[Shackleford]

Hmm. Each suggestion has validity.

 

[Nolen Ryba]

Hmm. Each suggestion has validity.

Nope; just mine.