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### Upper Level Math Classes - Useful/Interesting

Posted: **Sun Mar 22, 2020 10:56 pm**

by **PhysRiches**

What types of math courses would be useful for a physicist?

Subjects like Real Analysis, Topology, and Complex Analysis are some of the foundations of pure mathematics and a requirement for most mathematics majors. But, are these courses or any others (abstract algebra, advanced linear, non-euclidean geometry, etc.) useful for a physicist? If not useful are there some that were interesting when you studied them?

### Re: Upper Level Math Classes - Useful/Interesting

Posted: **Mon Mar 23, 2020 6:12 am**

by **jabennett2194**

I was a math physics double major.

Real analysis is the most PURE of the maths and I don't think I have or ever will use it in my physics hahaha

Topology is awesome, but I'm not at a high enough level to use it yet. It is very helpful for HEP-th and CMT though. Go ahead and search key words like cohomology or homotopy in arXiv or InspireHEP to get an idea. Some physics words that involve topology: topological quantum field theory, Chern number, Chern-Simons theory, topological invariants, topological insulators, many others.

Abstract Algebra (if you view that as a primer for subjects like Lie Groups, Lie Algebras, and Representation Theory) is the most important from my personal experience. I use Lie theory and Rep theory all day in HEP-th and it is of course also extremely important for CMT (think about symmetries).

I REALLY wish I was forced to take Complex Analysis. It was an elective and I skipped it to do an independent study in algebraic topology. This (contour integration, residue theorem, etc.) is very important for quantum field theory.

Linear algebra is rather important for quantum mechanics if I remember correctly, I haven't taken pure QM in way too long. I'm screwed for next semester hahah

### Re: Upper Level Math Classes - Useful/Interesting

Posted: **Mon Mar 23, 2020 12:07 pm**

by **geekusprimus**

Numerical methods all the way. Easily the most useful extra math classes I've taken.

### Re: Upper Level Math Classes - Useful/Interesting

Posted: **Thu Mar 26, 2020 11:24 am**

by **ztruwk**

I agree with all said above. You'll want to take complex analysis for QFT. You'll definitely want to take abstract algebra if you're going into any particle/high energy type subject, but you'll probably want it regardless. I don't see how taking an advanced linear course could ever hurt.

If you ever want to take GR, taking a course with differential geometry (which is probably what is covered in non-Euclidean geometry) first will do you so well. I took GR in undergrad without having ever encountered that stuff before, and man, the physics is hard enough, knowing a bit about the math first would be really helpful.

### Re: Upper Level Math Classes - Useful/Interesting

Posted: **Thu Mar 26, 2020 12:55 pm**

by **jabennett2194**

ztruwk wrote: ↑Thu Mar 26, 2020 11:24 am

I agree with all said above. You'll want to take complex analysis for QFT. You'll definitely want to take abstract algebra if you're going into any particle/high energy type subject, but you'll probably want it regardless. I don't see how taking an advanced linear course could ever hurt.

If you ever want to take GR, taking a course with differential geometry (which is probably what is covered in non-Euclidean geometry) first will do you so well. I took GR in undergrad without having ever encountered that stuff before, and man, the physics is hard enough, knowing a bit about the math first would be really helpful.

I agree with comp. analysis for path integral QFT and abstract algebra for theoretical physics generally.

But I've found pure mathematician diff. geo. courses do not do a good job of teaching semi-reimmanian geometry. I recommend taking an undergrad GR course and using this textbook

http://pages.pomona.edu/~tmoore/grw/Res ... RWBook.pdf instead. Moore is a pedagogical genius. Chapter's 3-6 are golden index notation intro material.

### Re: Upper Level Math Classes - Useful/Interesting

Posted: **Thu Apr 02, 2020 12:26 am**

by **ztruwk**

jabennett2194 wrote: ↑Thu Mar 26, 2020 12:55 pm

I agree with comp. analysis for path integral QFT and abstract algebra for theoretical physics generally.

But I've found pure mathematician diff. geo. courses do not do a good job of teaching semi-reimmanian geometry. I recommend taking an undergrad GR course and using this textbook

http://pages.pomona.edu/~tmoore/grw/Res ... RWBook.pdf instead. Moore is a pedagogical genius. Chapter's 3-6 are golden index notation intro material.

Man, thank you. This is what I needed. I'm going to take a look at that myself before heading to grad