Is analysis necessary to know topology and differential geometry?

[jesse73]

The physics books that cover mathematical topics also tend to add more physics insight to the problem. A book on mathematical methods for physics will make sure you know why eigenvalues are important in QM while a book focused on just mathematical concerns won't even mention QM.

There is a difference in a linear algebra or math class for mathematicians course and a linear algebra class meant for engineers or physicist. There is a even difference in a QFT class taught by particle theorist and one taught by condensed matter theorist. You should try to take the one more directly related to your goals so you get a higher rate of return for your time invested.

 

[R136a1]

I'm pretty sure GR specialists who strongly suggested I take these classes know much more than you do when it comes to the field, with all due respect of course.

WannabeNewton is one of our best posters when it comes to GR. I wouldn't immediately dismiss his advice like this.

https://www.youtube.com/watch?v=obCjODeoLVw

And a good knowledge of pure math can sometimes completely destroy your love and ability to do physics. I have significant difficulties in solving physics problems because I think way too mathematical about them.

 

[Student100]

So you're basing the utility of the math knowledge and intuition on solving back of the book problems?

No, I believe he's basing his argument on the utility of pure math in garnering any kind of advantage in physics over someone who just studies the utility of math.

But maybe my reading comprehension is folly.

 

[-Dragoon-]

Hah. I clearly said "a formal study of pure math". You haven't seen a true pure math course in PDEs before have you? It's nothing like the kind engineers and physics students generally take. You haven't done nearly enough to know the difference between math as used in physics and math as used in a pure math text as this thread clearly shows. There's nothing wrong in learning what you want to learn but don't go around making it seem like studying Papa Rudin, Lee's topological manifolds, Rosenlicht etc. will make your grad school endeavors somehow easier than someone who just stuck to mathematical methods books and the likes constructed primarily for utility in the line of physics.

The main problem I have is actually finding books geared towards physicists that go beyond very basic and trivial stuff like Fourier transforms, contour integration, Cauchy's theorem, special functions, and basic PDE methods. I've taken the math methods for physicists course in the department, and despite getting an A+, I still feel I don't know any "serious" mathematics.

The only book I've found thus far that does exactly what you said is "geometrical methods of mathematical physics" by Schutz, but it really only does serious treatment differential geometry. Before graduate school, I at least want to have a very deep understanding of complex analysis, basic group theory, topology, and differential geometry. I enjoy doing math for the sake of math and it's mostly curiosity, but you mean to absolutely tell me that having a deeper understanding of these topics than your typical graduate student won't be a boon at all? So my professors simply have no idea what they are talking about?

 

[R136a1]

The main problem I have is actually finding books geared towards physicists that go beyond very basic and trivial stuff like Fourier transforms, contour integration, Cauchy's theorem, special functions, and basic PDE methods. I've taken the math methods for physicists course in the department, and despite getting an A+, I still feel I don't know any "serious" mathematics.

The only book I've found thus far that does exactly what you said is "geometrical methods of mathematical physics" by Schutz, but it really only does serious treatment differential geometry. Before graduate school, I at least want to have a very deep understanding of complex analysis, basic group theory, topology, and differential geometry. I enjoy doing math for the sake of math and it's mostly curiosity, but you mean to absolutely tell me that having a deeper understanding of these topics than your typical graduate student won't be a boon at all? So my professors simply have no idea what they are talking about?

Group theory: mathematics courses usually focus on finite groups, while physicists usually use infinite matrix groups. For this reason, math courses on group theory do not tend to be very useful.

Topology: physicists tend to focus on very nice spaces such as manifolds. A topology course deals with very weird and strange spaces, and many of it is not very useful at all outside of mathematics. A deeper understanding of compactness and connectedness is useful though.

Differential geometry: this could be useful to physicists as some kind of foundational thing. But still, I fail to see how a rigorous proof of Whitney's embedding theorem could benefit you. Plus a differential geometry text in the pure math setting will strive to stick to index-free, coordinate-free methods throughout and this will not fly in GR.

Complex analysis: I don't see why physicists need to bother with the many proofs in this course. Sure, it is a very elegant course and the proofs are beautiful. But I doubt it will make you a better physicist. Physicists who spend most of their time worrying about whether interchanging series and integral is allowed don't produce much physics.

 

[Jorriss]

Group theory: mathematics courses usually focus on finite groups, while physicists usually use infinite matrix groups. For this reason, math courses on group theory do not tend to be very useful.

Finite groups matter too! I think an algebra course on group theory could be fairly useful so long as one doesn't stick around for ring and field theory (as those have not been as useful in my experience).

 

[R136a1]

Finite groups matter too!

I'm sure they do. But once they get into Sylow theorems and simple groups, I think it's a bit less useful.