Should I study Topology or Group Theory?

[MostafaAlkady]

Hello! I'm a physics graduate who is interested to work in Mathematical Physics. I haven't taken any specialized maths courses in undergrad, and currently I have some time to self-learn. I have finished studying Real Analysis from "Understanding Analysis - Stephen Abbott" and I'm currently deciding which topic should I be studying in the next few months. I'm deciding between Topology and Group Theory. Topology seems so interesting and I've always wanted to study it but didn't have the time. Also Group Theory is ubiquitous in physics and super exciting on its own. I'll be joining grad school this next fall so maybe I can study one of them on my own and postpone the other until I start grad school (so maybe I can audit the course there instead of studying on my own). So which one do you think I should self-study now and why?
Thank you for your time!

 

[romsofia]

If you enjoy analysis, then I'd go for topology, specifically "point-set topology". However, I'm not a mathematician, and I haven't studied that book from Abbott, so I'm not sure how deep he goes in analysis, and the overlapping proof styles.

If you're tired of analysis, then go for group theory! It's a change of pace, and cyclic groups are a pretty fun concept when tied to some other modulo concepts.

Neither of these will really prepare you for anything you'll face in graduate school, they should just be for the joy of mathematics. If you want to study things that'll get you "ready" for your graduate physics course, then math wise you should be hitting functional analysis, PDEs, and getting comfortable manipulating series ala perturbation theory.

Either way, I've always studied math for the joy of math, and graduate school is a fun place where you have a bunch of other people interested in these areas who are always up to chat about some wild math concept you learned the night before from my experience.

 

[Orodruin]

While both are useful in physics, I would recommend group theory first as it tends to be usable in many many different fields. Symmetry arguments are often central and group theory is the language used to describe them.

 

[dextercioby]

Well, I would suggest also going thoroughly through the first 4 chapters of Munkres' Topology textbook. It cannot hurt. The trick is that a solid preparation in Lie Groups must include topological aspects, too.

 

[MostafaAlkady]

If you enjoy analysis, then I'd go for topology, specifically "point-set topology". However, I'm not a mathematician, and I haven't studied that book from Abbott, so I'm not sure how deep he goes in analysis, and the overlapping proof styles.

If you're tired of analysis, then go for group theory! It's a change of pace, and cyclic groups are a pretty fun concept when tied to some other modulo concepts.

Neither of these will really prepare you for anything you'll face in graduate school, they should just be for the joy of mathematics. If you want to study things that'll get you "ready" for your graduate physics course, then math wise you should be hitting functional analysis, PDEs, and getting comfortable manipulating series ala perturbation theory.

Either way, I've always studied math for the joy of math, and graduate school is a fun place where you have a bunch of other people interested in these areas who are always up to chat about some wild math concept you learned the night before from my experience.

I enjoyed analysis so much. The book was pretty basic (probably the easiest analysis book out there), but I have studied it thoroughly and done most of the exercises. However I guess I'd like to try group theory now because I know so little about it and I hear it's ubiquitous in physics.

What are the prerequisites for functional analysis? And do you know of any good textbook for self-study?

 

[MostafaAlkady]

While both are useful in physics, I would recommend group theory first as it tends to be usable in many many different fields. Symmetry arguments are often central and group theory is the language used to describe them.

I guess I'm going to go for it. I have the book "Group Theory in a nutshell for physicists - A.Zee" in mind, but I would appreciate if you know any other book that was useful for your study.

 

[MostafaAlkady]

Well, I would suggest also going thoroughly through the first 4 chapters of Munkres' Topology textbook. It cannot hurt. The trick is that a solid preparation in Lie Groups must include topological aspects, too.

That's a good idea. Hopefully in the next summer after I get to know some decent group theory first, and then I can work on linking both later on.

 

[romsofia]

What are the prerequisites for functional analysis? And do you know of any good textbook for self-study?

I didn't have to self study it, luckily, so I can't comment on a book that would help. However, having self studied other things, the textbook isn't going to make or break you anymore. The most important part of learning, imo, is feedback. Find an author who you can understand and derive the things in the text, then look for problem sets from universities.

Ultimately, to be confident in your self studied abilities you have to test your knowledge. So, find a textbook you think you'll enjoy (whether it be how they write, their layout, etc), find some problem sets, and get cracking.

Prerequisites for my course were linear algebra, and PDEs.