# Real Analysis/Group Theory in Physics Undergrad?

I was wondering how people felt about taking courses in Real Analysis and/or Group Theory during an undergraduate leading towards physics graduate school in a possibly theoretical field.

Right now I am going into my third year in engineering science at the University of Toronto and have so far followed almost exactly what a physics specialist here at UofT is required to take (plus a lot more). They are not required to take any advanced pure math courses above PDEs and complex analysis, and that is why I am asking the question.

I could take real analysis and group theory, but then I would have to drop a physics course in a topic such as solid state physics or nuclear and particle physics which I am really interested in. I know math is important, but I would like to take more courses in physics, so I can get a good feel of what I am interested in. At the same time, I don't want to be missing out on important math skills I will need for later.

If it helps I'm considering in 4th year taking courses such as General Relativity, High Energy Physics, Advanced QM, Condensed Matter Physics and a course on Fusion Energy.

PS. I should mention I posted a similar thread a couple days ago. I decided to make my question a bit more specific and repost it here.

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I posted on your other thread so I'll just make a similar post. In QFT classes in grad school you'll learn the group theory necessary they won't assume you have taken the class but it does help if you have previous exposure. The ideas discussed in your group theory class according to the U of T math calender are:

Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagranges theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.

Most of this will be unnecessary to you since they are essentially used as "tools" in QFT instead of abstract objects (although this is becoming the case in more theoretical work).

Which Real analysis are you considering? U of T offers four Real Analysis courses: http://www.artsandscience.utoronto.ca/ofr/calendar/prg_mat.htm [Broken]

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If I took real analysis it would be Introduction to Real Analysis:
"Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension."

Since I am in engineering, I have to follow my program outlines. For third year winter semester, I got to choose 4 electives from a list of ~20 electives. Only the introduction to real analysis is on that list. I'm assuming that maybe I might not have the complete prereqs to get into the true real analysis course, Real Analysis 1. However, I could probably talk to my program's department and convince them to let me take it (engineering science student's have a lot of freedom, and we are assumed to be able to handle almost anything).

I guess I am saying that there is the possibility I could take Real Analysis 1:
"Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, Lp spaces. Applications to probability."

Similarly, there's also the possibility I could take the more advanced Groups course, Groups, Rings and Fields:
"Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hlder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic."

I didn't realize this until now, but I could also take a course called Introduction to Topology:
"Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Topology of function spaces. Fundamental group and covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point theorem. "

How useful would this topology course be?

If I took real analysis it would be Introduction to Real Analysis:
"Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension."

Since I am in engineering, I have to follow my program outlines. For third year winter semester, I got to choose 4 electives from a list of ~20 electives. Only the introduction to real analysis is on that list. I'm assuming that maybe I might not have the complete prereqs to get into the true real analysis course, Real Analysis 1. However, I could probably talk to my program's department and convince them to let me take it (engineering science student's have a lot of freedom, and we are assumed to be able to handle almost anything).

I guess I am saying that there is the possibility I could take Real Analysis 1:
"Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, Lp spaces. Applications to probability."

Similarly, there's also the possibility I could take the more advanced Groups course, Groups, Rings and Fields:
"Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hlder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic."

I didn't realize this until now, but I could also take a course called Introduction to Topology:
"Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Topology of function spaces. Fundamental group and covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point theorem. "

How useful would this topology course be?

Have you taken a rigorous proof course? If you haven't Real Analysis I and Group, Rings and Fields will be very difficult. Proving is much like an art and takes a lot of time to become accustomed to.

The most impotant math courses I think you can take as a a future physics graduate are:

- Complex Analysis (most important)
- Introduction to Differential Geometry (most important)
- Groups and Symmetries (medium importance, used in Yang-Mills gauge theories (QFT) and string theory, you'll learn the mathematics in graduate school in your classes)
- Introduction to topology (below medium importance; used in Yang-Mills gauge theories but the full depth of the course won't be entirely useful. The Jones polynomial (knot theory) is used in string theory so are topological invariants. Once again, taking the whole course will probably give too much depth)
- Real Analysis (I'd think it would be most beneficial for you to take a course in functional analysis i.e calculus of variations which is incredibly useful to a physicist which happens to be this course: http://www.math.toronto.edu/lguth/Math1000Y.html [Broken])

Talk to the user named wisvuze, he's a mathematics specialist undergrad at U of T and he can gauge if you're capable of taking these courses and how useful they'd be to you.

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Intro to topology is below medium importance to you?? Wow...
You use topology in general relativity (for differential geometry) and in quantum mechanics (for functional analysis and the like). Topology is extremely useful!!

Out of Real analysis I, abstract algebra and topology, I would say that topology is the most useful course...

Intro to topology is below medium importance to you?? Wow...
You use topology in general relativity (for differential geometry) and in quantum mechanics (for functional analysis and the like). Topology is extremely useful!!

Out of Real analysis I, abstract algebra and topology, I would say that topology is the most useful course...

My opinion is worthless I was just trying to keep the post alive. I though differential geometry would be more relevant but you know more.

Groups and Symmetries is an applied abstract algebra course, the real abstract algebra course is Groups, Rings and Fields.

Intro to topology is below medium importance to you?? Wow...
You use topology in general relativity (for differential geometry) and in quantum mechanics (for functional analysis and the like). Topology is extremely useful!!

Out of Real analysis I, abstract algebra and topology, I would say that topology is the most useful course...

This being a post about math courses at my own school.. I felt inclined to say something.. but alas, I know nothing about physics and cannot give definitive advice. Although, the topology below medium importance thing did sound off key.. topology? geometry? I'm fairly sure these are things you will need to know about a lot in physics

My opinion is worthless I was just trying to keep the post alive. I though differential geometry would be more relevant but you know more.

Groups and Symmetries is an applied abstract algebra course, the real abstract algebra course is Groups, Rings and Fields.

Topological considerations are key ingredients when studying differential geometry

My opinion is worthless I was just trying to keep the post alive. I though differential geometry would be more relevant but you know more.

Groups and Symmetries is an applied abstract algebra course, the real abstract algebra course is Groups, Rings and Fields.

MAT301H1 "Groups and Symmetries" is definitely not an "applied" abstract algebra course. The difference is that MAT347 "Groups, Rings, Fields" is a more "stepped up" version, and that MAT347 covers more than just group theory ( as the name suggests ). But MAT301 is just a regular math course on groups, as much as a math course can be.
In fact, if you like MAT301, you can "continue" your abstract algebra education by taking MAT401 http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm#MAT401H1. By taking this after 301, you will have "caught up" to what someone taking MAT347 might've studied

you can see for yourself: http://www.math.toronto.edu/lgoldmak/301F09/301F09.html [Broken]

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I was wondering how people felt about taking courses in Real Analysis and/or Group Theory during an undergraduate leading towards physics graduate school in a possibly theoretical field.

Why not buy a physics book that uses mathematics, and read that instead? The problem with taking a math class is that it will teach you a bunch of junk that you may not even remember if your primary motivation is physics.

If you enjoy math by itself, I would say both real analysis and group theory are good additions to your knowledge, and will help in approaching advanced physics topics. If you do not intrinsically enjoy math, then read a graduate level physics book that uses mathematics at that level, and plan to take math courses accordingly.

I was wondering how people felt about taking courses in Real Analysis and/or Group Theory during an undergraduate leading towards physics graduate school in a possibly theoretical field.

One problem with mathematicians is they tend to start thinking in terms of mathematics, when a lot of the math courses are just more general versions of things done at a lower level. (By the way, I am guilty of this, so I'm not pointing fingers, although I'm more aware than many math folk, I think, of my tendency.)

Perhaps in very advanced physics, a thorough command of differential geometry/topology, group theory, etc is required. But I feel it is unlikely that one can't survive a graduate level physics course without those classes in math - in fact, I think a lot of math is introduced as you go. That's the case in engineering courses as well.

What of topology that you do not get in basic real analysis do you really need to do physics? Heck, a basic point-set topology course probably teaches more point-set topology than many mathematicians themselves really need, depending on what they work on.

Let's not forget that nicholls wants to take physics courses in greater abundance to better discover interests within physics. For non-mathematicians, often this is the best strategy, and then if some somewhat generalized constructions are required to do research, they'll take the math courses necessary in, perhaps, their first year of grad school or last year of undergrad, or even beyond.

As Kevin_Axion has given a good summary of the mathematics I think you should be learning, I'll say no more, aside from possibly recommend some PDEs at an advanced level - I think these come up in studying GR, etc.

Out of Real analysis I, abstract algebra and topology, I would say that topology is the most useful course...

Really? How much can you really do with topology without knowing the other two, especially if your goal is differential geometry...at the very least, mastery of linear algebra is to be expected, although I guess that doesn't require abstract algebra.

Alright, I appreciate the responses. I think I am going to hold on real analysis for now, the applications to physics seem more limited, thus if there is anything I really need to know I will probably just pick it up in a physics course along the way.

I looked at some more fourth year courses that I am interested in. Once course, Relativity 1, suggests studying the topic of calculus of variations. In what type of course would I find this in?

Also, a lot of you mention both topology and differential geometry as being similar or related in some way. There are two courses available to me to take on these subjects. One is the Introduction to Topology I mentioned earlier. The other is Introduction to Differential Geometry:
"Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature."

I can only take one of the courses, as taking both leads to conflicts in my schedule. Thus, if I were to take one of the courses, which one would be the most useful to me? Especially If like I said, I will be taking courses in topics such as General Relativity, Advanced QM, Plasma Physics, High Energy Physics, Relativistic Electrodynamics and Condensed Matter Physics.

This being a post about math courses at my own school.. I felt inclined to say something.. but alas, I know nothing about physics and cannot give definitive advice. Although, the topology below medium importance thing did sound off key.. topology? geometry? I'm fairly sure these are things you will need to know about a lot in physics

Topological considerations are key ingredients when studying differential geometry

MAT301H1 "Groups and Symmetries" is definitely not an "applied" abstract algebra course. The difference is that MAT347 "Groups, Rings, Fields" is a more "stepped up" version, and that MAT347 covers more than just group theory ( as the name suggests ). But MAT301 is just a regular math course on groups, as much as a math course can be.
In fact, if you like MAT301, you can "continue" your abstract algebra education by taking MAT401 http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm#MAT401H1. By taking this after 301, you will have "caught up" to what someone taking MAT347 might've studied

you can see for yourself: http://www.math.toronto.edu/lgoldmak/301F09/301F09.html [Broken]

I see. It's a good thing you posted since I was just judging based on the course description.

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