Which Is Better for Physics: Abstract Algebra or Set Theory?

[saybrook1]

I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set theory? Thank you!

 

[micromass]

Are there any other options?

Set theory is close to useless in physics. So only take it if you're particularly interested in it (it is a rather beautiful theory!)

Abstract algebra comes up occasionally in physics. In particular, groups and representations are important. The problem is however that abstract algebra is more about finite groups and a first course doesn't deal very much with the groups which are actually used in physics. In that sense, abstract algebra will only be useful to you to understand the basic concepts and definitions of groups. So while more useful than set theory, I don't consider it very useful on its own.

Again, this answer supposes you want to choose a math class based on how useful it is to physics. If you're interested also in the intrinsic mathematical beauty, then that changes a lot.

 

[saybrook1]

Yeah, I'm really just interested in taking a math course that will be useful for physics. Other choices include ring-field theory, discrete mathematics, advanced number theory, COM AN-BOUND VAL(not sure what this is?), ADV DIFF EQUAT, MODERN ALGEBRA, ADV NUMERICAL AN, and modern analysis... Thanks again for your reply

 

[saybrook1]

I like the teacher for math control theory and com an-bound val if you think those are helpful at all

 

[micromass]

Yeah, I'm really just interested in taking a math course that will be useful for physics. Other choices include ring-field theory, discrete mathematics, advanced number theory, COM AN-BOUND VAL(not sure what this is?), ADV DIFF EQUAT, MODERN ALGEBRA, ADV NUMERICAL AN, and modern analysis... Thanks again for your reply

I think something like "Advanced differential equations" or "numerical analysis" might be more suitable. Numerical analysis will be useful because it will (likely) ask you to program stuff in MATLAB (or similar) and this is always a very useful tool in research.
Differential equations show up everywhere in physics, so it's clear why this is important. Even if the course is about existence and uniqueness theorems, I think it's more applicable that set theory or abstract algebra.

 

[micromass]

I like the teacher for math control theory and com an-bound val if you think those are helpful at all

Maybe you should ask the teach what "com an-bound val" is, because I don't have a clue.

 

[saybrook1]

Awesome, will do. Thanks a ton; really helped me out a lot!

 

[Hypersphere]

I'd guess that "com an-bound val" stands for complex analysis and boundary value problems, or something similar. On the other hand, it is not clear to me whether a course with that name would focus on (useful) complex methods for solving differential equations, or on equations with solutions satisfying the Cauchy-Riemann conditions. The latter is a tad formal, and not that useful in physics as far as I know.

Anyway, "advanced differential equations" and "numerical analysis" sound most directly applicable, but as always it is a matter of the syllabi.

 

[micromass]

I'd guess that "com an-bound val" stands for complex analysis and boundary value problems, or something similar. On the other hand, it is not clear to me whether a course with that name would focus on (useful) complex methods for solving differential equations, or on equations with solutions satisfying the Cauchy-Riemann conditions. The latter is a tad formal, and not that useful in physics as far as I know.

Anyway, "advanced differential equations" and "numerical analysis" sound most directly applicable, but as always it is a matter of the syllabi.

Ah! Complex Analysis, of course!

Complex analysis is also quite useful in physics. Being able to calculate integrals using contours and residues comes up a lot. However, this is actually more often taught in "math methods" courses, which I think are better for physicists.

 

[saybrook1]

Great, thanks again! I think I am going to take "methods in mathematical physics". I may also take numerical analysis or advanced D.E's if I'm feeling ambitious!

 

[samgrace]

Are there any other options?

Set theory is close to useless in physics. So only take it if you're particularly interested in it (it is a rather beautiful theory!)

Abstract algebra comes up occasionally in physics. In particular, groups and representations are important. The problem is however that abstract algebra is more about finite groups and a first course doesn't deal very much with the groups which are actually used in physics. In that sense, abstract algebra will only be useful to you to understand the basic concepts and definitions of groups. So while more useful than set theory, I don't consider it very useful on its own.

Again, this answer supposes you want to choose a math class based on how useful it is to physics. If you're interested also in the intrinsic mathematical beauty, then that changes a lot.

Hello, I am a physics student studying set theory in his spare time, set theory is completely useless in the physics sense? What about the cartesian product of two sets and their respective ordered pairs that are mapped into Real/2-dimensional Euclidian space? Particularly if the two elements of the ordered pairs are scalars or a 2-dimensional vectors? Surely this applies to Classical mechanics when working with basic trajectories? Generally this can be done for 3-vectors or 4 vectors, if you do the cartestian product of three or four sets.
Real analysis can then be done on this 3D Real/Euclidian space, which is where calculus comes from. Abstract algebra can be used to describe geometry, which when combined with calculus allows differential geometry, which is essential to Relativity. It also describes Hilbert space for QM.
The reason I am studying fundamental maths is so that I can understand physics beyond the normal level, and I feel this would be impossible without understand these areas of maths.

 

[homeomorphic]

It may be useful to have some of the concepts of set theory, but that doesn't mean taking a whole class about it. However, if you are going to be more on the math side of physics, it might be good to know more set theory, just to have a good math foundation. Still, even most mathematicians probably can't rattle off the Zermelo-Fraenkel axioms off the top of their heads, so I don't think you need to go too much in depth with it.

Real analysis can then be done on this 3D Real/Euclidian space, which is where calculus comes from.

You have it backwards. Real analysis comes from calculus, historically, and a lot of physicists and engineers get by pretty well without it.

Abstract algebra can be used to describe geometry, which when combined with calculus allows differential geometry, which is essential to Relativity. It also describes Hilbert space for QM.

There are abstract algebraic ways to describe geometry, but it doesn't "allow" differential geometry. There's a good portion of differential geometry that's just an extension of multi-variable calculus. You need abstract algebra if you want to understand connections on principal bundles or Cartan geometry, but those aren't really necessary to study GR. Hilbert spaces are functional analysis, not abstract algebra. You could view the vector space axioms as abstract algebra, but it's just one definition from abstract algebra, and at any rate, the defining completeness property of Hilbert spaces is a very analytic thing.

 

[mathwonk]

after all is said and done, the only useful math topics for physics, or anything else, are ones you understand. so if you don't take it, it won't be useful.