Recommended mathematical courses for a Physics major

[torito_verdejo]

Hi, you all,

I am an undegrad Physics student and I'm choosing my optional courses for my third year. I'm looking for advice (or opinions, if you prefer) since I'm not sure what of the following mathematical, optional courses would be more "beneficial" (I know this term is abstract) to my physics formation. Note that this course will already follow four semesters of real analysis, two of complex analysis, one on linear algebra, and another on functional analysis. Note too that I am eager to study abstract algebra on my own (as I already have) so I don't fear not "being ready" for the course.

What do you think I should choose among the following? (If you can, separate your answer in what to choose according to its "beauty", and what to choose according to its applicability to physics):

1. Galois theory
2. Algebraic geometry
3. Algebraic topology

All three are first courses on each subject. Thank you in advance.

 

[fresh_42]

Forget Galois theory. It's nice but more or less useless in physics. Algebraic geometry is as ambitious as algebraic topology is. However, algebraic geometry is ring theory, hence useless in physics (duck and run ...), and algebraic topology is closest to physics, e.g. in cosmology and differential geometry. You probably won't see the connections to physics, but they exist. We still do not know how our universe looks like, we have to integrate over a lot of paths in physics, and it is important to know whether e.g. there is a singularity in between. Algebraic topology provides the tools which are necessary to classify the manifolds which occur in physics.

 

[Dr. Courtney]

Have you had a good course in numerical methods yet?

 

[PeroK]

Hi, you all,

I am an undegrad Physics student and I'm choosing my optional courses for my third year. I'm looking for advice (or opinions, if you prefer) since I'm not sure what of the following mathematical, optional courses would be more "beneficial" (I know this term is abstract) to my physics formation. Note that this course will already follow four semesters of real analysis, two of complex analysis, one on linear algebra, and another on functional analysis. Note too that I am eager to study abstract algebra on my own (as I already have) so I don't fear not "being ready" for the course.

What do you think I should choose among the following? (If you can, separate your answer in what to choose according to its "beauty", and what to choose according to its applicability to physics):

1. Galois theory
2. Algebraic geometry
3. Algebraic topology

All three are first courses on each subject. Thank you in advance.

Are you sure you shouldn't be a math major?

 

[torito_verdejo]

Have you had a good course in numerical methods yet?

Actually I didn't and, now that I think of it, it would certainly be extremely useful. I'll think of it further, as there is also a course in numerical analysis at my disposal. Nevertheless I must confess that this mathematical courses I'll be taking are choosen more for pleasure than for usefullness, and I miss abstract algebra. This said, I really appreciate your idea and I will consider it.

 

[torito_verdejo]

Forget Galois theory. It's nice but more or less useless in physics. Algebraic geometry is as ambitious as algebraic topology is. However, algebraic geometry is ring theory, hence useless in physics (duck and run ...), and algebraic topology is closest to physics, e.g. in cosmology and differential geometry. You probably won't see the connections to physics, but they exist. We still do not know how our universe looks like, we have to integrate over a lot of paths in physics, and it is important to know whether e.g. there is a singularity in between. Algebraic topology provides the tools which are necessary to classify the manifolds which occur in physics.

Thanks for the detailed answer, it really helped me. I guess I'll leave Galois theory as a summer reading. Concerning algebraic geometry, I thought it would be applicable to theoretical physics simply and naively because it's something I can't stop hearing about (and it sounds sexy, hahaha). I trust you though, on your telling me algebraic topology will be more nourishing for my physics. Actually this semester we are being mortified with hamiltonian formalism by the hand of Arnold and when I saw "symplectic manifold" on the syllabus of alg. topology I suffered from a PTSD episode. Hahaha. I'm kidding, I like the formalism, but it is very sudden for a second year undergrad.

 

[berkeman]

Nevertheless I must confess that this mathematical courses I'll be taking are choosen more for pleasure than for usefullness

Doesn't that seem a bit misplaced? I guess it's best if you can have both (pleasure and career-oriented), but putting pleasure before career pursuit seems misplaced to me.

 

[fresh_42]

Nevertheless I must confess that this mathematical courses I'll be taking are choosen more for pleasure than for usefullness, and I miss abstract algebra.

This sounds as if Galois theory is on the list again. It is not too difficult, closed in itself and really beautiful. At the end you cannot use it anywhere, but you can answer the three ancient classical problems: squaring a circle, doubling a cube, and trisection an angle. They are all impossible, but Galois theory can tell you why. However, the old Greeks are not very attractive these days.

I would still choose algebraic topology. It has the amount of abstract algebra you are thinking of, but on its way will tell you useful things about boundary operators (aka derivatives) and that cohomology theory can actually be useful. E.g. here's what I found on algebraic topology on Wikipedia (translated via Google)

In physics, too, the Chern classes have been increasingly used since around 2015 and are also explicitly called this (which was not the case before), since now, not only in high-energy physics, but also increasingly in solid-state physics, new differential topological aspects are dealt with: In addition to older Umlauf "statements in physics, such as the Aharonov-Bohm effect of quantum mechanics or the well-known Meissner-Ochsenfeld effect of superconductivity, are used by Chern classes in physics primarily for the differential topological classification of Umlauf singularities, especially in the so-called quantum Hall-Effect or with the so-called topological superconductors or topological insulators.

 

[fresh_42]

Concerning algebraic geometry, I thought it would be applicable to theoretical physics simply and naively because it's something I can't stop hearing about (and it sounds sexy, hahaha).

Algebraic geometry sounds geometrical, but not in the sense of real numbers. It deals with ideals of polynomial rings. It is the zeros of those polynomials which provide the geometric part. Does this sound interesting:

A Gröbner basis is a finite generating system for an ideal $I$ in the polynomial ring $K [X_{1}, \ldots, X_{n }]$ over the field $K$, which is particularly suitable for deciding whether a given polynomial belongs to the ideal or not.

 

[mpresic3]

Of the three courses you list, AlgebraicTopology is most relevant to physics, assuming you go in for general relativity, field theory, or maybe high-energy theory. A numerical analysis course would be most useful if you intend to intern at a laboratory during summers. This is likely not the last time you can take algebraic topology, galois theory, or algebraic topology. You may get a chance in graduate school. If you have a research advisor, (s)he is probably a good source to ask