The Top Undergraduate Math Courses for Physics

[Joshua L]

Hey all, I am currently a freshman double majoring in Physics and Math at Purdue University. I have been recently creating a schedule that orders all of the courses I need to take and those that I want to take for the rest of my undergraduate endeavors. I know that it's rather early to do this, but I need to. If I craft my schedule in a certain way quite soon, I may able to avoid taking classes this summer, graduate in four year, and most importantly, save money. I love physics and math; so if I feel as though that there is a course I need to enroll in, I will do so unquestionably.

So here is the question: what are the most important undergraduate math courses for a student who wishes to focus on theoretical physics and go to graduate school for a PhD?

(I know that Linear Algebra is considered quite important, but I would much appreciate it if you can list more than one. Perhaps five or six?)

I would love to hear your lists. Thanks a bunch.

 

[micromass]

What kind of theoretical physics?

 

[Cake]

Single and multivariable calculus are mandatory. As well as ordinary and partial differential equations. Linear algebra is very necessary. And a mathematical methods of physics class or a class on analysis are necessary as well. I think topology and analysis are required to tackle general relativity. Hope that helps :D

 

[Joshua L]

I hope to work with either general relativity or quantum field theory (or maybe both?).

I also heard that Lie Groups are important, yet I can't find it as its own separate course. Could it be that Lie Groups are covered in a more broad course?

Thanks for the replies so far!

 

[MarneMath]

You may not encounter Lie groups as an undergraduate. When I first met them (and not by choice) the required knowledge was analysis, group theory and topology. However, I think there exist books out there geared towards physics students that cover this. Nevertheless, I still expect you to only come across a course like Lie Groups if you need it or if you're into Algebra.

 

[Joshua L]

Would Lie Groups be a part of Modern or Abstract Algebra then?

 

[micromass]

Would Lie Groups be a part of Modern or Abstract Algebra then?

Not at all. I would classify Lie groups as part of differential geometry, and not as abstract algebra. So if you ask me what classes are important to appreciate the mathematics of GR and QFT, I'd say:

- Differential geometry of curves and surfaces
- Topology
- Differential geometry of manifolds
- Riemannian geometry
- Lie groups
- Representation theory
- Linear algebra (the more you take, the better. Be sure to get a course that'll do dual spaces and maybe even tensors)
- Complex variables

If you still want more, then maybe you can take analysis. The problem with analysis is that the first course is just a more rigorous version of calculus, and this will be almost entirely useless to you. It will take some time before you get into the useful analysis stuff (like Fourier series or functional analysis).

 

[Joshua L]

This is great to know.

How much abstract algebra would I need to know?

My university offers a two semester path or a four semester path.

Is Galois Theory important at all?

**Also, what exactly is representation theory? What typical undergrad course would that correspond to?**

 

[micromass]

This is great to know.

How much abstract algebra would I need to know?

My university offers a two semester path or a four semester path.

I don't think you need to know any abstract algebra. The only things which might be useful are the basics of group theory (basics means: everything in the appendix of Hall's book on Lie groups). I don't really think you need a course for that, you can easily learn it yourself.

Even if you take abstract algebra, I think four semesters would be definitely overkill.

Is Galois Theory important at all?

Not at all.

**Also, what exactly is representation theory? What typical undergrad course would that correspond to?**

Roughly, representation theory let's you take a group and express it as a matrix group in various ways. Representation theory is covered in abstract algebra. But there's a catch! When I took abstract algebra, the representation theory we did was representations of finite groups, and nothing else. And this is not what you would be interested in as far as I know. You would be interested in representations of infinite Lie groups such as SL or the Lorentz group. A typical abstract algebra course will not talk about these groups, which is why I think an abstract algebra course will not be useful to you. So which course do you want to take? It depends on your university. If they have a course on Lie groups, then they might do the necessary representation theory. Don't be surprised if it turns out to be a grad course though.

 

[Joshua L]

Thanks for the great reply.

One more last question:

Purdue offers the course Introduction to Differential Geometry and Topology
It is the only course that includes differential geometry so I'm definitely taking that.

However, it includes topology, a subject that I also may need to know.
So, should also take the course Elementary Topology the semester after I take the long-named course above?

 

[micromass]

Thanks for the great reply.

One more last question:

Purdue offers the course Introduction to Differential Geometry and Topology
It is the only course that includes differential geometry so I'm definitely taking that.

However, it includes topology, a subject that I also may need to know.
So, should also take the course Elementary Topology the semester after I take the long-named course above?

What the content of both classes?

 

[Joshua L]

Introduction to Differential Geometry and Topology
Smooth manifolds; tangent vectors; inverse and implicit function theorems; submanifolds; vector fields; integral curves; differential forms; the exterior derivative; DeRham cohomology groups; surfaces in E3., Gaussian curvature; two dimensional Riemannian geometry; Gauss-Bonnet and Poincare theorems on vector fields.

Elementary Topology
Fundamentals of point set topology with a brief introduction to the fundamental group and related topics, topological and metric spaces, compactness, connectedness, separation properties, local compactness, introduction to function spaces, basic notions involving deformations of continuous paths.

Introduction to Algebraic Topology
Singular homology theory; Eilenberg-Steenrod asioms; simplicial and cell complexes; elementary homotopy theory; Lefschetz fixed point theorem.

 

[micromass]

Yeah, taking both differential geometry and elementary topology is advisable. It's also better to take elementary topology before the differential geometry course (in fact, the elementary topology course should be a prereq to the differential geometry course).

The algebraic topology course is less necessary. Sure, it has quite some applications in physics, but they tend to be more esoteric and advanced. Only take it if you have no other choice. (And be sure to take elementary topology before it).