Choosing Math Directed Study Topics

[Elwin.Martin]

Alright I'll give you a basic overview of what I've taken really quick:
Calc I-Diff Eqs (Boyce and DiPrima, nothing special for my Calc courses)
Linear Algebra (Strang) and a course in Abstract Algebra (Dummit and Gallian)
Discrete (Rosen) and an elementary Stats class

Since my school is not especially strong on math, but has two or three strong professors, my adviser is having me take up to three directed studies next semester before I transfer out. My goal is to double major in Physics/Math somewhere.

Here are some topics/subjects I want to cover:
PDE
Analysis (Real and Complex)
Point Set Topology
Differential Geometry
Lie Groups/Lie Algebra
Clifford Algebra
Division Algebras!

and probably some other topics I can't come up with off the top of my head.

Now, I've been told that some of the topics are probably waaaaaaaaay above my head, but I mentioned them to give you an idea of my direction. Like, I believe he said I had a long way to go before I can get to Clifford Algebra, but Lie Algebra he thinks I can get through after Analysis and maybe some more Abstract Algebra. He told me that it would be very difficult (at a small school) to find a differential geometer...so that's something I'm probably going to have to put off and so forth.

The professors who I can work with are specialized as follows: the first did a lot of research into harmonic analysis, but feels he would be willing to teach me Point Set Topology along with someone else doing Analysis, he taught me Linear Algebra and is my adviser. The second is a surprisingly well published mathematician who did his PhD under Rudin and will teach anything Analysis related, maybe Lie Algebra with the recommendation of my adviser. The third just got his PhD from a relatively small school, but he does a lot of interesting work in Combinatorics and Abstract Algebra (he's teaching my Abstract course now).

Does anyone have any recommendations on what I should ask to study? I'm asking to do a Real Analysis direct study (so I can cover material at a different depth than the course offered) and Abstract Algebra directed study (maybe in Groups?), but the option of doing Topology is really appealing as well. If anyone comes up with something of a reasonable level they think I should consider, let me know. I'm looking at mostly "advanced undergraduate" and early graduate material.

Thanks in advanced for your help! I'm really hoping I can make things work out and have an awesome set of courses next semester.

 

[Elwin.Martin]

Having (im)patiently waited 24 hours, I will use my bump.

Surely one of the viewers of this thread has something to add about their experiences in directed studies or their recommendation for courses based on my experience.

I'll add some additional information that might be useful to you in helping me with my options:
There aren't many professors in "active" research here, but there is one who studies the Beam Equation, primarily, and he's been publishing fairly regularly who I could work with on some topics, though he's less friendly.

There's an applied professor who is also publishing, but she's less related to the topics I'm considering. She has name recognition, or so I'm told as she is well publish and her father is a famed mathematician (at least in Europe, specifically Ukraine). I'm not really that concerned about it, but my adviser said it might help to have that recognition and I may find something. Lately she's worked on "multilinear interpolation error", but I don't know what that is or if I could maybe find some sort of numerical analysis work I could do with her?

Hope this wasn't a completely empty bump and that someone can give me some advice, again any and all advice would be great! I don't really know the extent of my options and I'm not sure within those options what I should be considering. Since my end goal is a PhD in Physics (Theory), I really want to have a good foundation for my mathematics, though I know most math is taught along with the physics (I've heard that most people learn how to use tensors *in* GR).

Thanks again to anyone who responds,
Elwin

 

[micromass]

Maybe it is best if you do some real analysis. The benifit of doing real analysis is that you can orient it how you want. That is: you can focus on topology if you want to. So if you do real analysis, then you can learn topology concurently. You can also focus more on measure theory and integration topics, or on functionals, etc.

After you know topology and real analysis, you can use this to learn differential geometry (which inclused Lie thingies).

Abstract algebra is also a nice idea, but I don't know if it will be that useful in physics.

I'd focus on real analysis and topology right now. And maybe take a small complement in abstract algebra.

 

[Elwin.Martin]

Maybe it is best if you do some real analysis. The benifit of doing real analysis is that you can orient it how you want. That is: you can focus on topology if you want to. So if you do real analysis, then you can learn topology concurently. You can also focus more on measure theory and integration topics, or on functionals, etc.

After you know topology and real analysis, you can use this to learn differential geometry (which inclused Lie thingies).

Abstract algebra is also a nice idea, but I don't know if it will be that useful in physics.

I'd focus on real analysis and topology right now. And maybe take a small complement in abstract algebra.

That sounds like a good idea, thanks :) I think this is probably my best option, next time I meet with my adviser I will see if he agrees.

Why would the abstract algebra be of less help? The main reason for wanting to cover more is that my present course is rather, erm, weak and is only based in Groups (though I don't know that this is an issue).

 

[micromass]

That sounds like a good idea, thanks :) I think this is probably my best option, next time I meet with my adviser I will see if he agrees.

Why would the abstract algebra be of less help? The main reason for wanting to cover more is that my present course is rather, erm, weak and is only based in Groups (though I don't know that this is an issue).

It will be of less help since I know very little applications of abstract algebra in physics. I know that topology, analysis, differential geometry is very helpful. But I know of no applications of abstract algebra. Group theory is useful in physics, but I don't think that it is more important thant the other topics.

 

[micromass]

That said, if you want to study things like Lie algebra's/groups, Clifford algebra's and division algebra's, then abstract algebra does have its uses. But you need to study really much of abstract algebra before you can begin to apply it to the things you want to study.

If you want to put your time in it, very good. Abstract algebra is a really elegant theory. But do focus on more analytical topics first.

 

[Elwin.Martin]

That said, if you want to study things like Lie algebra's/groups, Clifford algebra's and division algebra's, then abstract algebra does have its uses. But you need to study really much of abstract algebra before you can begin to apply it to the things you want to study.

If you want to put your time in it, very good. Abstract algebra is a really elegant theory. But do focus on more analytical topics first.

I'm just trying to have a foundation in math so that if I choose a more math intense branch of Physics for grad school, I'll be prepared. I enjoy math as well, it's just not my primary interest.

I should have time to delve into things later, I'm rushing just a little bit at the moment. I'm in my first year of school, but I feel behind since I can't even take Physics III at my school without filling out special paperwork. When I transfer, I'm going to probably lose credit for a lot of the material, but I plan on learning what I learn well so that when I reach those topics again it will be familiar (though not necessarily easier).

Thanks for the advice, again.
Elwin

 

[Anonymous217]

Like the others, I highly recommend real analysis. It's practically the foundations for many of the courses you're interested in and by the end, you'll have a better idea of the more advanced subjects you're interested in.

If you want to get into lie groups or even further into topology (along the algebraic route it seems), you'll need to know abstract algebra. And as a mathematician (I can't speak for physicists), it's immensely useful to know abstract algebra; like real analysis, it's one of the core subjects that all (pure) math majors just need to know.

If you're interested in more physics-oriented math, differential equations and numerical methods subjects are highly useful. I think if you do go the topology/algebra/analysis route (largely pure-math oriented), you might be able to get into QFT and lie groups, but that's a long ways off and you'll need to know quite a few "pure" subjects before you can see their use in physics.

 

[Elwin.Martin]

Like the others, I highly recommend real analysis. It's practically the foundations for many of the courses you're interested in and by the end, you'll have a better idea of the more advanced subjects you're interested in.

If you want to get into lie groups or even further into topology (along the algebraic route it seems), you'll need to know abstract algebra. And as a mathematician (I can't speak for physicists), it's immensely useful to know abstract algebra; like real analysis, it's one of the core subjects that all (pure) math majors just need to know.

If you're interested in more physics-oriented math, differential equations and numerical methods subjects are highly useful. I think if you do go the topology/algebra/analysis route (largely pure-math oriented), you might be able to get into QFT and lie groups, but that's a long ways off and you'll need to know quite a few "pure" subjects before you can see their use in physics.

^^; I'm in QFT now and I'm just kind of accepting the math as I go along at times and not going into some of it rigorously. I'm a step below Peskin and Schroeder (Ryder, if you're familiar) this semester, but next semester I may be doing Peskin and Schroeder along with an additional text that I don't know very well.

I think the topology/algebra/analysis route is going to be fun and hopefully challenging. I don't know what level I should do the Algebra at...I'm not sure who I'm doing it with, so I may have a bit of say in what text we end up using. I will have finished Gallian's text as far as Groups, Rings, and Fields (maybe not Fields, depends on how the semester goes). What would be a reasonable step up from that? I feel like Gallian's text is a good introduction, but does not offer as much substance as what I need so I've been reading Dummit on my own as well. Would Serge Lang's book be too far reaching? I've just heard good things, I have no idea how difficult it is.

 

[Anonymous217]

If you finish Gallian or Fraliegh's abstract algebra texts, then try Lang's Algebra next. Lang's will be extremely in-depth and rigorous. Whenever he says something is "obvious", for example, he really means that you need to spend a few hours on your own to get it.
You need to be dedicated to go through the text. Lang's is not something you can read superfluously unless you already know the material (and even then, you'll learn quite a few new things). A lot of your understanding from Lang will come from working out problems and I'm not sure how much time or whether you can be dedicated enough to get through Lang.

However, since you seem to be more oriented on physics, I don't think you need to be that rigorous in your abstract algebra knowledge. I have a few physics friends and whenever they use Hilbert spaces or lie groups, I just scoff at how sketchy it is.

Also, it's safe to say that if you don't already know abstract algebra or real analysis, then you're probably not learning QFT. (at any level rigorous beyond a Wikipedia article, that is) Just a mathematician's point of view. ;)

 

[Elwin.Martin]

If you finish Gallian or Fraliegh's abstract algebra texts, then try Lang's Algebra next. Lang's will be extremely in-depth and rigorous. Whenever he says something is "obvious", for example, he really means that you need to spend a few hours on your own to get it.
You need to be dedicated to go through the text. Lang's is not something you can read superfluously unless you already know the material (and even then, you'll learn quite a few new things). A lot of your understanding from Lang will come from working out problems and I'm not sure how much time or whether you can be dedicated enough to get through Lang.

However, since you seem to be more oriented on physics, I don't think you need to be that rigorous in your abstract algebra knowledge. I have a few physics friends and whenever they use Hilbert spaces or lie groups, I just scoff at how sketchy it is.

Also, it's safe to say that if you don't already know abstract algebra or real analysis, then you're probably not learning QFT. (at any level rigorous beyond a Wikipedia article, that is) Just a mathematician's point of view. ;)

I appreciate your suggestion to use Lang's book, I'll probably seek a second opinion though as I don't want to jump into something above my head.

While my goal is to do research in physics, I don't see myself as being especially physics oriented at the moment. I want to take math that will be useful to me, yes, but that doesn't mean I won't get anything out of using Spivak over using Stewart for Calculus.

My goal is to be able to use the math that physicists seem to almost abuse in an elegant and correct manner, so I would really prefer to have a rigorous abstract algebra background. If it weren't silly, I'd do a double PhD in math and physics (assuming I could handle it, which I'm not saying I could).

However I disagree about one thing pretty strongly.
I'm pretty sure Ryder is QFT...

Here's an abbreviated chapter outline for the second chapter;

Chapter 2-Single particle relativistic wave equations
2.1 Relativistic notations
2.2 Klein-Gordon Eq.
2.3 Dirac Eq.
2.3.a SU(2) and the Lorentz group
2.3.b SL(2,C) and the Lorentz group
2.4 Prediction of antipartciles
2.5 Construction of Dirac spinors: algebra of gamma matrices
2.6 Non-relativistic limit and the electron magnetic moment
2.7 The relevance of the Pioncare Group:spin operators and the zero mass limit
2.8 Maxwell and Proca eq.
2.9 Maxwell's eq. and differential geometry

There are some more math intensive sections through out the book as well; for example, the entire 10th chapter is about topological objects in fields. The book is easier than Peskin & Schroeder mostly in the sense that he works out his examples explicitly and has a less ambitious list of topics, but I wouldn't say it's not QFT. Griffiths's book on elementary particles? That I would agree to not being QFT.

I didn't say I was doing *well*, or that it was easy, but I did say I was taking it. It's pretty frustrating keeping up with some of the math so far even though I'm only getting into the third chapter, and while this is a gentler approach than Peskin and Schroeder, most people I've asked say it's a good book for an undergraduate to work on after some more advanced q-mech and before attempting P&S. If you're familiar with David Tong, he recommends reading it concurrently with P&S in his lecture series published through PIRSA (it's pretty well put together, in my opinion).

My professor basically said to worry about mostly calculation based work at the moment and trying to understand the text. He wanted me to skip the portion on the Poincare group, but I'm still trying to read through it. I don't think there's anything wrong with having a bit of difficulty in the rigorous bits mathematically for a while, I've got the time to work through it. Just because I haven't formally taken analysis yet doesn't mean I know *nothing*, I'm just not good (yet).

 

[Anonymous217]

I appreciate your suggestion to use Lang's book, I'll probably seek a second opinion though as I don't want to jump into something above my head.

There's nothing wrong with picking up the book at your library and looking through it, especially through Chapter 1. The beginning covers a lot of abstract algebra you probably already know (groups, cyclic groups, normal subgroups, ...), just at a really fast paced level. It's actually possible for someone to use Lang's Algebra as their first course in Abstract Algebra. Not saying that it won't be completely impractical, but that it's still "possible", since the text doesn't assume pre-requisites.

Reading a couple of the pages would give you a better hint than any suggestion a person could give you. After all, you're the one who's aiming to take it so I'd imagine that you're the best person at judging whether or not you can handle a certain text.Also, try not to take my QFT comment too seriously. It was just a playful jab at physicists going over their heads when using mathematics. ;)

 

[Elwin.Martin]

There's nothing wrong with picking up the book at your library and looking through it, especially through Chapter 1. The beginning covers a lot of abstract algebra you probably already know (groups, cyclic groups, normal subgroups, ...), just at a really fast paced level. It's actually possible for someone to use Lang's Algebra as their first course in Abstract Algebra. Not saying that it won't be completely impractical, but that it's still "possible", since the text doesn't assume pre-requisites.

Reading a couple of the pages would give you a better hint than any suggestion a person could give you. After all, you're the one who's aiming to take it so I'd imagine that you're the best person at judging whether or not you can handle a certain text.


Also, try not to take my QFT comment too seriously. It was just a playful jab at physicists going over their heads when using mathematics. ;)

Thanks! I'll be sure to check the library for it and then use that in making my decision.