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Looking for books on group theory and differential geometry

  • Thread starter Theaumasch
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Main Question or Discussion Point

My university doesn't offer many courses on theoretical physics (I'm studying applied physics), but because I might want to get my masters degree in theoretical physics, I want to read into some of the math and physics.

What books would you recommend to a student who has had linear algebra on the level of Anton and Rorres, calculus on the level of Adams and some introductory differential geometry during an introductory course on general relativity?
 

Answers and Replies

  • #2
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Do you want rigorous math books that treat the subject from the perspective of a mathematician, or do you want books that talk about groups, etc in terms of physics?

I'll be honest, Im a math and physics major, and thus far no physics class Ive taken has required the math rigor that a math class provides, and most physics books that I have come across that do make use of abstract math tend to present the physics at a level far higher than what I can understand at the moment.

My opinion is that, if you learn the rigorous and "weird" abstract math NOW, then when it comes time to apply is in the context of physics, you will be able to spend most of your time understanding the physics, and not have to spend as much time trying to learn the math AND the physics.

With that in mind, try Fraleigh's "A First Course In Abstract Algebra." Some will suggest other fine books on algebra (like Dummit and Foote), but if you are self teaching and if this is gonna be your first trek into real, rigorous, abstract math then those books will probably be a bit much for a first time reader. Fraleigh is about the easiest formal textbook to learn Group THeory from. Once you master that, then you can definitely get a whole lot out of deeper and more rigorous books.

Dont have a suggestion on Diff Geo specifically, but my guess is that any serious treatment of Differential Geometry will require some familiarity with Real Analysis. So pick up Ross's "Elementary Analysis." Same idea as the algebra book. People will suggest other books that arefar more in depth, and far more rigorous, but again if you are new to rigorous math and you are trying to self teach yourself, books like Rudin might be a bit much. Again, Ross is the easiest. Once you master Ross, you can get a lot out of Rudin.

You'll probably want a serious treatment on Linear Algebra as well (Anton is elementary, and Im not familiar with the level of Rorres). Friedberg is a good place to start. Same idea as above. Master this first, then move on to others.
 
  • #3
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Thanks for the comment! I searched for all the books in my university's library and only the book by Friedberg wasn't found :) I guess I'll take a look today.
 
  • #5
Fredrik
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I've read two of those books by Isham: The ones about quantum mechanics and differential geometry.

About the QM book: It's a very good presentation of QM that's tells you more than most books about what the theory actually says, but very little about how to calculate stuff. So it's about understanding the theory, not about how to use it. I think it can be studied by people who haven't studied QM before, but I think it's intended mainly for people who have taken one of the standard introductory courses already, e.g. a course based on Griffiths. I highly recommend it, regardless of what you have studied before.

About the differential geometry book: This is very nice introduction to some aspects of differential geometry. I really liked how he covered tangent spaces for example. It also has a very nice introduction to Lie groups, Lie group actions, and fiber bundles. The level of mathematical rigor isn't as high as in a math book. He will often not bother to prove that a function has all the properties we need it to have. If we e.g. need a smooth bijection, he'll just prove that it's a bijection.

The goal isn't to teach the differential geometry that you need for general relativity. It's to teach the differential geometry you need to understand Yang-Mills theory. I think he has successfully accomplished what must have been one of his goals: to make his book the best place to start a journey towards Yang-Mills. But I think it ends at a weird place. I would have wanted to see two more chapters or so that cover differential forms, integration on manifolds, and field theory Lagrangians, so that the reader actually gets to the point where he understands the Yang-Mills Lagrangian.

The last chapter, about connections and curvature in fiber bundles, isn't as well written as the earlier chapters. I got the impression that he was in a hurry to finish the book. It takes a lot more work to understand what he's saying there, but it's not just because of the writing. It's the most difficult subject in the book, and it's probably still easier to read about it in this chapter than in another book. All things considered, it's a very good book, and I highly recommend it to everyone who wants an introduction to the things I mentioned above.

However, I think a person who wants to learn the differential geometry needed for general relativity should study Lee instead. Read his "Introduction to smooth manifolds" to learn about manifolds, tangent spaces, tensors, differential forms and integration on manifolds (and also Lie groups if you're interested in that), and read his "Riemannian manifolds: an introduction to curvature" to learn about connections, parallel transport, geodesics and curvature.

For group theory, consider the book by Brian Hall. It focuses on matrix Lie groups, the corresponding Lie algebras, and representation theory. By focusing on matrix Lie groups, he's able to avoid differential geometry altogether.

For linear algebra, the most popular books around here are the ones by Axler and Hoffman & Kunze. I have only read the former, and I think it's really good, and covers precisely the topics I'm interested in (the stuff you need for quantum mechanics). I've been told that Hoffman & Kunze covers a wider range of topics and that their explanations are more detailed. I don't like Anton, because he doesn't define linear operators until around page 300. I find that completely ridiculous.
 
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  • #6
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The very best introduction to group theory I have seen is Pinter's "A book of abstract algebra". And it's quite cheap too. The positive side of the book is that it's very intuitive and that it doesn't read like a textbook: it's like the author is telling you a story. There is also a wealth of exercises which are not to hard and not to easy (and which I encourage you to make all).

You won't need to do all the book, just the first half of the book should be sufficient for an introduction of group theory. The rest of the book is spent on field theory and Galois theory.

For a second course in group theory, you will be best with Dummit and Foote. This books contains much information in group theory and other algebras. But like somebody else already mentioned: it's not suitable for a first course...


For the differential geometry. This is probably a little hard. The most differential geometry books will allready assume that you know topology and real analysis. A book which will NOT assume this is "A comprehensive introduction to differential geometry" by Spivak. It's really a great and comprehensive book. Although the book "calculus on manifolds" by Spivak is considered a prerequisite. So I don't know if you're ready for it...

Another book which assumes no topology or real analysis is "Differential Geometry of Curves and Surfaces" by Do Carmo. It's not as comprehensive as Spivak's book, but it does the job...
 
  • #7
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here's a free intro to diff geom that i recommend. Shifrin was a student of the great differential geometer S.S. Chern:

http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf [Broken]


here are my free algebra books, not as good as Shifrin, but still free:

http://www.math.uga.edu/~roy/ (items #3, #6)


I made a very quick survey of other much more beautifully formatted free books online, but did not find one that is as fully explained as mine. I did not look long, there are surely many of them. I apologize for the primitive type fonts in my works, but they do seem to help people willing to read them in old style type

If you are willing to spend money, I recommend M. Artin's Algebra, as much better than mine, and costing about $50, a bargain really.
 
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  • #8
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In the post link below, I try to pull out the minimal set of math and physics ideas, backed up by the actual history, that underlie the knowledge needed to navigate from junior level math/physics through graduate school and beyond, including the current methods in theoretical physics.

[Included are texts/references on algebra, topology, geometory and topology for the theoretical physicist in an uploaded Word document].

I list and review a core set of the best, clearest books and literature to this end, often including what you should get from each book/article. I probably would have saved about a decade, and lots of money had I had a "syllabus" like this.

https://www.physicsforums.com/showthread.php?t=553988

Thanks,

A. Alaniz
 
  • #9
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