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Math Becoming a Mathematical Physicist

  1. Jul 22, 2009 #1
    I hear all these topics like Lie Algebra, Symplectic Geometry, Hopf Algebra, Clifford Algebra, Quantum Groups, Homology, Poisson Algebra, Semi-Riemannian Geometry and the list goes on..

    As of the moment, I honestly have no idea what they are. But probably these topics are related or are even sub-fields of each other. Now as I've browsed through books/papers discussing these topics, I can say that such kinds of stuff (mathematically rigorous and related to physics in an abstract manner) really interest me. I already find it a pastime to browse through such books in my university library even though I don't really understand anything my eyes set on. Maybe people will find this weird or odd. But doing this gives me some kind of motivation and excitement to tell myself "wow, I want to say to myself that Iunderstand this stuff". ^^

    I want to get to a level where I can understand and do research in these topics. My present mathematical background is of a beginning graduate student in physics. This includes analysis, vectors, ODE, some PDE, complex analysis, special functions, linear algebra, calculus of variations, and some tensors. Most of these topics are quite "calculational" or algebraic in approach. The paths towards these topics are quite clear and one can just follow the standard prerequisites in sequence. I want to make a transition from this kind of mathematics to a more geometric and abstract one like the topics I mentioned above. But for the "higher" topics above, I simply don't know where to start and what path/s of prerequisites to follow. Can someone help me with this?

    Thank you very much!
  2. jcsd
  3. Jul 22, 2009 #2
    Of the things on your list, I would advise getting a decent understanding of Lie algebras first, since this is a good place to start for learning about Hopf algebras and quantum groups.

    Good books for learning about Lie algebras include

    * Fulton & Harris: Representation Theory: A First course,
    * Georgi: Lie Algebras in Particle Physics,
    * the book by Cahn (which is available here:
    http://phyweb.lbl.gov/~rncahn/www/liealgebras/book.html ) and
    * Humphreys: Introduction to Lie algebras and Representation Theory.

    (You could spend a year working through these books - also maybe you'll want to know about Lie Groups too, but I guess maybe you already learnt about them because you know about PDEs?).

    Quantum groups are Hopf algebras, so you can put those two together.
    They're part of an attempt to create a non-commutative analogue of Lie groups as used in classical mechanics. I like the book "Quantum Groups" by Christian Kassel for learning about them, and eventually I guess you can read Drinfeld's paper "Quantum Groups".
    The reason why I recommended learning about Lie algebras first is that quantum groups are usually obtained by deforming either Lie algebras or Lie groups.

    I think it's best to learn Homology once you've got some motivation because otherwise it's a bit dry. It comes up a lot in algebraic geometry, which isn't on your list.

    Have you learnt about Riemannian geometry already? Also, if you want to get used to abstraction, perhaps you could learn about category theory.

    Your list is pretty broad, so I guess maybe different people will help with the other parts which I don't know anything about. It could take years for you to learn all that! (But I guess most people are quicker than me). At the moment my personal favourite book for looking at and saying ""wow, I want to say to myself that I understand this stuff" is the book "Representation Theory and Complex Geometry" by Chriss and Ginzburg.
  4. Jul 23, 2009 #3
    These are the kinds of topics we studied in my undergraduate and graduate programs in pure math. Especially lie algebras- you will study these in abstract algebra. I suggest getting a minor or if you can a double major in pure math.
  5. Jul 24, 2009 #4
    Physics courses would cover them too. I guess you could learn about Lie algebras in a course on particle physics, semi-Riemannian and symplectic geometry in a course on GR, and quantum group stuff in a course on quantum integrable systems.
  6. Jul 24, 2009 #5
    I wouldn't rely on physics courses. Some Lie group/algebra theory is usually covered in a good QFT course, but you essentially just learn what the representations of the SU(n) groups are. The most beautiful thing on Lie groups - the geometric picture - is rarely covered.

    I also doubt that symplectic geometry is covered in any way in a GR course (you would more likely encounter this in a String Theory course).

    If you really want to do Mathematical physics and go in the direction of String Theory or QFT (and some topological/geometrical methods therein), I would highly recommend taking Differential Geometry, Lie groups and Topology first, and then you'll see what your interests really are.
  7. Jul 24, 2009 #6
    Thanks Tiger99 for the advice on Lie Groups and some of the other stuff. Um yeah I haven't learned about Riemannian Geometry yet, but I guess it proceeds after I learn differential geometry. As for algebraic geometry, what are the prerequisites for it? Is it the same as geometric algebra? Because one of the professors around here is a guru in geometric algebra and he has been promoting it a lot to almost everyone.. just a thought.

    Just to make it clear, starting with Lie algebras will take me on the way to Hopf algebras and quantum groups. Do Lie algebras and Lie groups go together in one package as their common names have?

    Hi Martin, thanks also. Now starting with differential geometry, topology and Lie groups will lead me to some of the "higher" topics I mentioned. Can I learn these three simultaneously? Or should I take them in some order?

    Oh, and I don't really wanna go into string theory. ^^ So I guess I'll be going for the QFT one. I would be rather vague because I still don't know a lot, but I feel that I want to go in a direction that deals with the deeper mathematical structure of "what's already working" like QM or GR or QFT, etc.

    Is symplectic geometry also used in fields other than string theory?
  8. Jul 24, 2009 #7

    Lie groups and Lie algebras are very closely related, but you can study one without knowing much about the other. (I don't know much about Lie groups). Lie groups are differentiable manifolds with a group structure, and then the tangent space at the identity of the group is a Lie algebra. But I think the study of Lie groups tends to be more geometric, whereas the study of Lie algebras relies a lot on linear algebra and can be quite abstract.

    I don't know much about symplectic geometry but I think (??) it was originally developed for classical mechanics (in the Hamiltonian formulation) so I guess it must have a wide range of applications.

    I think geometric algebra and algebraic geometry are very different. Algebraic geometry involves the study of varieties (the zero sets of polynomials) and is quite nice in the beginning because you can draw graphs and so forth and be thoroughly grounded in geometric intuition, but then it evolves to be extremely abstract.
    The prerequisite for studying it is commutative algebra (rings, ideals, Spec, ...) - there's a book called "Commutative Algebra with a View Towards Algebraic Geometry" by Eisenbud. (As a disclaimer I actually know very little about algebraic geometry and nothing about geometric algebra.)

    That said, if there's an expert in geometric algebra at your university who wants to teach people about it, then you should probably take advantage of that! They'll be able to teach you much more quickly what is important and guide you straight towards the cutting edge of research, so you can see if you like that area by trying it out.
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