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How and do graduate students know a LOT?

  1. Sep 26, 2008 #1
    Hi guys... I'm just wondering, I'm taking this course about group theory in physics. The lecture sometimes really goes berserk.. by berserk i mean the professor starts talking about cohomology, complex manifolds... Sometimes he says somethings that I've never heard about (for instance, Young—Baxter equations, torsors and fibrations). The prerequisites for that class is merely a year of graduate quantum mechanics (no QFT). In that class I can really feel that sometimes... pretty much only one or two people are really following...


    So, my question is, what exactly is generally expected from a grad students (specially the ones going in theory)? should he/she know a good deal of differential geometry and topology? Where do they generally learn those things? just reading books? Or do they just dive right into crazy ideas and try to understanding the most complicated stuffs without any background (like in this class)?
     
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  3. Sep 26, 2008 #2
    I'm not in grad school, but I knew very early on that if I wanted to do theory I'd better have a math major on the side. Group theory is basically abstract algebra. As a grad student, I'm told to know abstract algebra, basic topology, basic differential geometry, and analysis. Otherwise have fun picking this stuff up in grad school.
     
  4. Sep 26, 2008 #3
    I think grad students are expected to "fill in the blanks" as they see fit. So, maybe you don't have to learn some complex geometry, but if you do, you'll be much better off doing it.

    It's best to hold yourself to your own standards. You should expect more from yourself than what others expect from you. If you meet your own (should be high) standards, you don't have to worry about anything else.

    The way I see it, eventually you will need to learn advanced geometry and topology concepts for theoretical physics. Start now and you'll get that much better with it when it comes time to start your thesis and your adviser doesn't have to assign you Milnor or Munkres.
     
  5. Sep 27, 2008 #4
    "Disciplined Minds" by Jeff Schmidt tells you exactly what you need to know. Basically, you need to know how to do the questions on the exam papers from previous years. Your "beserk" professor is throwing everything at you to try to frighten you out of physics. There are too many pigs at the trough, and some kicking is needed to keep back the crowd :-) Just look like you think you know what's going on, and make sure you can do all the questions from previous exam papers. This doesn't mean reading many books -- just the set textbooks that tell you how to do questions from exam papers. Schmidt tells a story about a student who was (rightly) dismayed by this "hoop jumping" approach and read lots of books instead - he got kicked out because he didn't know the right tricks for doing the usual exam questions.
     
  6. Sep 27, 2008 #5

    Vanadium 50

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    He or she is expected to, as said before, fill in the gaps, and also to interrupt the professor if he or she isn't following the material.

    The grad student experience is the transformation from "student" to "colleague" and colleagues will interrupt if they don't understand. (Students, however, will do it more gracefully!)
     
  7. Sep 27, 2008 #6

    Moonbear

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    Indeed. OR, grad students should have the skill set that allows them to identify topics in lecture that they do not know or understand, and look up the details independently. I think of graduate level lectures more as overviews or "tours" of a subject, while the real learning is done independently based on the sights you stopped to see in lecture that the professor basically told you are important ones to know at your level.
     
  8. Sep 27, 2008 #7

    mathwonk

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    you are expected to ask questions, and make notes of things said that you do not understand and go look them up afterwards. i.e. you are supposed to keep learning. it also helps to recognize nonsense and cynicism and ignore it, as epitomized in some earlier posts here.

    a manifold is just a smooth curve or surface or solid of some dimension or other on which one can use calculus. a complex manifold allows one to use complex calculus.

    it helps to know about integrating forms like Pdx + Qdy over paths. such a thing is called a one form, as a gadget like Adxdy + B dydz + C dzdx is called a 2 form and is integrated over a 2 dimensional surface.

    In general objects that are integrated over k dimensional solids are called k forms. There is a differentiation tht takes k forms to k+1 forms, e.g. d(Pdx + Qdy)
    = (∂Q/dx - P/dy) dxdy. This is defined to make the statements of the usual theorems of Green, Stokes etc, uniform.

    A closed k form is one that has zero integral over the boundary of small k+1 cubes. Then cohomology measures the failure to have zero integral over larger k dimensional solids

    I.e. two closed k forms are cohomologous precisely if they give the same integral over every k dimensional solid. Dually two k dimensional solids are homologous if any two closed forms have the same integral over them.

    It is a related fact that ∂^2 = 0 on any form. This allows abstract generalizations of cohomology to be defiend in any setting where there is an algebraic map with this property. i.e. this just says you have an operation whose image lies in its kernel.

    we then define the associated "homology" or cohomology" group to measure the failure of the image to equal the full kernel. i.e. the homology group equals kernel/image.

    any differential operator with d^2 = 0, yields such a group, namely those functions or fields satisfying d = 0, modulo those which are d of something. so cohomology groups measure the extent to which the necessary condition dG=0 for solving some diffeq dF = G, is sufficient.
     
  9. Sep 27, 2008 #8
    wow...i almost understand that. is that where i keep seeing that chain partially ordered by d that i can never quite visualise?
     
    Last edited: Sep 27, 2008
  10. Oct 3, 2008 #9
    I apologize for the lack of responses. Thank you all for the comments. I suppose I'll just have to spend more time (than a couple hours per day plus a full weekend ) to learn all these stuffs. Enough ranting I suppose. Now... if anyone can explain to me what the heck a quantum group is...and what they are good for...
     
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