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Real Analysis vs Differential Geometry vs Topology

  1. Apr 19, 2008 #1
    I would just like to know which of these math courses is best suited for physics. I have taken advanced calculus and linear algebra, so I've seen most of the proofs one typically sees in an intro analysis course (ie. epsilon delta etc.). I intend to do work with a lot of Quantum Field Theory, and maybe try out general relativity (though not to specialize in it, just to see why its not comptable with qm). I don't intend to study string theory, atleast not as part of my work. I'll also mostly be dealing with macroscopic, e&m, and perhaps stat mech.

    I am sure all three are beautiful math subjects, and I independently intend to learn all of them. But for practical reasons, I can only choose to study one. I heard diff geometry is used often and not just in GR. I have rarely heard analysis or topology being applied to physics.
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  3. Apr 19, 2008 #2


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    All three are important. Both Real Analysis and Differential Geometry lead to Topology. If you can, take all three: RA teaches about point-set topology, measure theory and integration, metric spaces and Hilbert (&Banach) spaces, and .....; DG is, in many respects, GR without the physics, and Topology is about the structure of spaces -- including those used in current physics research.

    Talk to your professors, and fellow students. Are the teachers good ones? Do the courses help with physics, or are they geared toward math students. Find out all you can before making any decisions. Good luck.

    Reilly Atkinson
  4. Apr 19, 2008 #3
    The answer is definitely Differential Geometry, especially when you want to do QFT, where it is widely used. It will also give you much insight in other subjects (apart from the obvious GR), like classical mechanics, electrodynamics, advanced QM,... it's everywhere.

    Real analysis might be also useful, but it depends on what exactly is in the syllabus. Measure and integration theory aren't that interesting for physicist, but theory of Banach and Hilbert spaces, spectral theory and distributions are frequently used, not only in QM.

    I wouldn't consider topology, if you're not planning to do string theory.
  5. Apr 19, 2008 #4


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    Can you even take differential geometry without having taken topology? I thought that was generally required especially if its a grad class.
  6. Apr 19, 2008 #5


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    I thought Einsteins idea was to translate physics into differential geometry. analysis and topology are more like foundational underpinnings for differential geometry.

    so i would take the diff geom and learn whatever analysis and topology are needed to understand it.

    as spivak says in his great differential geometry book, when he discusses pde, "and now a word from our sponsor".
  7. Apr 19, 2008 #6
    Elementary diff geometry. Nothing higher than R^3. Im going to be a junior.
  8. Apr 19, 2008 #7
    This thread has made me reconsider what I plan on doing as far as math courses go. I'm a freshman Physics major and I plan on minoring in math, and there are several "tracks" I can go down to pursue this. Right now what I'm most interested in doing in grad school is plasma physics, but that can change of course. Currently I'm on the Geometry and Topology track, but I haven't taken any classes towards it yet so I can still change.

    The classes I need to take for Geometry and Topology are Diff. Geometry, Intro Topology, and Intro Algebraic Topology (along with 3 other 3000+ level classes, one of which would have to be Analysis 1 because the Topology classes require it).

    Would I be better served by doing the Differentials and Dynamics track, where I'd end up taking Partial Diff Eq 1 and 2, and Dynamics and Bifurcations 1, along with 3 other classes. Should I do something like Partial Diff Eq 1 and 2, Dynamics and Bifurcations 1, and then maybe an undergrad and grad level Diff Geo. class with 1 other class (Hilbert Spaces maybe?). Would Analysis still be useful?
  9. Apr 20, 2008 #8
    I am no expert but it seems that Differentials and Dynamics would be better suited to plasmsa physics. That being said thouh, could you do the Differentials and Dynamics track and use your three elective classes to take Analysis 1, Intro to Topology and Diff. Geo?
  10. Apr 20, 2008 #9


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    I was unaware that QFT used Differential Geometry. Could you explain when it pops in?
  11. Apr 20, 2008 #10


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    I don't think it should matter to him if it's geared to mathmemticians or physicists, I think it will be best if it were geared to maths grads, cause you need to first grasp the definitions and theorems especially if for example the topology course is your first.
    usually maths courses taught by physicists lack rigour and without rigour you don't really undersatnd what's going on there, and you cannot really appreciate the applications.
  12. Apr 20, 2008 #11
    Yup, I could do that. Thanks for the advice.
  13. Apr 22, 2008 #12
    Real Analysis especially Functional Analysis is what you want to study for quantum mechanics. Luckily this will give you intuition to study topology which will give you the necessary back ground for manifold theory. To me it really depends on what kind of courses you are talking about. If you have the ability to take a graduate level course in Real Analysis (measure theory and/or functional analysis) then I would say do that. But, if you are taking about functions of a single real variable verses a course on differential geometry in R^3 (both undergrad courses), then I would say differential geometry will be the most beneficial. I don't understand why topology would be desirable (unless you mean topology at a very high level) unless you need it to talk about measures on locally compact topological spaces or advanced manifold theory (which I think a half a year or a year of topology at the graduate level would be enough for those topics).
  14. Apr 22, 2008 #13
    I would take that then and audit a course on functional analysis.
  15. Apr 23, 2008 #14
    Given his background, a course in functional analysis might as well be taught in Klingon.

    To the OP, personally, I think you ought to take the analysis course. If you're going to be doing anything quantum, you'll need functional analysis at some point; so you'll need intro analysis (i.e. baby rudin) as well as measure theory and some Lp space theory before you get there.

    I think that undergraduate differential geometry courses are a waste of time; it's multivariable calculus, that's all. I understand the importance of building intuition, but it doesn't even prepare you for the difficult part of differential geometry.
  16. Apr 23, 2008 #15


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    Again, when do L^p spaces pop up in QM? I'm very curious.
  17. Apr 23, 2008 #16
    It comes up when you talk about fourier transformations of Lebesgue-Integrable functions. The question is which functions have fourier transformations and which functions are a fourier transformation of function. You need to carefully discuss L^1 and L^2. But, also your states are elements of L[tex] $ ^2(-\infty,+\infty)$ [/tex], and your operators--position, momentum, and observables--are from some domain to L[tex] $^2(-\infty,+\infty)$ [/tex]. So, knowing some facts such as L^P completeness or L^P continuity, et cetera, et cetera would help.

    I think an audit is entirely reasonable if you have time, but I wouldn't expect to get too much into QM in the class. Rather, that is something you will have to do on your own with functional analysis.
    Last edited: Apr 23, 2008
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