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    Hi. I'm about to start the second year of my maths degree. I've already chosen my modules but am starting to re-think them a bit. Has anybody done any differential geometry? Would you recommend it? (I like pure stuff like group theory, analysis, algebra...)

    Also, has anybody done any second year stats (i.e bayesian etc..)? I'm not a big fan of stats but have heard it gets better after the first year. Any comments appreciated.

    Cheers.
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    (Original post by Dan R)
    Hi. I'm about to start the second year of my maths degree. I've already chosen my modules but am starting to re-think them a bit. Has anybody done any differential geometry? Would you recommend it? (I like pure stuff like group theory, analysis, algebra...)

    Also, has anybody done any second year stats (i.e bayesian etc..)? I'm not a big fan of stats but have heard it gets better after the first year. Any comments appreciated.

    Cheers.
    What topics of differential geometry are included? I could answer if you pointed me to a syllabus. If you like pure stuff then I think you'll like it.
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    Yeah sure, here's a link to the syllabus .

    Main topics seem to be parametrized curves, frenet formulae, hypersurfaces, geodesics...

    I don't really know that much about differential geometry in general so any information/ comments/ experiences would be a big help.
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    RichE would you say Differential Geomtery is a difficult branch of Mathematics compared to other branches?
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    (Original post by Dan R)
    I don't really know that much about differential geometry in general so any information/ comments/ experiences would be a big help.
    Have you met any multivariable calculus in your first year - chain rule, stokes theorem etc. This will probably be the nearest thing to diff geom which you've met.

    Diff geom is essentially the study of manifolds - a 1-manifold is a curve, a 2-manifold a surface etc. etc. - and the point is that you can, at least locally, put co-ordinates on these curves/surfaces/... to describe them and work out their curvatures, areas, geodesics (= shortest paths on the surface) and so on.

    A hyper-surface is a manifold in R^n given by one equation like a sphere in R^3 etc. Under the right conditions f(x,y,z)=c (which is a level set) will define a surface in R^3.

    A vector field is like a flow on the surface - think a fluid moving around on the surface. The topology of a closed (no boundary) surface determines what flows there can be on the surface with what stationary points.

    A curve has two important intrinsic properties namely its curvature and torsion - they are related by the Serret-Frenet formulae - and they also determine the curve up to rigid motions.

    The Weingarten map contains all the useful information about a surface near a point - where it curves most and by how much.

    The actual intrinsic (=metric) geometry of the surface is measured by the First Fundamental Form which says just how you have bent the co-ordinate plane to make that patch of surface. Things like the curvature (as Gauss showed) and geodesic equations (being metric) can be written in terms of it.

    A minimal surface is one, like a soap bubble, which has a minimal area whilst having a given boundary.

    I don't know if there is too much or too little detail here. Hope it helps. If you have further questions do post them.
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    (Original post by RichE)
    I don't know if there is too much or too little detail here. Hope it helps.
    That's pretty thorough, cheers! I'm starting to get an idea of what it's all about now. Can't say I understand everything you said, but it sounds like it could be interesting. One question though- you say it's a little like multivariable calculus, so does it class as a pure or applied subject? Is it fairly proof based?
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    (Original post by Dan R)
    That's pretty thorough, cheers! I'm starting to get an idea of what it's all about now. Can't say I understand everything you said, but it sounds like it could be interesting. One question though- you say it's a little like multivariable calculus, so does it class as a pure or applied subject? Is it fairly proof based?
    It's pure more than applied - but amongst pure mathematicians geometers do have something of a rep for being hand-wavey. So it might not be as focused on rigour as a course in real analysis say; but that's reasonable given the ideas being discussed are somewhat more complicated.
 
 
 
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