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Currently trying to find a topic for a roughly fifteen page paper on Differential Geometry with presentation, with the rough target being a second year graduate student audience.

I was looking in particular for some interesting problem that can be efficiently solved using differential-geometric techniques, or at least something suitably categorical. Any ideas? I was considering discussing categorical generalizations of Lie algebras.

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The four-vertex-theorem or brachistochrone curves.

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How about the stories of world discoverers in the fifteenth centuries era, why lots of them did not end up where they though they would be. What is the definition of "straight" on a sphere? Relate it to the world which is not flat. Put it together with loxodromes (you should know what that is!) and find a mathematical mechanism that allows you to set up a path with given starting coordinates on the sphere and desired end coordinates; path having a fixed angle with due north (they only had compass remember?). There comes in your Diff Geometry theory. Lots of path from the past were rhumb lines, but that had its own problems. And why were loxodromes easier to follow than great circles? This is something you can explore with Diff geometry and relate to your audience

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This sounds more like something for undergraduates, not second year graduate students. @Cameron: You don't tell us what your background (or the background of your audience) is in terms of graduate-level differential geometry. Since you mention Lie algebras, maybe you might study the invariant connections and curvature on symmetric spaces. – Ted Shifrin Nov 18 '13 at 17:25
Seriously, that all depends how deep you want to go into spherical material. A good source for this would be McCleary's Differential viewpoint on Geometry. A book used for professional math studies. It can be broken down to various levels accordingly. – imranfat Nov 18 '13 at 17:57
As I said, this is undergraduate level. McCleary's book is a low-level undergraduate text. I have taught out of it and it is a more elementary level than I ordinarily teach our juniors and seniors that material. – Ted Shifrin Nov 18 '13 at 19:18
I don't know why you're making a federal case here. The OP specifically said second-year GRADUATE level. At this stage, I'd like to know more about his background and that of his intended audience. – Ted Shifrin Nov 18 '13 at 19:41
I deleted my comment... – imranfat Nov 18 '13 at 20:57

You could make an introduction to Morse theory finishing with the Reeb theorem, which states that a compact manifold admitting a Morse function with only two critical points is homeomorphic to a sphere. This would be the first three sections plus theorem 4.1 of Milnor's book. If that were too short, you can go further in any of the multiple directions Morse theory has developed.

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