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I was asked to give a talk to a mixture of undergraduate and graduate students on a topic of my choosing from differential geometry. The students will mostly not be in geometry; however, I believe I can assume they know the definition of a manifold, vector fields and integral curves as well as differential forms along with some other basic concepts.

I am looking for some suggestions of concepts in geometry, which can be geared towards a not-so-geometrical audience, that are really attention grabbing. The talk will be pretty informal and an hour long.

The talk can be in other areas of geometry too (Riemannian, Sympletic,...) but accessible enough that I can explain what I need to before getting to the crux, or the cool part, of the talk

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  • $\begingroup$ There are enough topics of this sort for finitists to question if they all exist. Talk about something that you think is accessible for your audience, and that you are personally interested in. Prepare to know a lot more of the material than what you are planning on talking about. If you want topics read some abstracts from the dozens of active "junior/student" weekly geometry seminars in universities around the world. $\endgroup$ – PVAL-inactive Oct 1 '14 at 1:34
  • $\begingroup$ Questions about choice of topics for study/research/exposition are in my opinion way too broad/opinion-based for this site. I voted to close as too broad. If nothing else this is a CW/big-list question. $\endgroup$ – PVAL-inactive Oct 1 '14 at 1:36
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I'd be quite careful about your background assumptions. Very many undergraduates, at least in the United States, don't learn the definition of a manifold during their degree-so your assumptions are only accurate if they've all taken an advanced geometry course.

As for advice, a first idea is to talk about the theorema egregium. The independence of a manifold from its embedding is one of the most important conceptual contributions of the twentieth century in geometry. For a different tack, you might consider focusing the talk around transversality and all the spectacular results that follow from it, along the line of Guillemin and Pollack. I find this stuff immensely more beautiful than the better known homological techniques that get the same results, and it's very amenable to pictures, to legitimate proofs in a short time span, and is again conceptually of the utmost significance.

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