I suggest something different! $(H^1)^* = \text{BMO}$. – Jonas Teuwen Jan 8 '12 at 18:15 We're only now getting into Lebesgue measure, so that's probably a bit too advanced. Thanks, though. – FPP Jan 8 '12 at 18:25 How about giving a rigorous discussion of the Banach-Tarksi paradox and explaining why it doesn't violate conservation of area. Also might discuss why it is impossible to construct a set function on the power set of $\mathbb{R}$ satisfying the four properties: (i) It should be defined for all sets in the power set; (ii) for an interval it gives its length; (iii) it is countable additive; (iv) it is translation invariant. (It isn't known whether there is one satisfying (i)--(iii), but continuum hypothesis implies that there isn't). – William Jan 8 '12 at 18:48 PS. I only mention these because I did a little presentation on that when I was an undergrad :). – William Jan 8 '12 at 18:49 Hah, it's funny that you mention Banach-Tarski. Unfortunately, someone already grabbed that topic before I even had a chance to do some research on it :(. Good suggestion, though, thanks. That's the kind of thing I'm looking for (but maybe related to non-standard analysis or ordinals/cardinals). – FPP Jan 8 '12 at 19:07