The Geometry of 3-Manifolds, a lecture by Curt McMullen. This is one of the Science Center Research Lectures; in which Harvard professors talk about their research to the interested public; the series doesn't appear to have a web page, but here's a list of videos available online in that series; these include Benedict Gross and William Stein on solving cubic equations. There are non-mathematical things too, but this is at least nominally a math blog, so I won't go there.
McCullen apparently also gave this lecture at the 2008 AAAS meeting in Boston, and has a couple other video lectures available online.
And now I want a do(ugh)hnut-shaped globe with just North and South America on it. This is a fanciful example of what an "armchair Magellan" might suspect the world looked like if humans had reached the North and South poles starting from somewhere in the Americas but had never crossed the Atlantic or Pacific; they might suspect that the cold area to the north and the cold area to the south are actually the same. McMullen uses to illustrate that tori and spheres are not the same, since loops on the sphere are contractible but loops on the torus are not. The lecture, which leads up to telling the story of the Poincaré conjecture, begins by using this as an example of how topology can distinguish between surfaces.
Finally, here's an interesting story, which may be well-known to some people but wasn't to me: Wolfgang Haken, one of the provers of the Four-Color Theorem, may have intended that (famous, computer-assisted) proof to be a "trial balloon" for a brute-force proof of the Poincare conjecture.