I know this is not a video, but it is well worth reading: http://www.ams.org/notices/200503/fea-weil.pdf
A wonderful illustration of biology. Must read for every high school student. http://highered.mcgraw-hill.com/sites/dl/free/0072886161/323275/samplech04.pdf
I think the double slit result captures the weirdness of particle/wave duality the best. It is just so non-intuitive.
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This set of videos give a real glimpse of what mathematics looks like at the professional level. It is rough going, but if you go through all six videos at least you can understand the statement of the Hodge Conjecture: http://www.youtube.com/watch?v=gIi92JSZ9J4
http://www.youtube.com/watch?v=lWZ2Bz0tS-s Videos like this make learning fun, and they gently introduce some very deep ideas. I challenge even the most dull person to view this entire series and not come away buying "Godel Escher Bach" http://www.amazon.com/Godel-Escher-...=1368574816&sr=8-1&keywords=godel+escher+bach
You can be hot and a mathematician <iframe width="640" height="360" src="http://www.youtube.com/embed/LSxqpaCCPvY?feature=player_detailpage" frameborder="0" allowfullscreen></iframe>
Lisa Randall never dissapoints. Very soft science, more on the sociology of science, but well worth listening to. <iframe width="640" height="360" src="//www.youtube.com/embed/yDHta5Fq93M?feature=player_detailpage" frameborder="0" allowfullscreen></iframe>
I have become a big fan of Khovanov Homology: <iframe width="640" height="390" src="//www.youtube.com/embed/filQcRSBhFw" frameborder="0" allowfullscreen></iframe> It is really interesting how he used "Categorification" http://en.wikipedia.org/wiki/Categorification and extremely important concept in modern mathematics where we go from set theory to categories. Khovanocv abstracts relatively simple invariant, the Jones Plolynomial, in order to get a far more general invariant using Categorification. It also shows in a crystal clear the power of Grothendieck's methods. Really, imo it is also one of the most accessible abstract forms of mathematics since much of the math is constructive (algorithmic) and aided by simple diagrams. My interest in this is the interesting applications it has to quantum computers, and the ties to quantum mechanics in general: <iframe width="640" height="390" src="//www.youtube.com/embed/8nA17Id4JyU" frameborder="0" allowfullscreen></iframe> <iframe width="640" height="390" src="//www.youtube.com/embed/vykoKInjuPY" frameborder="0" allowfullscreen></iframe>