Here's a really interesting paper on knot theory. Specifically, a categorification of the Jones polynomial, which remains totally mysterious to me. I don't understand much, but my mind is already blown by his interpretation of knots as sorta-like cobordisms. Now cobordisms aren't actually scary: a typical example is a pair of pants. Generally a cobordism is like the shape you might get from a rubber sheet or soap film or something, stretched between two wiry shapes at either end.

The idea with knots is that the crossing

\ /
 /
/ \

somehow stands for the saddle-surfacey cobordism that goes from

\   /
 | |
/   \

to

\_/
 _
/ \

and suddenly the bizarre Jones polynomial skein relations make just a little bit more sense.