Understand the geometry of a black hole horizon by falling into one:
For #sciencesunday this great video by +Tony Darnell
conveys a fantastic amount of visual information in a compact way.
Black holes are simultaneously the most simple and most complex possible objects in the universe. Simple because the no-hair theorem guarantees their lack of any distinguishing characteristics besides charge and angular momentum, and complex because they represent the maximum density of internal states – the maximum entropy – that can ever be squeezed into a given volume of space. And surprise: the number of states is proportional to the area of the horizon, not the volume.
To compress a lot of info myself, here is a lecture by a guy who made a significant contribution to understanding the nature of that entropy, Andy Strominger:http://physics.berkeley.edu/events/Colloquia/movies/col.streaming.3-12-12.mov
Black holes have rapidly become arguably the most important system to understand in all of physics, just like the harmonic oscillator. Andy recalls how as a grad student he was tasked to understand the strong nuclear force by turning it into harmonic oscillators. Now Andy's the professor and he tells his PhD students to understand the strong force by turning it into a black hole.
Its an amazing anecdote to summarize some of the deep mathematical relationships that have emerged between black hole physics, including their speculative high energy completion to string theory, and many much more accessible scientific questions such as QCD, superconductors, and classical fluid dynamics.