Solid, liquid, gas – we see these phases of matter all around us. But physicists have discovered exotic phases with strange properties, such as superfluids and superconductors. What kinds of phase transitions happen in an extremely cold, thin sheet of matter? What does this have to do with the field of math known as topology? In this talk we’ll see how three physicists won a Nobel Prize for answering this question. This talk will be accessible to all undergraduates (in particular, no physics background will be required).