Billiards in polygons, flat surfaces, and dynamics in moduli spaces

We will start with a discussion of billiards, including Sinai billiards and Ehrenfest wind-tree model. The study of billiards would naturally lead us to flat surfaces and to dynamics in the moduli space of Abelian differentials. I will finish by trying to explain, why the recent Magic Wand Theorem of Eskin-Mirzakhani-Mohammadi and Filip is so astonishing.