Questions about polygonal billiards lead to the study of moduli spaces of translation surfaces or "strata". There are intriguing parallels between strata and homogeneous spaces and there has been a fruitful migration of ideas from homogeneous dynamics to "Teichmuller dynamics" on strata. One pivotal question is the extent to which Ratner theory applies to strata. I will discuss positive results of Eskin and Mirzakhani and describe negative results obtained with Barak Weiss.
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