Dynamics in Teichmüller spaces is a subject that goes back to the work of Thurston. For the past fifteen years, this field has grown considerably, with the contribution of many leading mathematicians. The outstanding recent work in rigidity of the horocycle flow, due to Eskin and Mirzakhani, is an absolutely fantastic advance that open new directions of research. This result was announced in 2012. Part of the current work uses their ideas to solve problems that were out of reach until then.

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