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Abstract: Much is known about the dynamics of the mapping class group on different spaces: Teichmüller space, the space of singular measured foliations, the space of geodesic currents. However, very little is known about its effective dynamics. In this talk I will discuss work in progress that aims at clearing up this picture. Applications to counting problems on surfaces, including a partial solution to an open problem of Wright, will also be discussed. No previous knowledge of any of these topics will be assumed.