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Flat Surfaces

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to be isomorphic to the moduli spaces of holomorphic one-forms. One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmuller geodesic flow and under the linear group action. In particular, the Teichmuller geodesic flow plays the role of a time acceleration machine (renormalization procedure) which allows to study the asymptotic behavior of interval exchange transformations and of surface foliations. This long survey is an attempt to present some selected ideas, concepts and facts in Teichmuller dynamics in a playful way.