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11/12/2007, 14:30 — 15:30 — Room P9, Mathematics Building

Marcelo Viana, *Instituto de Matemática Pura e Aplicada (IMPA)*

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Dynamics and geometry of flat surfaces

Flat surfaces carry a metric whose curvature vanishes identically, outside a finite set of conical singularities. Geodesics correspond to straight lines, in appropriate local coordinates, and so the local picture is completely trivial. However, the global behavior of geodesics is suprisingly rich! I'll discuss some main notions, and two results in this field: the Masur-Veech unique ergodicity theorem, and the Zorich-Kontsevich asymptotic flag conjecture, proved by Avila and myself, following important partial results by Forni.