Mathematics Colloquia and Seminars

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I will explain how to construct Lagrangian surfaces in the 4-ball bounding a given Legendrian knot. In fact, I will show that we can construct one such surface per each cluster seed, where cluster seeds are combinatorial pieces of data that can be entirely recovered from the boundary knot. This resolves the surjectivity part of the ADE Conjecture for Lagrangian fillings. At the core of the construction is a new idea studying simple curves in a smooth surface and the polygons they bound.