The associahedra form a family of convex polytopes which realize the Tamari lattices of planar binary trees. They have numerous algebraic, combinatorial, topological and geometrical properties. For instance, they allow one to recognize loop spaces, they encode the notion of an associative algebra up to homotopy, and their toric varieties provide us with a non-commutative version of the moduli spaces of genus 0 stable curves. In this talk, I will explain what is the problem of the diagonal of the associahedra algebraically with the operadic calculus, and combinatorially in terms of cellular decomposition. I will show why this problem is crucial: its resolution allows us to consider the product of A_infinifty-categories (symplectic topology), the tensor product of string field theories (Zwiebach), and to perform new homological computations of fiber spaces. This talk will be as elementary as possible including meaningful pictures. (Joint work with Naruki Masuda, Hugh Thomas and Andy Tonks available at ArXiv:1902.08059.)

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