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We shall discuss associative binary structures and analyze why such a structure
is not a homotopy invariant. This naturally leads to higher associative structures, often
called $A_\infinity$ structures, which are prevalent in topology, geometry and mathematical
physics. We shall discuss a recognition principle for $A_\infinity$ spaces. We will explain
why certain polytopes, which are of great combinatorial interest on its own, govern maps
between $A_\infinity$ spaces. The talk will be self-contained.
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