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Beschreibung:
This thesis presents several developments related to the associahedron. All results are motivated by two specific problems. The first one, which was completely solved in this work, concerns some polytopal realizations of associahedra (Chapter 1), while the second one is about the existence of polytopal realizations of multi-associahedra. Although this second problem was not solved in the thesis, it served as an starting point for very interesting results connecting subword complexes in the study of Gröbner geometry and cluster complexes in the theory of cluster algebras (Chapter 2). These results provide a new approach and new perspectives for problems related to multi- associahedra and, in a more general context, to generalized multi- associahedra. For example, we use this approach as a tool to produce polytopal realizations for small explicit examples (Chapter 3). The thesis is subdivided into three chapters. The first chapter is focused on geometric realizations of the associahedron, and is joint work with Francisco Santos and Günter M. Ziegler [15]. We show that three systematic construction methods for the n-dimensional associahedron (as the secondary polytope of a convex (n+3)-gon by Gelfand, Kapranov and Zelevinsky, via cluster complexes of the root system A_n by Chapoton, Fomin and Zelevinsky, and as Minkowski sums of simplices by Postnikov) produce substantially different realizations, for any choice of the parameters for the constructions. The cluster complex and the Minkowski sum realizations were generalized by Hohlweg and Lange to produce exponentially many distinct realizations, all of them with normal vectors in $\\{0,\pm1\\}^n$. We present another, even larger, exponential family, generalizing the cluster complex construction --- and verify that this family is again disjoint from the previous ones, with one single exception: The Chapoton--Fomin--Zelevinsky associahedron appears in both exponential families. The second chapter is joint work with Jean-Philippe Labbé and Christian Stump [14]. We ...