The AGR family (Ferber & Gutknecht, 1998; Ferber et al. 2004) has been one of the first to give a simple and detailed account of what an OCMAS should be, and has been the conceptual basis for the MadKit platform (MadKit 2004). The Moise+ has also shown its importance in the field by providing a general framework for groups, roles, goal-driven agents and norms (Hübner, Sichman & Boissier, 2002)....

The groupoid of projectivities, introduced by M. Joswig [17], serves as a basis for a construction of parallel transport of graph and more general Hom-complexes. In this framework we develop a general conceptual approach to the Lovász conjecture, recently resolved by E. Babson and D. Kozlov in [4], and extend their result from graphs to the case of simplicial complexes.

Recently a new homotopy theory for graphs and simplicial complexes was defined (cf. [3, 4]). The motivation for the definition came initially from a desire to find invariants for dynamic processes that could be encoded via (combinatorial) simplicial complexes. The invariants were supposed to be topological in nature, but should at the same time be sensitive to the combinatorics encoded in the c...

iii Acknowledgments iv Chapter

Let G be a chordal graph and I(G) its edge ideal. Let β(I(G)) = (β0, β1, . . . , βp) denote the Betti sequence of I(G), where βi stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex ∆ of dimension p whose f -vector f(∆) = (f0, f1, . . . , fp) coincides with β(I(G)).

We consider heat kernels on different spaces such as Riemannian manifolds, graphs, and abstract metric measure spaces including fractals. The talk is an overview of the relationships between the heat kernel upper and lower bounds and the geometric properties of the underlying space. As an application some estimate of higher eigenvalues of the Dirichlet problem is considered.

In this article, we explain how spherical Tits buildings arise naturally and play a basic role in studying many questions about symmetric spaces and arithmetic groups, why Bruhat-Tits Euclidean buildings are needed for studying S-arithmetic groups, and how analogous simplicial complexes arise in other contexts and serve purposes similar to those of buildings. We emphasize the close relationship...