Combinatorics of 3D Maps – 3DMaps

Combinatorial maps, which are graphs embedded on 2D surfaces (and which can be naturally related to triangulations of such surfaces), have been successfully studied in various domains of computer science. The purpose of this proposal is to study 3D maps, seen as a natural generalization of 2D combinatorial maps. While combinatorial maps are canonically associated to the so-called random matrix models, 3D maps are, in the same way, canonically associated to tensor models. The proposal is divided into three main objectives. The first objective is focused on the study of combinatorial properties of particular tensor models which were recently shown to play a crucial role in theoretical physics. The second objective deals with the involved problematics of counting 3D triangulations – finding, for example, natural subfamilies of 3D triangulations which are exponentially bounded. Our third objective focuses on the search of analytically controlled random metric spaces of dimension 3.