Recent progress on the combinatorial diameter of polytopes and simplicial complexes

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in $n^{Theta(d)}$ and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter of simplicial complexes and abstractions of them, in preparation

The geometry of hyperbolic lines in polar spaces

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have applications in group theory, in the theory of Lie algebras and in graph theory

Hidden geometries in networks arising from cooperative self-assembly

Multilevel self-assembly involving small structured groups of nano-particles provides new routes to development of functional materials with a sophisticated architecture. Apart from the inter-particle forces, the geometrical shapes and compatibility of the building blocks are decisive factors in each phase of growth. Therefore, a comprehensive understanding of these processes is essential for the design of large assemblies of desired properties. Here, we introduce a computational model for cooperative self-assembly with simultaneous attachment of structured groups of particles, which can be described by simplexes (connected pairs, triangles, tetrahedrons and higher order cliques) to a growing network, starting from a small seed. The model incorporates geometric rules that provide suitable nesting spaces for the new group and the chemical affinity $\nu$ of the system to accepting an excess number of particles. For varying chemical affinity, we grow different classes of assemblies by binding the cliques of distributed sizes. Furthermore, to characterise the emergent large-scale structures, we use the metrics of graph theory and algebraic topology of graphs, and 4-point test for the intrinsic hyperbolicity of the networks. Our results show that higher Q-connectedness of the appearing simplicial complexes can arise due to only geometrical factors, i.e., for $\nu = 0$, and that it can be effectively modulated by changing the chemical potential and the polydispersity of the size of binding simplexes. For certain parameters in the model we obtain networks of mono-dispersed clicks, triangles and tetrahedrons, which represent the geometrical descriptors that are relevant in quantum physics and frequently occurring chemical clusters.Comment: 9 pages, 8 figure