Graph theory in higher dimensions

Graph theory is a modern and highly active branch of mathematics with an increasingly important impact on other areas, including computer science, geometry, number theory, topology, probability, and statistical mechanics.

A mainstream trend in graph theory aims at generalising results from graphs to hypergraphs. Such gen- eralisations tend to be much harder than their graph analogues, or even provably impossible, and so despite the career-long efforts and deep machinery of generations of graph theorists, we will never be able to extend all of graph theory to hypergraphs. But it is definitely worth doing so for those statements likely to have an impact on other disciplines.

This proposal takes this viewpoint as a starting point. It departs from the mainstream in two ways. Firstly, the generalisations we seek are driven by concrete applications to other areas of mathematics highlighted by the objectives set below. Secondly, rather than a purely combinatorial approach to graphs and hypergraphs, we take a topological viewpoint: when graphs are viewed as 1-dimensional simplicial complexes, natural topological extensions of definitions and theorems to higher-dimensional cell-complexes -the alter ego of hypergraphs- suggest themselves.

One of the ambitions of this project is to advance the development of a unified theory of low-dimensional topological combinatorics that parallels the growth of graph theory into a discipline and has a long-lasting impact on other disciplines. To avoid the risk of getting lost in abstract theory building, concrete objectives are set out below to ensure that the theory grows into the right directions and bears fruit within the time-frame of the project. Planarity, and its higher-dimensional analogues, plays an important role throughout providing some common ground for the various objectives. These objectives are both important and timely, being strongly related to seminal recent advances.