Hypermaps: polynomials, dualities and minors

This project is concerned with higher order interactions in networks. Mathematics has traditionally modelled networks using "graphs". Graphs provide a highly powerful structure for modelling pairwise interactions of objects. However, effective modelling of many applications requires capturing simultaneous interactions of multiple objects, rather than just pairs. In such applications "hypergraphs" or "hypermaps" are used. Despite such higher order interactions being pervasive, their systematic study is only just receiving due attention and there is a pressing need to develop the corresponding combinatorial theory. In this project we will introduce a theoretical underpinning for these higher order interactions by developing combinatorial, topological and algebraic tools and structure for them. Motivated by applications in statistical physics, biomathematics and topology, our focus is on the "hypermap polynomials" that lie at the intersection of these applications. We develop a theoretical framework for hypermap polynomials by undertaking a systematic study of hypermap minors, dualities, and associated algebraic and combinatorial structures.