Hypermaps: polynomials, dualities and minors

Most networks in the real world involve more than two things interacting at once, but the mathematics used to model them has mostly been limited to pairs. This project develops the theory of hypermaps—mathematical structures that capture simultaneous interactions among multiple objects, such as three or more genes influencing a disease, or several species competing in an ecosystem. Current graph-based models miss these higher-order connections, leaving a gap in the combinatorial theory needed to understand them systematically. The researchers will build new algebraic and topological tools, focusing on hypermap polynomials that link applications in statistical physics, biology, and topology. This is fundamental science with no immediate practical application. However, similar foundational work on graphs later enabled modern network science, which now underpins everything from social media algorithms to power grid reliability. A deeper understanding of hypermaps could eventually lead to better models for systems where interactions are inherently collective—such as protein complexes, neural circuits, or ecological webs—rather than merely pairwise.