Take a regular polygon, and draw lines between its vertices such that the whole polygon is now tiled with non-overlapping triangles. This is called a triangulation. A powerful observation is that no matter which triangulation we choose, we will always be able to reach any other triangulation by replacing diagonals one by one. This structure can be vastly generalised to the notion of cluster combinatorics, which describes how to mutate between structures by an evolving set of rules, and appears in a variety of interesting settings across algebra, geometry, topology and physics.

The Villum Young Investigator project tackles the central questions: What do all of these instances of cluster combinatorics have in common? How can we formally compare them, and find invariants that can help distinguish if we are dealing with the same mutation rules in different settings? Sira's original scientific key idea is to characterise such rules in an abstract setting by describing the mutation procedure without an ambient context, and to apply ideas from topology to study their “shapes”. Different instances of the same set of rules must have the same shape, so this will allow us to compare such processes even if they occur in very different contexts, building bridges between areas of mathematics and physics and facilitating the transfer of machinery from one area to another.

Sira is currently senior lecturer at the University of Glasgow, but will be pursuing her project at the Department of Mathematics, Aarhus University, where she will start on 1 May 2022, in an associate professor position. Sira will be hiring a PhD student and two postdocs on her Villum grant.