Cluster algebras and their combinatorics have emerged as an important feature throughout mathematics. The project addresses the fundamental question: How can we formally compare their various guises? We will exploit the answer to both advance our understanding of cluster algebras and for explicit computation of important examples from geometry and topology.
The project focuses on metrics and completions of triangulated categories. The two main objectives are to exploit recent breakthroughs in the theory of metrics on triangulated categories to answer open questions in the representation theory of algebras, and to push their development to the next level.
In the 1950s, Øystein Ore proved that a finite group is cyclic if and only if its lattice of subgroups is distributive. The project aims to establish an analogous programme for triangulated categories: it will show that distributivity of the lattice of thick subcategories of a triangulated category has strong implications both for the properties of the triangulated category itself and for the algebraic and geometric objects that underlie it.
