Mutations and geometric models

Representation theory follows the long-standing tradition of studying complex problems via linearisation, usually by studying the actions of algebraic structures on vector spaces. Linearisation is such a natural idea that representation theory interacts strongly with many branches of mathematics, e.g. combinatorics, geometry and topology. At the cutting edge, representation theory is often studied using homological algebra and formulated in the language of triangulated categories.
Kontsevich introduced Calabi-Yau (CY) triangulated categories in the context of mirror symmetry from mathematical physics: the n-CY property is a homological symmetry condition enjoyed by n-dimensional CY manifolds occurring in string theory. CY categories have become ubiquitous in mathematics, particularly in algebraic geometry, symplectic geometry and in representation theory, particularly cluster theory. Due to their physical origin, n-CY categories have been extensively studied for n=2,3.
However, as a homological condition, the n-CY property can be studied for any integer n. This means there are many more examples of n-CY categories than those of physical origin. Many categories naturally arising in modular representation theory, singularity theory, Fuss-Catalan combinatorics, and higher homological algebra are n-CY for n different from 2 and 3.
The systematic study of CY categories in these wider contexts is in its infancy and many aspects remain poorly understood. This project will provide methods to address this shortfall using mutations and geometric models. The goal is to develop homological and geometric methods for representation theory and the theory of CY categories by

developing mutation and tilting theories for CY categories;
constructing and describing CY categories using surface models;
applying geometric and homological techniques to Cambrian combinatorics and higher homological algebra. 

Structural understanding of categories is often gained by studying their 'building blocks' (generators). Broadly speaking, there are two types of generator: projective and simple objects. Projective objects play a central role in tilting theory, which permits comparison of disparate objects such as representations of the Kronecker algebra and coherent sheaves on the projective line. On the other hand, the theory of simple objects is more complicated and less well developed but more widely applicable because they can appear in the absence of projective objects. One crucial aspect in the study of generators is mutation, which relates different generators of the same type by switching summands. This project will develop mutation and tilting theory in new contexts.
Geometric models, originating in triangulations of surfaces in cluster theory in the early 2000s, provide powerful tools for the structural understanding of categories by permitting the use of combinatorial and topological methods. They have highlighted unexpected applications and connections between different branches of mathematics, such as representation theory, symplectic geometry and theoretical physics. This project will develop new geometric models which will aid our understanding of CY structures, advance higher homological algebra, and develop novel representation theoretic concepts coming from Cambrian lattice theory.
The project will produce useful methods applicable in modular representation theory, singularity theory and higher homological algebra. It will:

provide tools for computing homology, constructing convex-geometric invariants and studying geometric spaces of stability conditions in novel settings;
contribute towards a finite type classification theorem in higher homological algebra, akin to  Gabriel's theorem, the jewel in the crown of classical representation theory;
yield much needed data for the building of a common framework for the multitude of mutation theories now being widely applied.