Dynkin-like Categories: Root Systems, Clusters, and Generalized Associahedra

My proposal is about continuous generalizations of the connections among three concepts: root systems, cluster structures, and generalized associahedra. The three concepts, in the finite discrete setting, are partially classified by the Dynkin diagrams: An, Bn, Cn, Dn, E6, E7, E8, F4, and G2. The same diagrams classify many other objects throughout mathematics and physics. I have worked on a continuous cluster structure of type A and am working on a continuous generalized associahedron of the same type. The proposed project is to understand what parts of the finite discrete connections among the three concepts extends to the continuum for types A, B, C and D. Generally, I work with representations of continuous quivers. Representations of a Dynkin species (generalized quivers) are related to the roots of the corresponding root system. A special triangulated category obtained from the representations classify both the cluster structure and generalized associahedra of the same type. Studying continuous species is a logical place to start exploring the continuous connections. Continuous versions of root systems, representations, and generalized associahedra appear in the study of Fock spaces, quantum groups, and particle physics. This project will help us understand how these are all related.