Cluster structures in geometry: Construction, investigation, applications to cluster combinatorics

• Applicant: Privatdozentin Dr. Anna Felikson
• Subject Area: Mathematics
• Term: from 2011 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 213999450

The project concerns interplay between the theory of cluster algebras and different geometrical objects such as reflection groups, triangulations of orbifolds, decompositions of surfaces into 3-holed spheres etc. The main directions of the proposed research are the following:• Combinatorics of cluster algebras.• Geometry of Coxeter groups and their reflection subgroups.• Geometric structures on 2-dimensional surfaces, their transformations and parametrizations of Teihmüller spaces.These three parts, though described separately below, make one project, thanks to deep links between them, some already discovered, some still in investigation. The main goals of the project are:• Construction of the structure theory for cluster algebras of finite mutation type;• Obtaining estimates of growth rates for all cluster algebras;• Construction of cluster-like structures on 2-dimensional surfaces and generators of Coxeter groups, applications to topological invariants of 3-manifolds.

• DFG Programme: Research Grants