The project focuses on the study of combinatorial structures arising in algebra and geometry, with an emphasis on the theory and applications of cluster algebras. Cluster algebras, discovered by the investigator in collaboration with A.Zelevinsky, have found applications in several mathematical disciplines, including representation theory, Teichmueller theory, discrete dynamical systems, discrete potential theory, Lie theory, tropical geometry, and enumerative and geometric combinatorics. The investigator develops general structural theory of cluster algebras and related combinatorial constructions, and applies it to the study of concrete classes of cluster algebras arising in various applications.
This project is motivated by several classical areas of mathematics listed above. The main tools come from combinatorics, including combinatorial topology, algebraic and geometric combinatorics, and the machinery of quiver mutations. Combinatorics deals with discrete objects such as finite sets, graphs, permutations, partial orders, etc. Many continuous phenomena allow for a discrete representation, lending themselves amenable to combinatorial methods of study. It is often the case that identical or similar combinatorial structures underlie seamingly unrelated mathematical entities, revealing hidden connections between them and allowing to transfer insights and techniques from one discipline to another. A case in point is the theory of cluster algebras, which are the main focus of this project.
