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, are a class of commutative rings which have found applications in several mathematical disciplines, including representation theory, Teichmueller theory, discrete dynamical systems, total positivity, 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.
The original motivation for this project comes from several classical areas of mathematics listed above. The main tools come from combinatorics, including combinatorial topology, algebraic and geometric combinatorics, and the theory of root systems. 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 transport insights and techniques from one discipline to another. One case in point is the theory of cluster algebras, which are the main focus of this project.
The main outcomes of this project concern advances in our understanding of combinatorial structures arising in algebra and geometry. The most significant progress was achieved in the study of cluster algebras. These algebras, discovered by the principal investigator in collaboration with Andrei Zelevinsky, have found applications in several mathematical disciplines, including representation theory, Teichmueller theory, integrable systems, total positivity, Lie theory, and enumerative and geometric combinatorics. More recent applications include string theory and symplectic geometry. The project developed the general structural theory of cluster algebras and related combinatorial constructions, and advanced our understanding of important concrete classes of cluster algebras arising in applications. The concepts and techniques developed within this project stimulated substantial amount of research activity, as reflected on the online Cluster Algebras Portal maintained by the principal investigator. The Mathematical Sciences Research Institute in Berkeley, CA, will host a semester-long program on Cluster Algebras in Fall 2012. The project also yielded new results in enumerative geometry, namely new combinatorial formulas for counting plane algebraic curves. The broader impacts of the project include: supervision of several graduate students and several postdoctoral researchers, all of whom continued on to successful academic careers; lecture courses given by the investigator at summer schools in China, Denmark, Israel, and the United Kingdom; public lectures and interviews; and extensive editorial and other professional service.
