Algebraic Combinatorics

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The project focuses on the study of combinatorial structures arisingin algebra and geometry, with an emphasis on the theory and applicationsof cluster algebras. Cluster algebras, discovered by the investigator incollaboration with A.Zelevinsky, are a class of commutative rings whichhave found applications in several mathematical disciplines, includingrepresentation theory, Teichmueller theory, discrete dynamical systems,total positivity, Lie theory, tropical geometry, and enumerative andgeometric combinatorics. The investigator develops general structuraltheory of cluster algebras and related combinatorial constructions, andapplies it to the study of concrete classes of cluster algebras arisingin various applications.The original motivation for this project comes from several classicalareas 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.