Algebraic Combinatorics

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.