The proposed research focuses on cluster algebras, a class of commutative rings discovered by the PI in collaboration with S. Fomin. This theory arose as an attempt to create an algebraic framework for the study of two classical fields: theory of total positivity, and representation theory of semisimple Lie groups. Since its inception, the theory of cluster algebras found a number of exciting connections and applications: quiver representations, non-commutative geometry, Seiberg dualities, discrete integrable systems, Teichmuller theory, etc. The PI explores the structural properties of cluster algebras, and their connections and applications. He also develops the theory of quivers with potentials and their representations, motivated among other things, by the theory of superpotentials in theoretical physics.
This project has roots in two classical areas of mathematics: representation theory and the theory of total positivity. Representation theory provides mathematical tools for studying symmetry, while total positivity is a remarkable property of matrices (square arrays of numbers) that generalizes the familiar notion of positive numbers. Both theories find numerous applications in physics, chemistry and other sciences, as well as numerous connections with other mathematical disciplines. In fact, representation theory serves as the mathematical foundation of quantum mechanics, while total positivity is a major tool for explaining oscillations in mechanical systems. The cluster algebras lying in the heart of this project provide a new algebraic framework for the study of these disciplines, making their formalism more explicit and understandable.