The project focuses on cluster algebras discovered by the investigator in collaboration with S.Fomin. Cluster algebras are commutative of a special kind designed to provide an algebraic framework for the study of total positivity and canonical bases in semisimple groups and their representations. The investigator studies structural properties of cluster algebras and their quantum deformations. This study uncovers unexpected connections with such diverse subjects as the structural theory of Kac-Moody algebras, representation theory of finite dimensional algebras, geometry of moduli spaces, and the theory of superpotentials. One of the main instruments of the study is polyhedral combinatorics.
This project is motivated by two classical areas of mathematics: representation theory and the theory of total positivity. Representation theory is a mathematical approach to studying symmetry; more specifically, it encodes the symmetry properties of various physical and biological systems that occur in nature. 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 in 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. During the last decade, deep connections were found between the two fields, and the scope of their applications was greatly extended. This project explores the modern framework of representation theory and total positivity, with the goal of making its formalism more explicit and understandable.
