Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

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The project focuses on cluster algebras discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras arecommutative of a special kind designed to provide an algebraicframework for the study of total positivity and canonical bases insemisimple groups and their representations. The investigator studiesstructural properties of cluster algebras and their quantumdeformations. This study uncovers unexpected connections with suchdiverse subjects as the structural theory of Kac-Moody algebras,representation theory of finite dimensional algebras, geometry ofmoduli spaces, and the theory of superpotentials. One of the maininstruments 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 variousphysical and biological systems that occur in nature. Total positivityis a remarkable property of matrices (square arrays of numbers) thatgeneralizes the familiar notion of positive numbers. Both theoriesfind numerous applications in physics, chemistry and other sciences,as well as in other mathematical disciplines. In fact, representationtheory serves as the mathematical foundation of quantum mechanics,while total positivity is a major tool for explaining oscillations inmechanical systems. During the last decade, deep connections werefound between the two fields, and the scope of their applications wasgreatly extended. This project explores the modern framework ofrepresentation theory and total positivity, with the goal of makingits formalism more explicit and understandable.